OPTIMIZATION OF HULL SHAPES FOR WATER-SKIING  
AND WAKEBOARDING 
 
 
 
Except where reference is made to the work of others, the work described in this thesis is 
my own or was done in collaboration with my advisory committee.  This thesis does not 
include proprietary or classified information. 
 
 
 
 
_______________________________ 
Robert L. Daily 
 
 
Certificate of Approval: 
 
 
_____________________________ 
Jay Khodadadi 
Professor 
Mechanical Engineering 
_____________________________ 
Peter Jones, Chair 
Associate Professor 
Mechanical Engineering 
_____________________________ 
A. J. Meir 
Professor 
Mathematics and Statistics 
_____________________________ 
Stephen L. McFarland 
Dean 
Graduate School 
OPTIMIZATION OF HULL SHAPES FOR WATER-SKIING  
AND WAKEBOARDING 
 
Robert L. Daily 
 
 
 
A Thesis 
Submitted to 
the Graduate Faculty of 
Auburn University 
in Partial Fulfillment of the 
Requirements for the 
Degree of 
Master of Science 
 
 
Auburn, Alabama 
December 16, 2005 
 
 iii
OPTIMIZATION OF HULL SHAPES FOR WATER-SKIING  
AND WAKEBOARDING 
 
Robert L. Daily 
 
 
 
Permission is granted to Auburn University to make copies of this thesis at its discretion, 
upon request of individuals or institutions and at their expense. The author reserves all 
publication rights. 
 
 
 
 __________________________________ 
 Signature of Author 
 
 __________________________________ 
 Date of Graduation 
 
 iv
THESIS ABSTRACT 
OPTIMIZATION OF HULL SHAPES FOR WATER-SKIING  
AND WAKEBOARDING 
 
Robert L. Daily 
Master of Science, December 16, 2005 
(B.S., Auburn University, 1999) 
 
143 Typed Pages 
Directed by Peter Jones 
 
This study attempts to define an optimal hull shape for a boat towing either a skier or 
a wakeboarder.  Two methods for determining a free surface deformation (both hull and 
wake shape) given an applied pressure disturbance are compared.  The methods derive 
from the same potential flow, but vary in how the pressure disturbance is defined; one 
uses a Fourier type approximation, the other a piecewise constant interpolation. 
Using the results from the comparison of the two methods, an approach for simulating 
the wake shape given different hull shapes (via the pressure distribution) is developed.  
Using this approach the hull shape is optimized based on two wake parameters: height 
and slope.  Examples of this optimization are presented for different towing scenarios.  
The resulting hulls and wakes are then examined for realism. 
 
 v
ACKNOWLEDGEMENTS 
The author would like to thank his advisor Dr. Peter Jones for his guidance through 
the highs and lows of this research.  He would also like to thank the other committee 
members, Dr. Jay Khodadadi and Dr. A. J. Meir, for their instruction both in classes and 
when problems arose during research, not to mention their patience through this process. 
The author additionally wishes to express his gratitude to his entire family for their 
support throughout his entire school career and for instilling a deep love for the activities 
motivating this research.  Also, he wishes to thank Ms. Susan Wooden for her support 
and encouragement. 
 
 vi
Style manual or journal used: ASME Journal of Dynamics Systems, Measurement, and 
Control.
Computer software used: Microsoft Word 2003. 
 
 vii
TABLE OF CONTENTS
List of Figures..................................................................................................................... x 
List of Tables ...................................................................................................................xiii 
Nomenclature................................................................................................................... xiv 
1. Introduction................................................................................................................. 1 
2. Mathematical Derivations........................................................................................... 7 
2.1. Basic Assumptions and Relationships ................................................................ 7 
2.1.1. Conservation of Mass ................................................................................. 9 
2.1.2. Kinematic Boundary Condition ................................................................ 10 
2.1.3. Dynamic Boundary Condition .................................................................. 11 
2.1.4. Far Field Boundary Condition .................................................................. 11 
2.2. Velocity Potential.............................................................................................. 12 
2.3. Sinusoidal Pressure Patches.............................................................................. 14 
2.3.1. Surface Integral......................................................................................... 15 
2.3.2. Wave Number Integral.............................................................................. 21 
2.3.3. Wave Angle Integral................................................................................. 31 
2.3.4. Evaluation Grid......................................................................................... 41 
2.4. Constant Pressure Patches................................................................................. 45 
2.4.1. Surface Integral......................................................................................... 47 
2.4.2. Wave Number Integral.............................................................................. 48 
 
 viii
2.4.3. Wave Angle Integral................................................................................. 48 
2.4.4. Evaluation Grid......................................................................................... 49 
2.5. Numerical Issues............................................................................................... 51 
2.5.1. Sinusoidal Pressure Method...................................................................... 52 
2.5.2. Constant Pressure Method ........................................................................ 59 
3. Method Validation .................................................................................................... 63 
3.1. Sinusoidal Pressure Basis Function .................................................................. 63 
3.2. Simplified Realistic Hull Pressure.................................................................... 67 
3.3. Sinusoidal Pressure Centerlines........................................................................ 69 
4. Method Issues ........................................................................................................... 73 
4.1. Constant Pressure Method Symmetry............................................................... 73 
4.2. Sinusoidal Pressure Method Symmetry............................................................ 75 
4.3. Sinusoidal Pressure Method Oscillation ........................................................... 77 
4.4. Issues With Inversion........................................................................................ 82 
5. Hull Optimization ..................................................................................................... 88 
5.1. Event Definition................................................................................................ 88 
5.2. Typical Hull Shape ........................................................................................... 92 
5.3. Test Cases ......................................................................................................... 94 
5.4. Jump 1 Test Results .......................................................................................... 97 
5.4.1. Resulting Pressure and Hull Shape........................................................... 98 
5.4.2. Resulting Wake....................................................................................... 100 
5.5. Jump 2 Test Results ........................................................................................ 102 
5.5.1. Resulting Pressure and Hull Shape......................................................... 103 
 
 ix
5.5.2. Resulting Wake....................................................................................... 105 
5.6. Slalom 1 Test Results...................................................................................... 106 
5.6.1. Resulting Pressure and Hull Shape......................................................... 107 
5.6.2. Resulting Wake....................................................................................... 108 
5.7. Slalom 2 Test Results...................................................................................... 109 
5.7.1. Resulting Pressure and Hull Shape......................................................... 110 
5.7.2. Resulting Wake....................................................................................... 111 
5.8. Wakeboard 1 Test Results .............................................................................. 112 
5.8.1. Resulting Pressure and Hull Shape......................................................... 113 
5.8.2. Resulting Non-Optimal Pressure and Hull Shape................................... 114 
5.8.3. Resulting Non-Optimal Wake................................................................. 115 
5.9. Wakeboard 2 Test Results .............................................................................. 116 
5.9.1. Resulting Pressure and Hull Shape......................................................... 117 
5.9.2. Resulting Wake....................................................................................... 118 
5.10. Results Summary ........................................................................................ 119 
6. Conclusions and Suggestions for Future Research................................................. 121 
6.1. Conclusions..................................................................................................... 121 
6.2. Suggestions for Future Research .................................................................... 123 
References....................................................................................................................... 126 
 
 
 x
LIST OF FIGURES
Fig. 1 Nautique Ski 196 ...................................................................................................... 2 
Fig. 2 Monterey 268 SS SuperSport ................................................................................... 3 
Fig. 3 Super Air Nautique 210............................................................................................ 4 
Fig. 4 Coordinate Frame and Uniform Flow ...................................................................... 8 
Fig. 5 Sinusoidal Pressure Patch Definition ..................................................................... 15 
Fig. 6 Complex Plane and Line Integrals.......................................................................... 23 
Fig. 7 Sinusoidal Pressure Hull Discretization ................................................................. 42 
Fig. 8 Constant Pressure Hull Discretization.................................................................... 46 
Fig. 9 Sinusoidal Pressure Combined Integrands ............................................................. 55 
Fig. 10 Sinusoidal Pressure II Integrand with Slow Oscillations ..................................... 56 
Fig. 11 Sinusoidal Pressure Final Integrands.................................................................... 58 
Fig. 12 Constant Pressure Original Far Field Integrand ................................................... 60 
Fig. 13 Constant Pressure Far Field Integrand with Slow Oscillations ............................ 61 
Fig. 14 Constant Pressure Far Field Final Integrand ........................................................ 62 
Fig. 15 Sinusoidal Pressure Basis Function Test Pressure ............................................... 64 
Fig. 16 Sinusoidal Pressure Basis Function Test Wake.................................................... 64 
Fig. 17 Sinusoidal Pressure Basis Function Test Hull...................................................... 65 
Fig. 18 Simplified Realistic Hull Test Pressure................................................................ 67 
Fig. 19 Simplified Realistic Hull Test Wake.................................................................... 68 
 
 xi
Fig. 20 Simplified Realistic Hull Test Hull ...................................................................... 69 
Fig. 21 Sinusoidal Pressure Basis Function Centerlines................................................... 71 
Fig. 22 Single Constant Pressure Patch Wake.................................................................. 74 
Fig. 23 Corresponding Constant Pressure Patch Wakes................................................... 74 
Fig. 24 Total Constant Pressure Patch Basis Function ..................................................... 75 
Fig. 25 Sinusoidal Pressure Patch Wakes......................................................................... 76 
Fig. 26 Loss of Data as More Fourier Terms are Used..................................................... 78 
Fig. 27 Oversampled Pressure Basis Function ................................................................. 79 
Fig. 28 Oversampled Wake Basis Function...................................................................... 80 
Fig. 29 Oversampled Wake Basis Function Surface ........................................................ 81 
Fig. 30 Original Test Surface............................................................................................ 83 
Fig. 31 Pressure Resulting From Original Test Surface ................................................... 84 
Fig. 32 Pressure Resulting From Rotated Test Surface .................................................... 85 
Fig. 33 Increasing Pressure Oscillation ............................................................................ 87 
Fig. 34 Ski Jumping Example and Course Layout ........................................................... 89 
Fig. 35 Slaloming Example and Course Layout ............................................................... 90 
Fig. 36 Wakeboarding Examples...................................................................................... 91 
Fig. 37 Wake Parameters .................................................................................................. 92 
Fig. 38 Basic Planing Hull Shape ..................................................................................... 93 
Fig. 39 Examples of Planing Boats................................................................................... 95 
Fig. 40 Jump 1 Minimization............................................................................................ 98 
Fig. 41 Jump 1 Pressure and Hull Shape ........................................................................ 100 
Fig. 42 Jump 1 Wake ...................................................................................................... 102 
 
 xii
Fig. 43 Jump 2 Minimization.......................................................................................... 103 
Fig. 44 Jump 2 Pressure and Hull Shape ........................................................................ 104 
Fig. 45 MasterCraft X-Star ............................................................................................. 105 
Fig. 46 Jump 2 Wake ...................................................................................................... 106 
Fig. 47 Slalom 1 Minimization ....................................................................................... 107 
Fig. 48 Slalom 1 Pressure and Hull Shape...................................................................... 108 
Fig. 49 Slalom 1 Wake.................................................................................................... 109 
Fig. 50 Slalom 2 Minimization ....................................................................................... 110 
Fig. 51 Slalom 2 Pressure and Hull Shape...................................................................... 111 
Fig. 52 Slalom 2 Wake.................................................................................................... 112 
Fig. 53 Wakeboard 1 Maximization ............................................................................... 113 
Fig. 54 Wakeboard 1 Pressure and Hull Shape............................................................... 114 
Fig. 55 Wakeboard 1 Non-Optimal Pressure and Hull Shape ........................................ 115 
Fig. 56 Wakeboard 1 Non-Optimal Wake ...................................................................... 116 
Fig. 57 Wakeboard 2 Maximization ............................................................................... 117 
Fig. 58 Wakeboard 2 Resulting Pressure and Hull Shape .............................................. 118 
Fig. 59 Wakeboard 2 Wake ............................................................................................ 119 
 
 
 xiii
LIST OF TABLES
Table 1 Test Parameters.................................................................................................... 94 
Table 2 Typical Tow Boat Parameters.............................................................................. 96 
 
 
NOMENCLATURE
A
 
amplitude coefficient of a pressure patch (N/m
2
) 
B half?beam (m) 
E leading edge contribution tensor (-) 
*
E  final wave height basis tensor (-) 
I total number of longitudinal solution points in the wetted area (-) 
J total number of transverse solution points in the wetted area (-) 
L wetted length of the craft (m) 
M total number of longitudinal pressure patches (-) 
N total number of frequency multiples / transverse pressure patches (-) 
P pressure (N/m
2
) 
U vehicle speed (m/s) 
a pressure patch length (m) 
b pressure patch width (m) 
g acceleration due to gravity (m/s
2
) 
i longitudinal index of solution point (-) 
j transverse index of solution point (-) 
k solution point index (-) 
k
0 
fundamental wave number (1/m) 
m longitudinal index of pressure patch (-) 
 xiv
 
n frequency multiple of the pressure patch / transverse index of pressure patch (-) 
p pressure strip index (-) 
t time (s) 
u x coordinate of velocity (m/s) 
v y coordinate of velocity (m/s) 
w z coordinate of velocity (m/s) 
x forward coordinate (m) 
y port coordinate (m) 
z up coordinate (m) 
?
i  forward direction (-) 
?
j  port direction (-) 
?
k  up direction (-) 
v  total fluid velocity (m/s) 
? total flow potential (m
2
/s) 
? height of the fluid surface (m) 
?  coordinate of the pressure (-) 
?
j
? wave angle (rad) 
?  coordinate of the pressure (-) 
?
i
? fluid density (kg/m
3
) 
? craft disturbance potential (m
2
/s) 
 xv
 
 1
1. INTRODUCTION
Wake shape is an important factor to consider when designing recreational or 
professional water skiing towboats.  With the gain in popularity of wakeboards and other 
water sports, the ?ideal? wake has become much more complicated.  Most professional 
ski events are based around objects in the water: slalom buoys, jump ramp, etc [AWSA, 
IWSF].  For these events the competitor wants as little interference from the wake as 
possible, i.e. a flat wake.  Wakeboard events, on the other hand, rely on the wake for the 
competition.  Most events are based around using the wake as a launching platform for 
varying jumps [WWA, WWC].  For these events the competitors want a wake shaped to 
allow higher jumping.  The shape of the wake is affected by several boat characteristics 
and operating conditions such as trim, speed, and mean draft.  The largest design variable 
is the hull shape. 
Most current towboats are designed based on experience and past designs.  The 
potential for radical improvements to boat design exists in utilizing computer simulations 
to test designs before they are manufactured.  There are two philosophies to designing 
towboats; the first is pure design (either wakeboard or ski).  These boats are designed for 
the sole purpose of towing a specific piece of equipment.  Other considerations such as 
passenger room and comfort are secondary to towing.  The boat is designed to create the 
best wake for the specific equipment.  A boat with this design is usually reserved for 
tournament tow boats; they are too limited for general public enjoyment.  An example of 
 
this type of design is the Nautique Ski 196.  Note the closed bow and limited seating.  
This boat is not designed for passengers.  Also notice the engine compartment in the 
center of the boat; this configuration is known as an inboard engine.  Directly in front of 
the engine compartment is the ski pylon where the tow rope is attached.  The forces from 
the rope act near the center of the boat and so pull it from its course less.  It is not 
obvious, but the hull is also designed specifically with the wake in mind. 
 
Fig. 1 Nautique Ski 196 
The second philosophy is mixed design.  These boats are for more general 
recreational use.  The wake produced can be used for either skiing or wakeboarding, but 
is not ideal for either.  An example of this type of design is the Monterey 268 SS 
SuperSport.  This boat has an open bow and much more seating.  The engine 
compartment is at the rear of the boat in what is known as an inboard/outboard 
configuration.  There is no ski pylon interior to the boat to attach the tow rope; instead on 
the transom there is a hook.  Because the rope attach point is at the rear of the boat, when 
the skier swings wide it can cause the rear of the boat to also move in that direction.  The 
hull of this boat is designed more for passenger comfort than wake performance. 
 2
 
 
Fig. 2 Monterey 268 SS SuperSport 
Wakes shapes for this type of boat are very hard to optimize.  The main differences in 
operating conditions between skiing and wakeboarding are rope length (rider placement 
in the wake) and boat speed.  The goal of the design is to make the wake act differently 
based on these two conditions.  Recently some companies have started putting water 
tanks at the stern of boats such as the Super Air Nautique 210; these tanks increase both 
draft and trim when full allowing the boat to create a bigger wake for wakeboarding.  
Another feature of most pure wakeboarding tow boats is the tower which takes the place 
of the ski pylon.  The tow rope attaches much higher allowing for better jumps, etc. 
 3
 
 
Fig. 3 Super Air Nautique 210 
A design method using computer simulations would significantly reduce the cost and 
increase the performance of the hull design.  The traditional method to simulate fluid 
flow is computational fluid dynamics (CFD).  This method involves numerically solving 
the Navier-Stokes equations [Currie]: 
 () ()()
2
P
t
? ????
?
+????+?=+
?
u
uu
ugg
?f. (1.1) 
with ? and ? viscosity parameters of the fluid.  In general this is not an easy problem to 
even simulate.  One simplification that can be made is to assume inviscid and 
incompressible flow resulting in Euler?s equations: 
 () P
t
? ??
?
++?=
?
?
u
u
ug
f. (1.2) 
This simplification removes some terms from the differential equations, but they are 
still nonlinear and without a closed-form solution.  Several methods for numerically 
approximating the solution to this equation are available.  These methods work very well 
for fluids in enclosed spaces.  They provide in-depth knowledge of both the pressure and 
the velocity of the fluid throughout the entire volume in question.  Determining the wake 
 4
 
 5
shape does not require that knowledge, however; it only requires the free surface height.  
The height can be determined from the fluid pressure and velocity information but 
several problems with the CFD methods exist.  One problem is the boundary conditions: 
to numerically approximate the solution the conditions at the boundary of the volume in 
question must be known.  In enclosed spaces this is generally a condition based on the 
velocity of the fluid at a solid wall or a velocity or pressure profile at an inlet or outlet.  In 
open water this is not as easy to determine.  Related to the boundary conditions is a 
bigger problem, the free surface itself.  One boundary of the fluid is the free surface.  
CFD methods require the boundary location to be known.  However, for the wake shape 
problem the air-fluid boundary location is the unknown.  To solve this problem using 
CFD an initial free surface height must be guessed and then the solution iterated by 
varying the shape and position of the free surface until all known boundary conditions are 
met.  This would take an inordinate amount of time, particularly because the free surface 
consists of many different points each of which must be varied independently.  In 
conclusion, a different method needs to be employed to determine the wake shape a boat 
produces. 
Researchers have studied the flow around planing bodies for many years.  The initial 
research centered on determining equations for the flow by using different simplifications 
for the hull shape [Sottorf, Maruo 1951, Maruo 1967].  This research was used to 
calculate the loading on the hull and thus the horsepower requirements for the boat 
[Savitsky].  The research evolved to using numerical technique approximations instead of 
simplifying the hull [Wang, Shen, Doctors].  Two numerical approaches evolved; the first 
uses a series of vortices to represent the flow [Lai].  The more common approach is to use 
 
 6
pressure elements to represent the hull on the surface [Tong, Cheng].  This idea has 
proven successful for determining lift and drag for hulls [Wellicome]; it is only recently 
being used to also determine the free surface height behind the boat [Scullen]. 
The idea to determine the wake shape is to take a pressure distribution on a free 
surface with a known wake shape.  If this wake shape is linear in the pressure distribution 
then many of the pressures can be combined over the wetted area of the boat.  The 
pressures are varied so the deformation of the surface in the wetted area matches the hull 
shape.  The free surface height behind the wetted area from the resulting pressures is then 
the wake shape.  Ideally, the method could be inverted so that a given hull shape (or 
perhaps even desired wake shape) produces the pressure.  This research compares two 
pressure functions used as the basis for the pressure distribution.  It then extends the 
method into finding the wake created by using many pressure patches.  The method 
presented allows any pressure to be modeled and therefore any hull should be possible.  
Using the developed method an optimal hull shape for both skiing and wakeboarding 
given the constraints each activity is under is examined. 
 
2. MATHEMATICAL DERIVATIONS
As the introduction mentioned, traditional computational fluid mechanics methods are 
not appropriate for computing the wake a planing craft creates.  Instead, a technique that 
creates basis functions linear in the pressure amplitude will be utilized.  These basis 
functions are summed together with different pressures to create the hull shape of the 
craft.  The pressure that created this hull is then used to create the wake shape. 
2.1. BASIC ASSUMPTIONS AND RELATIONSHIPS 
To create the surface basis functions, the following assumptions are made about the 
fluid and craft: 
1. The craft is traveling at a constant speed (U).  This assumption is equivalent to a 
stationary craft in a flow with constant velocity. 
2. The fluid is inviscid. 
3. The fluid is incompressible. 
4. The fluid has infinite depth. 
5. The disturbance created by the craft is small compared to the constant velocity 
flow. 
The flow created under these conditions can be represented as a potential flow.  The 
potential of the total flow (?) is given by: 
 Ux??=? . (2.1) 
 7
 
where ? is the potential of the disturbance created by the craft.  x ( ) is the direction the 
craft is traveling, y ( ) is to the port side of the craft, and z ( ) is up. 
?
i
?
j
?
k
 
Fig. 4 Coordinate Frame and Uniform Flow 
y
x
U 
From the definition of a potential flow the total fluid velocity is determined by 
 
?? ?
uvw= ++ =??vijk . (2.2) 
The following notation will be used to denote derivatives throughout the paper. 
 
2
,
xxy
x xy
???
?= ?=
?
? ??
 (2.3) 
where ? is a generic scalar variable.  Along with these basic assumptions, several 
preliminary properties are required to derive the velocity potential. 
 8
 
2.1.1. Conservation of Mass  
The first property to examine for the flow is conservation of mass [Currie], 
 0
V
D
dV
Dt
? =
?
. (2.4) 
? is the fluid density, t is time, and V is the volume of interest.  Expanding this equation 
using Reynolds? Transport Theorem, 
 
()
t
VV
D
dV dV
Dt
?=?+??
??
v , (2.5) 
the conservation of mass equation (2.4) then becomes 
 ()()() 0
t
xy z
V
uvwdV?? ? ?
??
+ ++ =
?
?
?
. (2.6) 
This equation must be true for an arbitrary volume, so the integrand must always be zero, 
 ( ) ( ) ( ) 0
t
xy z
uvw?? ? ?+ ++=. (2.7) 
Expanding the derivatives in this equation gives 
 
( )
0
txy z xyz
uvw uuu??? ??++++ ++=. (2.8) 
Because the fluid is assumed incompressible ( 0
?
? = ) the conservation of mass equation 
is further simplified to 
 0
xy z
uvw+ +=. (2.9) 
In terms of the perturbation potential, the conservation of mass equation is 
 0
xx yy zz
? ??+ +=. (2.10) 
 9
 
2.1.2. Kinematic Boundary Condition  
The next property to examine is the kinematic boundary condition.  It states that the 
normal velocity of the surface (
Dz
Dt
) is equal to the change in height of the surface with 
respect to time (
D
Dt
?
) where ? is the height of the fluid surface.  This idea is written as 
 ()0
D
z
Dt
?? = . (2.11) 
Expanding this using the definition of a material derivative, 
 
txy
D
uvw
Dt
y
?
=?+?+?+ ?, (2.12) 
the kinematic boundary condition (2.11) becomes 
 ()()()()(
txy
D
zzuzvzwz
Dt
)
z
? ????=?+ ? + ? + ??. (2.13) 
The assumption that the craft is traveling at a constant speed implies the system is in 
steady state, 
 ( )0
xy z
uvw1?? ?=? ? + ? . (2.14) 
Written in terms of the perturbation potential, 
 0
xx xyyz z
U
z
? ??????=? + ? + ? ?
z
. (2.15) 
Because of the assumption that the perturbation potential (and therefore the change in 
surface height) is small, it is safe to linearize this equation.  Additionally, it is assumed 
that this equation applies on the undisturbed free surface (z = 0) instead of the true 
surface (z = ?).  Therefore, the kinematic boundary condition is 
 0
x
U? ?= + . (2.16) 
 10
 
2.1.3. Dynamic Boundary Condition 
The next boundary condition to examine involves the conservation of momentum 
(Bernoulli?s equation) applied on the surface of the fluid.  The Bernoulli equation states 
 
1
2
P
GF
?
+ ?????? =  (2.17) 
where F is the Bernoulli constant, G is the gravity term (Ggz=? ), and P is the pressure.  
On the free surface the Bernoulli equation is, 
 
(
222 2
1
2
2
xyz x
P
FU
)
g? ?? ? ?
?
=+ ++? + + . (2.18) 
To determine the Bernoulli constant, the Bernoulli equation is evaluated far upstream 
where the surface is undisturbed (? = 0) the pressure is atmospheric (P = 0) and the 
velocity is constant (
?
U=?vi).  The resulting Bernoulli constant is 
 
2
1
2
FU= . (2.19) 
Substituting this constant into the Bernoulli equation (2.18) and linearizing results in the 
dynamic boundary condition: 
 0
x
P
Ug? ?
?
=? + . (2.20) 
This equation will be used to determine the wave height, so the last term is left in.  In all 
other aspects it is assumed to apply on the undisturbed free surface, however. 
2.1.4. Far Field Boundary Condition 
The final boundary condition to consider is the far field boundary condition.  This 
condition derives from the assumptions about the fluid far away from the disturbance 
created by the craft.  In particular, the condition holds in two locations.  The first is far 
 11
 
below the craft ( z =??); the second is far ahead of the craft ( x =?).  The condition 
states that the velocity perturbation will not affect the vertical velocity of the fluid 
(0
z
? = ). 
2.2. VELOCITY POTENTIAL 
The conditions presented thus far apply to any velocity potential that meets the 
assumptions presented in section 2.1.  Wehausen and Laitone derive a velocity potential 
for regular waves that meets these assumptions and describes the fluid behavior created 
by applying a pressure to the free surface of the fluid.  This potential is 
 
()
() ()
() ()
2
2
2
020
2
3sec
0
2
2
00
11
,sec
2sec
sin cos cos sin sec
cos sec cos sec sin
kz
S
kz
ke
P
Ukk
kx ky dkd k e
kx ky d dd
?
?
?
?
?
????
?? ? ?
?? ??? ?
?? ? ?????
?
?
?
?
?
=
?
?
?
?
??????
????
?
?
??????
?
????
?
?
?? ? ?
?
g
g
 (2.21) 
where S is the wetted area of the applied pressure, ? and ? are the coordinates of the 
pressure, ? is the wave angle, k is the wave number, and k
0 
is the fundamental wave 
number defined as 
 
0
2
g
k
U
= . (2.22) 
To determine the surface height, the dynamic boundary condition (2.20) is used. This 
equation requires the partial derivative of the velocity potential with respect to the 
longitudinal direction on the free surface.  This partial derivative is 
 12
 
 
()
() ()
() ()
2
2
20
020
2
24
0
2
2
00
11
,
2sec
cos cos cos sin sec
sin sec cos sec sin .
x
z
S
k
P
Uk
kx ky dkd k
kx ky d dd
?
?
?
?
???
?? ? ?
? ?????
? ????
?
=
?
?
?
?
=
?
?
?
?
??+????
????
?
?
??????
?
????
?
?
?? ? ?
?
g
g ??
 (2.23) 
Solving the dynamic boundary condition (2.20) for surface height results in 
 
() ()
() ()
() ()
()
2
2
2
020
2
24
0
2
2
00
11
,,
2sec
cos cos cos sin sec
,
sin sec cos sec sin .
S
k
xy P
gk
kx ky dkd k
Pxy
kx ky d dd
g
?
?
?
?
???
?? ? ?
?? ??? ?
?? ? ?????
?
?
?
?
?
?
=
?
?
?
?
??+????
????
?
?
???? ???
?
????
?
?
?? ? ?
?
g
g
 (2.24) 
An equation for wave height has now been developed; all that remains is to define the 
pressure acting on the surface and evaluate the integrals. 
The double integral over the surface does not have an analytical solution for all 
pressures.  The solution could be solved numerically at this point, but that would involve 
solving four nested integrals numerically for each point in a solution grid.  This is very 
expensive computationally.  To reduce the computation time pressures are chosen that 
result in an analytical solution for the double integral over the surface. Two forms of 
pressure functions will be presented.  Both are linear in the pressure amplitude 
coefficients.  The first method creates a pressure that varies sinusoidally in the transverse 
direction (y, ?) and is piecewise constant in the longitudinal direction (x, ?).  The second 
method is piecewise constant in both the longitudinal and transverse directions. 
 13
 
2.3. SINUSOIDAL PRESSURE PATCHES 
The sinusoidally varying pressure method utilizes pressure patches that vary as sine 
waves in the transverse direction to approximate the true pressure [Cheng].  Multiple 
pressure patches over the same surface with different sine frequencies are added together.  
This is similar to a Fourier approximation of a function.  The total pressure is piecewise 
constant in the longitudinal direction.  In general, the form of the pressure is 
 () ()
1
(,) sin
2
N
n
n
n
PA BB
B
?
?? ? ? ?
=
??
+?
??
??
?
B?. (2.25) 
The frequency of the sine wave is modified by changing n; A
n
(?)  is the amplitude 
coefficient of the pressure patch, B is the half-beam, and L is the wetted length of the 
craft.  With this pressure, the surface equation (2.24) becomes 
 ()
() ( )
() ()
() ( )
0
1
1
0
1
1
,,,
2
;
1
sin,
2
1
,,, ;
2
B
N
nn
n
BL
N
n
n
B
N
nn
n
BL
ASxy dd
g
B yB
n
Ax yBxy
gB
yB
ASxy dd
Byg
?????
??
?
?
?
?????
??
=
??
=
=
??
?
?
?
? ??
?
? ??
?+=
?
??
??
?
?
??
?
?
?
?
?
??
?
?
??
 (2.26) 
with 
 
() () ()
() ()
()
2
2
2
020
2
24
00
2
2
0
1
,,, sin cos cos
2se
cos sin sec sin sec
cos sec sin .
n
nk
Sxy B kx
Bkk
ky dkd k k x
ky d
?
?
?
?
?
? ?? ?
??
?
? ?? ? ??
????
?
?
?
?
???
=+ ?? ?
?
? ???
?
??
?
?
?+ ??? ?
?? ?
?
?
???
?
??
?
?
??
?
g
g
?
?
 (2.27) 
 14
 
 
Fig. 5 Sinusoidal Pressure Patch Definition 
    x = -L 
    x = 0 
    x = -a 
m = 1 
m = 2 
m = M
y = B y = 0
y,? 
x,? 
a 
The first step in determining the wave height is to evaluate the double integral over 
the wetted area. 
2.3.1. Surface Integral 
The pressure is piecewise constant in the longitudinal direction, i.e. A
n
(?) is piecewise 
constant in ?: 
 15
a
 
() ()
,1
,2
,
,1
,
0; 0
;0
;2
;1
;2
;
0;
n
n
n nm
nM
nM
Aa
a
A Aamam
ALa La
L
?
?
?
? ?
?
?
?
?
<
?
?
?<<
?
? ?<<?
?
?
?
= ? <<? ?
?
?
?
?+<<?+?
?
?<<?+
?
?
<?
?
M
M
. (2.28) 
 
The length of a pressure patch is given by a, and m defines the particular longitudinal 
patch. Therefore, the surface height becomes 
 
() () ()
()
()
()
()
() ()
0
,1 ,2
1
2
1
2
, 1
1
,
1
,,,,
2
,,, ,,,
sin
2
,,, .
Ba
N
nn n n
n
Ba a
am
La
nm n nM n
am L a
N
La n
n
nM n
L
xy A S xy d A S xy d
g
ASxydA Sxy
n
Ax yB
B
ASxydd
g
????
??
d
??
? ?? ???
?
?? ? ?
?
?
=
?? ?
??
?+
?
+
?+
=
?
?
=+
?
?
++ ++
??
+
??
?
??
+?
?
?
?
?? ?
?
?
?
L L  (2.29) 
To simplify notation, it is assumed the transverse location is in the half breadth of the 
craft ( B yB???).  If this is not the case the last term of the equation drops out. The 
condition will be added back later.  An intermediate variable is used to evaluate the 
longitudinal integration: 
 ( )
*
1am??= +?. (2.30) 
Therefore, the generic longitudinal integral appears as 
 
()
( )
()(
1
0
*
,,, ,, 1,
am
nn
am a
Sxy d Sxy am d
)
*
? ?? ? ??
??
=??
??
. (2.31) 
 16
 
The integrand of this integral (2.27) is 
 
()()()
()()
()
()()
()
}
()()()
2
2
*
2
020
*
2
24 *
00
2
2
0
***
,
1
,, 1, sin
2s
cos 1 cos
cos sin
sec sin 1 sec
cos sec sin
,, 1, ,, , .
n
nnm
nk
Sxy am B
Bkk
kx am
ky dkd
kkxam
ky d
Sxy am S xy
?
?
?
?
?
???
??
??
?? ?
ec
? ??
????
????
?
?
?
?
???
?? = +
?
??
?
??
?
?
?+ ?
???
??
? ?
+?+?
? ?
???
??
?? =
??
?
g
g
g
 (2.32) 
Substituting this integrand back into the height equation (2.29) results in 
 
() () ()
() ()
()
() ()
()
00
*** * **
,1 ,1 ,2 ,2
1
*** * **
, , ,1 ,1
0
***1
,,
,
1
,,,
2
,, , ,, ,
sin
2
,, ,
1
,
2
B
N
nn n n
n
Ba a
nm nm nM nM
aa
N
n
n
nM nM
a
nm
xy A S xy d A S xy d
g
ASxy d A S xy d
n
Ax yB
B
ASxy dd
g
xy A
g
????
??
?? ? ?? ?
?
?? ? ?
?
?
??
=
?? ?
??
=
?
?
=+
?
?
++ ++
??
+
??
?
??
+?
?
?
=
?
?? ?
?
?
LL
()
() ()
0
***1
,
11
sin
2
,, , .
N
B n
NM
n
nm
nm
Ba
n
Ax yB
B
Sxy dd
g
?
?? ??
?
=
==
??
??
+
??
??
?
?
??
??
(2.33) 
The surface integrand (2.32) is split into two terms, the single and the double integral: 
 
()( ) ( )
*** ** *
,,,1,2
,, , ,, , ,, ,
nm nm nm
Sxy S xy S xy? ???=+??. (2.34) 
 17
 
with 
 
() ()
()()()
() ()
()()()
2
2
**
,,1 2
020
*
2
**2 4
,,2 0
2
*2
00
1
,, , sin
2sec
cos 1 cos cos sin
,, , sin sec
2
sin 1 sec cos sec sin .
nm
nm
nk
Sxy B
Bkk
kx am ky dkd
n
Sxy k B
B
kx am ky d
?
?
?
?
?
?? ?
??
????
?
?? ? ?
?
? ????
?
?
?
??
=+
??
?
??
?+ ? ???
??
??
=+
??
??
???+ ? ?
??
??
?
 (2.35) 
This implies the surface integral term of the wave height equation (2.33) is also split into 
two parts:  
 
() () () () (
,,1 ,,2
11 1
,, sin
2
NM N
nm nm n
nm n
n
gxy g xy g xy Ax yB
B
?
?? ?? ??
== =
??
??=+?
????
??
?? ?
)
+ (2.36) 
with 
 
() ()
()()
()
() ()
()()
()
0
20
2
,
,,1 22
020
*
*
2 2
0
,
4
,,2
2
*
0
2
0
,sin
s
cos 1 cos
cos sin
,sin sec
22
sin 1 sec
cos sec s
B
nm
nm
Ba
B
nm
nm
Ba
A
nk
xy B
Bkk
kx am
ky dkdd d
Ak
n
xy B
B
kx am
ky
?
?
?
?
?
??
ec? ?
??
?? ???
?
??
?
??
??
?
?? ?
?? ?
??
=+
??
?
??
??
?+ ?
??
???
??
=+
??
??
??
?+ ?
??
?
?? ??
?? ?
g
g
g
g
*
in .dd d????
?
 (2.37) 
These two equations are integrated with respect to the pressure coordinates (? and ?) 
resulting in 
 18
 
 
()
() () () ()
()
() () () ()
2
,
,,1 22
020
1,1 1,2 2,1 2,2
2
,0 3
,,2
2
1,1 1,2 2,1 2,2
,sec
4sec
,,,,
,sec
4
,,,,
nm
nm
nm
nm
A
k
xy
kk
I kIkIkIkdkd
Ak
xy
Jk Jk Jk Jk d
?
?
?
?
??
??
? ???
??
?
?
? ???
?
?
?
=
?
??+??
??
=
+??
??
?
g
g ?
 (2.38) 
with 
 
()
() ( ) ( )
()
()
() ()()
()
() ()
()()
,
22
00
,
2
0
1
2
1sin sin
,
1sin
1, 2
2
1, 2
1 cos sec cos sec
,
1secsin
2
1cos sin
cos sin .
n
ll
lj
j
n
ll
lj
j
kk
Ik
n
k
l
B
j
kk
Jk
n
k
B
xam yB
xam yB
??
?
?
?
?? ??
?
?
??
???
+?
+?
?
?
??
=
?+
=
=
??
=
?+
=+ ? +??
??
=+ +
m
m
 (2.39) 
It is important to note that when the pressure strip is rotated by the wave angle ?, 
1
?
?
 and 
2
?
?
 correspond to the leading and trailing edges of the strip respectively.  The wave 
height component equations (2.38) are written in terms of the leading and trailing edge 
contributions as 
 
() ( ) ( )
() () ()
, ,1 , ,1, , ,1,
, ,2 , ,2, , ,2,
,,
nm nm L nm T
nm nm L nm T
x yxyxy
x yxy
?? ?=?
xy
 (2.40) 
 19
 
with 
 
() () ()
() () ()
() () ()
()
2
,
,,1, 1,1 1,2
22
020
2
,
,,1, 2,1 2,2
22
020
2
,0 3
,,2, 1,1 1,2
2
,0
,,2,
,sec
4sec
,sec
4sec
,sec
4
,
nm
nm L
nm
nm T
nm
nm L
nm
nm T
A
k
x yIkI
kk
A
k
kd
x
kk
Ak
xy J k J k d
Ak
xy
?
?
?
?
?
?
? ??
??
??
? ??
??
????
?
?
?
?
?
?
?
??=+
??
?
??
??
?
??=+
??
=
??
??
?
??
() ()
2
3
2,1 2,2
2
sec , , .
4
Jk Jk d
?
?
?? ??
?
?
??+
???
 (2.41) 
The trailing edge is equivalent to the leading edge with the longitudinal coordinate (x) 
shifted by the pressure patch length (a). 
 
( ) ( )
21
,,x yxa??
??
=?y, (2.42) 
which implies 
 
( ) ( )
() ( )
,,1, ,,1,
,,2, ,,2,
,,
nm T nm L
nm T nm L
.
x yxay
x yx
??=?
=?y
 (2.43) 
Therefore, each of the wave height components (2.40) only depends on the leading edge 
contribution: 
 
() ( ) ( )
() () ( )
, ,1 , ,1, , ,1,
,,2 ,,2, ,,2,
,,
nm nm L nm L
nm nm L nm L
,
.
x yxyxay
x yxyx
?? ?=??
y
 (2.44) 
Overall, the total surface height is the sum of the heights made by each pressure patch 
and the static deflection from the pressure being applied, i.e. 
 () () () (
,
11 1
,,sin
2
NM N
nm n
nm n
n
gxy g xy Ax yB
B
?
?? ??
== =
)
? ?
=? +
? ?
? ?
?? ?
. (2.45) 
 20
 
The contribution of each pressure patch is written as a leading edge component and a 
trailing edge component.  The trailing edge component is identical to, but offset from, the 
leading edge component: 
 () ( ) ( )
,,,,nm nmL nmL
gxyg xyg xay?? ?? ??=?,?. (2.46) 
The leading edge component is split into two terms, a single and a double integral: 
 () ( ) ( )
,, ,,1, ,,2,
,
nmL nm L nm L
.x yxy???=+xy (2.47) 
The wetted area integral is now evaluated.  Two integrals remain: the wave number 
and wave angle integrals.  Only the double integral term contains the wave number 
integral.  To make the wave angle integration similar for both terms, the wave number 
integral is evaluated next. 
2.3.2. Wave Number Integral 
To evaluate the wave number integral in the double integral term of the wave height 
(2.41) a complex integration method must be employed.  The basic form of the integral in 
this method is 
 
1,
2
00
1, 2
sec
jj
k
EIdk
kk ?
?
=
?
?
j=. (2.48) 
With I
1,j
 from equation (2.39) this integral becomes 
 
() ( ) ( )
()()
2
0
0
1sin sin
1, 2
sec 1 sin
2
n
mm
j
j
kkk
E
n
kk k
B
??
?
??
+?
? ??
??
??
==
??
??+
??
??
?
dj. (2.49) 
 21
 
To evaluate this integral a complex variable and equation are defined: 
 
() ()
()
1
12
1
iQ
j
j
Qkiu
Qe
FQ
QksQk
?
?
?
= +
=? ?
??
 (2.50) 
where s is simply a variable substitution and k
1
 and k
2
 are values at which the function 
becomes singular.  To match the integrand, 
 
2
10
2
sec
(1)
2sin
sin .
j
kk
n
k
B
s
?
?
?
?
=
=??
=?
 (2.51) 
On the real axis this function evaluates to 
 () ( )
()
1
12
1
ik
j
j
ke
Fk
kkskk
?
?
?
=? ?
??
. (2.52) 
It can also be written in terms of sine and cosine functions,  
 () ( )
( ) ( )
()
1
12
cos sin
1
j
j
kik
k
Fk
kk skk
1
? ?
??
?
+
=? ?
??
, (2.53) 
instead of an exponential.  This equation is beginning to look like the integrand presented 
above.  In fact the integral in equation  (2.49) is 
 
() () () () ()
11
00
1sgn Im sgn Im
n
jj
E F kdk F kdk??
??
++ ??
j
? ??
=? ?
?
? ??
?? ??
?
?
?
j
. (2.54) 
This integral is divided into two parts: 
 ()1
n
jj
E ee
+ ?
= ?? (2.55) 
 22
 
with 
 23
kd 
() ()
1
0
sgn Im
jj
eF?
?
???
? ?
=
? ?
??
?
. (2.56) 
In the complex plane, the line from zero to infinity is divided into five parts.  Part one 
lies on the real axis and goes from the origin to immediately before the first singularity.  
Part two circles the first singularity with an infinitely small radius.  Part three is also on 
the real axis between the two singularities.  Part four circles the second singularity similar 
to part two.  Part five is on the real axis again and goes from the second singularity to 
infinity.  Adding two more parts extends this line to a closed contour.  Part six is a quarter 
circle from the real axis to the imaginary axis with an infinite radius, and part seven lies 
on the imaginary axis from infinity back to the origin.   
u 
 
Fig. 6 Complex Plane and Line Integrals 
Q - plane 
k
1
l
6
 
l
5
 l
3
 
l
2
 
l
1
 
R??
k
2
l
4
 
r?0r?0
k 
l
7
 
The real integral from zero to infinity is equivalent to an integral over segments one, 
three, and five: 
 
 24
j
Qd 
() ()
135
1
sgn Im
j
lll
eF?
?? +
++
? ?
? ?
=
? ?
? ?
??
?
. (2.57) 
Using the Cauchy Theorem, the line integral over a closed contour is zero, 
 ( ) 0
c
j
l
FQ
?
=
?
 (2.58) 
and the integral over the real segments written in terms of the other segments is 
 ( ) ( )
135 2467
j
lll llll
FQ FQ
?
++ +++
=?
??
j
?
0
. (2.59) 
Therefore, equation (2.57) becomes 
 . (2.60) 
() () () () ()
2467
1
sgn Im
j jjjj
llll
e F QdQ F QdQ F QdQ F QdQ?
??????
??
??
=? +++
??
??
????
To evaluate this total integral, each of the component integrals is studied separately. 
The first integral to examine is over the l
2
 segment.  The following parameters define 
the line to integrate over: 
 . (2.61) 
21
:, 0
ii
lQkre dQired r
??
???=+ = ?? ?
The integral therefore becomes 
 
() ()
( )
()
11
2
0
1
0
1112
lim 1
i
ik re
i
j
i
j
i i
r
l
kre e
FQdQ ired
kre ksk re k
?
?
?
?
? ?
?
?
?
+
?
?
+
=??
+? +?
??
. (2.62) 
Evaluating the limit, gives 
 () ()
()
11
2
0
1
12
1
ik
j
j
l
ik e
FQdQ d
sk k
?
?
?
?
?
=??
?
??
, (2.63) 
 
or 
 ()
()
()
11
2
1
12
1
ikj
j
l
ik e
FQdQ
sk k
?
?
?
?
?
=
?
?
. (2.64) 
In terms of sine and cosine functions the integral is 
 ()
()
()
()()
2
1
11 11
12
1
sin cos
j
j
l
k
FQdQ k i k
sk k
?
?
??
?
?
?
? ?
=?+
? ?
?
?
. (2.65) 
Taking the imaginary component of this integral leads to 
 ()
()
()
(
2
1
11
12
1
Im cos
j
j
l
k
FQdQ k
sk k
?
)
?
? ?
??
?
??
=
??
?
??
?
. (2.66) 
The next integral to consider is over the l
4
 segment.  The following parameters define 
this segment: 
 
42
:, 0
ii
lQkre dQired r
??
???=+ = ?? ?0. (2.67) 
The integral over this segment is: 
 () ()
( )
()
21
4
0
2
0
2122
lim 1
i
ik re
i
j
i
j
i i
r
l
kre
e
FQdQ ired
kreksk re k
?
?
?
?
? ?
?
?
?
+
?
?
+
=??
+? +?
??
. (2.68) 
This integral is split into two conditions, one where the singularity is positive and one 
where it is negative.  If the singularity is positive (k
2
 > 0), the limit evaluates to the l
2
 
component: 
 
()
()
()
21
4
0
2
12
1
ikj
j
l
ik e
FQdQ d
sk k
?
?
?
?
?
??
=
?
??
. (2.69) 
 25
 
This integral is solved as 
 ()
()
()
21
4
2
12
1
ikj
j
l
ik e
FQdQ
sk k
?
?
?
?
?
=
?
?
. (2.70) 
Written in terms of sine and cosine functions the integral is 
 ()
()
()
()()
4
2
21 21
12
1
sin cos
j
j
l
k
FQdQ k i k
sk k
?
?
??
?
?
?
? ?
=?+
? ?
?
?
. (2.71) 
If the singularity is negative (k
2
 < 0), the limit evaluates to 
 ( )
4
0
j
l
FQdQ
?
=
?
. (2.72) 
Therefore, the total imaginary component of this integral is 
 
()
()
()
()
4
2
21 2
12
2
1
cos ; 0
Im
0;
j
j
l
k
kk
FQdQ
sk k
k
?
?
?
?
?
?
??
>?
??
=
?
???
?
??
<
?
?
. (2.73) 
To simplify evaluation of this function, the inequality condition is converted into terms of 
wave angle over which this function will be integrated.  From equation (2.51) the 
singularity being positive (k
2
 > 0) implies 
 
(1) 0
2sin
(1)
0.
sin
j
j
n
B
?
?
?
? ?>
?
<
 (2.74) 
j is given, so this function solved explicitly in terms of the wave angle is 
 
1;0
2
2; 0.
2
j
j
?
?
?
?
=<<
= ?<<
 (2.75) 
 26
 
The singularity being negative (k
2
 < 0) implies 
 
(1) 0
2sin
(1)
0.
sin
j
j
n
B
?
?
?
? ?<
?
>
 (2.76) 
This function is also solved explicitly in terms of the wave angle as 
 
1; 0
2
2;0 .
2
j
j
?
?
?
?
= ?<<
=<<
 (2.77) 
The next integral to consider is the l
6
 integral.  The following parameters define the 
line to integrate over: 
 
6
:, 0
2
ii
lQRe dQiRed R
??
?
??== ???. (2.78) 
The integral therefore becomes 
 
() ()
()
1
6
2
10 2
lim 1
i
iRe
i
j
i
j
i i
R
l
Re e
FQdQ iRed
Re k sRe k
?
??
?
?
? ?
?
?
?
??
=??
? ?
??
. (2.79) 
Written in terms of sine and cosine functions this integral is 
 () ()
()
1
6
sin cos2
2
12
0
lim 1
Ri
i
j
j
R
ii
l
ie e
FQdQ d
kk
se e
RR
????
?
??
?
?
?+
?
??
=??
????
??
????
????
??
. (2.80) 
Much of the integrand is simplified by taking the limit, but one term remains: 
 () ()
()
1
6
2
sin cos
0
1lim
Rij
j
R
l
i
FQdQ e d
s
?
???
?
?
?+
?
??
=? ?
??
. (2.81) 
 27
 
This term depends on the value of 
1
?
?
.  If 
1
0?
?
=  the integral is solved as 
 ()
()
6
1
2
j
j
l
i
FQdQ
s
?
?
??
=
?
. (2.82) 
On the other hand, if  the integral evaluates to 
1
0?
?
?
 
( )
6
0
j
l
FQdQ
?
=
?
. (2.83) 
Therefore, taking the total imaginary component of this integral leads to 
 
()
()
6
1
1
1
;
Im
2
0;
j
j
l
FQdQ
s
?
?
?
?
?
?
?
??
??
??? 0
0
=
=
???
?
??
?
?
?
. (2.84) 
Once again, it is simpler to write the condition in terms of the wave angle.  From 
equation (2.39), this condition becomes 
 
( ) ( )
() ()
()
1
1cos sin
01c
1
tan .
xam yB
xam yB
xam
yB
? ??
? ?
?
?
=+ ? +??
??
=+ ? +??
??
?+ ?
=
m
m
m
 (2.85) 
Solving this condition explicitly for the wave angle produces 
 
( )
1
1
tan
xam
yB
?
?
?+ ???
=
?
??
m
?
. (2.86) 
The last integral to consider is over the l
7
 segment. The following parameters define 
this segment: 
 
7
:, 0lQidQid? ??= =?<?. (2.87) 
 28
 
This integral is 
 () ()
()
1
7
0
12
1
j
j
l
e
FQdQ d
iksik
??
?
?
??
?
?
?
?
=?
??
??
. (2.88) 
Through some manipulation, this is written as 
 
() ()
()
()
()
()
()
()
()
()
()
11
7 12
11
12
12
22 22
12 120
11 22
22 22
12 120
1
1
1.
j
j
l
j
kkee
FQdQ d
skk kk
kke kkei
d
skk kk
?? ??
?? ??
??
?
?
??
??
??
??
?
?
??
?
? ?
? ?
=? ? ? +
? ?++
? ?
??
??
?? ?
??
++
??
??
?
 (2.89) 
The variable substitution 
i
k
?
? =  is made producing the following integrals: 
 
()
() ()
()
11
1 1
122 2
00
122 2
00
1
sgn sgn .
1
i
i
i
i
k
i
k
i
ii
ee
ddgk
k
ke e
dk dkfk
k
?? ? ?
?? ??
??
???
??
??
? ?
??
??
?
? ?
??
?
==
++
==
++
??
??
i
?
 (2.90) 
These solutions involve the even and odd auxiliary functions to the sine and cosine 
functions [Abramowitz]: 
 
()
()
2
0
2
0
;0
1
;0
1
zt
zt
te
gz dtz
t
e
fz dtz
t
? ?
? ?
.
= ?
+
= ?
+
?
?
 (2.91) 
Using these integrals, the imaginary portion of the total l
7
 integral is 
 
()
()
()
()
()
()
()
()
7
1
111
12
2
221
12
1
Im sgn
sgn .
j
j
l
k
FQdQ kfk
skk
k
kfk
kk
?
?
??
?
??
???
??
=
???
?
?
??
?
?
?
?
?
?
 (2.92) 
 29
 
The total integral in equation (2.60) is divided into components along the same lines as 
these individual integrals: 
 30
j
 
1, 2, 3, 4, 5,jjjjj
eeeeee
? ?????
=++++. (2.93) 
 where 
 
() ()
() ()
() ()
() ()
2
4
6
7
1, 1
2, 1
3, 1
4, 5, 1
sgn Im
sgn Im
sgn Im
sgn Im .
jj
l
jj
l
jj
l
jj j
l
eF
eF
eF
ee FQdQ
?
?
?
?
???
???
???
?? ? ?
Qd
Qd
Qd
? ?
? ?
=?
? ?
? ?
??
? ?
? ?
=?
? ?
? ?
??
? ?
? ?
=?
? ?
? ?
??
? ?
? ?
+=?
? ?
? ?
??
?
?
?
?
 (2.94) 
 
The integrals solved above are then substituted into these equations producing 
 
()
()
()
()
2
0 2
1, 1 0 1
2
0
2
1
1
2
0
2,
3,
1sec
sgn cos sec
sec sin ( 1)
2
1; 0
2
2sin
sgn cos ;
2sin
sec sin ( 1)
2; 0
2
2
1; 0
2
0;
2; 0
2
s
j
j
j
j
j
j
k
ek
n
k
B
n
j
n
B
n
B
k
j
B
e
j
j
e
??
???
?
??
?
?
?
??
?
?
?
??
??
?
?
?
?
?
?? ?
?
?
?
?
?
=
+?
?
?
=<<
?
?
?
??
?
?
???
? ??
?
+?
= ?<<
?
?
??
=
?
?
?
= ?<<
?
?
?
?
?
?
?
=<<
?
? ?
?
?
=
()() ()
()
()
()
()
()
1
1
1
22
001
4, 1
2
0
1
5, 1
2
0
gn 1
1
;tan
2sin
1
0;tan
1sec sec
sgn
sec sin ( 1)
2
(1)
sgn
2 sin sin 2 sin
sgn
sec sin ( 1)
j
j
j
j
j
j
j
xam
yB
xam
yB
kfk
e
n
k
B
nn
f
BB
e
k
??
?
?
?
???
?
?
??
???
?? ?
?
??
?
?
?
?
??
?
??
?
?
?+ ???
?
=
??
?
??
?
?+ ???
?
?
??
?
??
?
?
=?
+?
??
????
??
??
??
??
=?
+?
m
m
.
2
n
B
?
 (2.95) 
At this point, the wave number integral of the double integral component in the 
leading edge contribution (2.41) has been solved using the complex integration method.  
The wave angle integral in both leading edge components still must be solved. 
2.3.3. Wave Angle Integral 
Each of the resulting functions from the previous section, equation (2.95), is 
integrated with respect to the wave angle in equation (2.41).  The second component in 
this equation is also integrated with respect to the wave angle.  To simplify this 
 31
 
integration, the second component is written in terms similar to the first component (after 
the complex integration).  The integrand of the second term (2.39) is 
 32
j
e 
()
1, 6, 6,
1
n
jj
Je
+ ?
=? ?  (2.96) 
with  
 
( ) ( )
()
2
01
6,
2
0
1cos sec
sec sin 1
2
j
j
j
k
e
n
k
B
? ?
?
??
?
?
??
=
+?
. (2.97) 
The total leading edge contribution (2.47) is 
 
() ()
()
2
,
,, 1,1 1,222
020
2
,0 3
1,1 1,2
2
,sec
4sec
sec .
4
nm
nmL
nm
A
k
x yI
kk
Ak
JJd
?
?
?
?
Ikd? ??
??
??
?
?
?
?
=+
?
++
??
?
 (2.98) 
Using the complex integration method (2.48) the wave number integration is simplified 
to 
 () ()
()
22
,,03
,, 1 2 1,1 1,22
,sec sec
44
nm nm
nmL
AAk
x yEEd J
??
d? ?? ?
??
=++ +
??
?. (2.99) 
This leads to a natural splitting of the leading edge contribution (2.99) into two parts 
 
,, 1 2nmL
TT? = +  (2.100) 
where 
 
22
,,03
1,2
sec sec
44
nm nm
jj
AAk
TEd
??
j
Jd?
??
=+
??
?. (2.101) 
These integrals are expressed in terms of the functions defined above (2.55), (2.93), and 
(2.96) as 
 
 
()
()
()
22
1, 2, 3, 4, 5,
22
1, 2, 3, 4, 5,
22
33
1, 6, 6,
sec sec 1
sec sec 1 .
n
jjj
jjjjj
n
jj
Ed e e e e e
eeeee d
Jd e e d
??
??
??
?? ?
?
?? ? ?
+++++
??
?????
+?
??
?
= ? ++++
?
?
?++++
?
??
=??
??
??
??
j
j
t
  (2.102) 
Each part is made up of a positive and a negative contribution: 
 
()
1
n
jj
Tt
+ ?
= ?? (2.103) 
where 
 
2
,, ,
1, 2, 3,
22 2
2
,,,03
4, 5, 6,
22
sec sec sec
444
sec sec sec .
44 4
nm nm nm
jjj
nm nm nm
jj j
AAA
te e
AAAk
ee ed
?
?
???
???
j
? ??
?? ?
????
?
?? ?
?
=++
?
?
?
+++
?
?
?
?
 (2.104) 
To simplify the integration, each of these integrals is written separately using 
equations (2.95) and (2.97).  The first integral is 
 
()
() ()
2
1, 1,
2
32
2
001
1, 1
2
2
0
sec
1 sec cos sec
sgn .
sec sin ( 1)
2
jj
j
j
j
Red
kk
R d
n
k
B
?
?
?
?
??
?? ??
? ?
?
??
??
?
?
??
?
=
?
=
+?
?
?
 (2.105) 
 33
 
The second integral has two forms depending on the value of j, and is 
 
()
()
2
2, 2,
2
2
1
2
2,1 1
2
0
0
2
1
0
2,2 1
2
2
0
sec
sec
cos
2sin 2sin
sgn ; 1
sec sin ( 1)
2
sec
cos
2sin 2sin
sgn ; 2.
sec sin ( 1)
2
jj
j
j
Red
nn
BB
R
n
k
B
nn
BB
Rd
n
k
d
B
?
?
?
?
??
????
??
?
?
??
????
??
??
?
??
??
?
?
??
?
??
?
=
??
??
??
==
+?
??
??
??
+?
?
?
?
? (2.106) 
The third integral?s integrand only has a value at one wave angle and so is zero: 
 
2
3, 3,
2
3,
sec
0.
jj
j
R ed
R
?
?
? ?
??
?
?
=
=
?
 (2.107) 
The fourth integral is 
 
()
()
()
2
4, 4,
2
32
2
001
4, 1
2
2
0
sec
1sec sec
sgn .
sec sin ( 1)
2
jj
j
j
j
Red
kfk
R d
n
k
B
?
?
?
?
??
???
? ?
?
??
??
?
?
??
?
=
?
=?
+?
?
?
 (2.108) 
The fifth integral is slightly more complicated.  It is 
 
()
2
5, 5,
2
1
2
5, 1
2
2
0
sec
sec ( 1)
sgn
2 sin sin 2 sin
sgn .
sec sin ( 1)
2
jj
j
j
j
Red
nn
f
BB
R d
n
k
B
?
?
?
?
??
????
?? ?
? ?
?
??
??
?
?
??
?
=
??
????
??
??
??
??
=?
+?
?
?
 (2.109) 
 34
 
One of the sign functions depends solely on the value of j and whether the wave angle is 
positive or negative.  To simplify the equation it is split into two parts; one for positive 
wave angles and the other for negative wave angles, i.e. 
 35
bj
 
5, 5, , 5, ,jaj
R RR
?? ?
=+ (2.110) 
with 
 
()
()
1
2
5, , 1
2
0
0
1
0
5, , 1
2
2
0
sec
(1)
2 sin 2 sin
sgn
sec sin ( 1)
2
sec
(1)
2sin 2sin
sgn .
sec sin ( 1)
2
j
aj
j
j
bj
j
nn
f
Bb
Rd
n
k
B
nn
f
BB
R d
n
k
B
?
?
????
??
??
?
??
????
??
? ?
?
??
?
??
?
??
?
??
?
??
??
=
+?
??
?
??
??
=?
+?
?
?
 (2.111) 
The last integral is 
 
() (
()
)
2
3
6, 6,
2
32
2
01
6,
2
2
0
sec
1 sec cos sec
.
sec sin 1
2
jj
j
j
j
Red
k
R d
n
k
B
?
?
?
?
??
???
?
?
??
??
?
?
?
?
=
??
=
+?
?
?
 (2.112) 
Similar terms are grouped to match the equations in the Cheng and Wellicome paper.  
The first of these groups is 
 
,0 , ,0
,1 1, 6,2
44 4
nm nm nm
jj
Ak A Ak
ef R R
???
j
? ?
=+
?
 (2.113) 
which becomes 
 ()
()
( )
32
2
01
,1 1
2
2
0
sec cos sec
11sgn
sec sin ( 1)
2
j
j
j
k
ef d
n
k
B
?
?
???
? ?
?
??
?
??
?
??
=? ? ?
??
+?
?
. (2.114) 
 
The next three terms are simply equations from above, 
 36
j
j
  (2.115) 
,1 4,
,2 5, ,
,3 5, ,
jj
ja
jb
el R
el R
el R
??
??
??
=
=
=?
which become 
 
()
( ) ( )
()
()
32
2
101
,1 0
2
2
0
1
1
2
,2
2
0
0
1
1
,3
2
0
sgn sec sec
1
sec sin ( 1)
2
sec
sgn
sin 2 sin
(1)
2
sec sin ( 1)
2
sec
sgn
sin 2 sin
(1)
2
sec sin (
j
j
j
j
j
j
j
j
fk
el k d
n
k
B
n
f
B
n
el d
n
B
k
B
n
f
B
n
el
B
k
?
?
?
?? ??
?
?
??
???
?
??
?
?
?
??
???
?
??
?
??
??
?
?
?
?
?
?
?
?
=? ?
+?
??
??
??
=?
+?
??
??
??
=?
+
?
?
0
2
.
1)
2
j
d
n
B
?
?
?
?
?
?
 (2.116) 
The positive and negative components of the leading edge contribution (2.104) are also 
written in terms of the Cheng and Wellicome functions: 
 
,, ,,0,
,1 ,2 ,3 ,1 2,
22 2
444 4 4
nm nm nm nm nm
jj j j j
AAAAkA
telelel ef
??? ??
??? ? ?
=+?+ +
j
R
?
. (2.117) 
This leads to the leading edge contributions (2.103) of 
 
() () ()
() ()
, ,
,1 ,1 ,2 ,2 ,3 ,32 2
,0 ,
,1 ,1 2, 2,2
11 1
.
44
nn n
nm nm
jjjjjj
nm nm
jj jj
Telelelelel
Ak A
ef ef R R
j
el
? ?
+? +? + ?
+? +?
??
=??+???
??
??
????
+??+??
????
 (2.118) 
To get the total leading edge contribution (2.100) this function is summed with j set to 
one and two.  Based on this, the total last term of the leading edge contribution is 
 
 () () ()
2
,,
2, 2, 2,1 2,1 2,2 2,2
22
1
11
44
nnn
nm nm
jj
j
AA
RR RR RR
??
+? +? +?
=
???
??=??+??
???
?
?
?
. (2.119) 
Using the functions in equation (2.106) this equation is 
 
()
() () ()
() ()
2
,
2, 2,
2
1
11
2
,
11
2
0 1
0
1
1
1
4
sec
cos cos
2 sin sin 2 sin
1sgn sgn
8
sec sin
cos
2
2sin
cos
2sin
1sgn
cos
n
nm
jj
j
n
nm
n
A
RR
nn
An BB
d
n
B n
k
B
B
n
B
n
?
?
?? ???
???
? ??
?
??
??
?
??
?
?
?
+?
=
+?
+?
?
+
+
??
??=
??
??
?
?? ??
?
?? ??
?? ???
??
?
??
?
?
??
??
?
????
?
??
??
??
+?
?
?
()
1
0
1
2
2 1
0
sec
cos
sin 2 sin
sgn .
sec sin
2
2sin
n
B
d
n
k
B
B
?
???
??
??
?
?
??
?
?
?
?
?
??
??
?
??
??
?
?
?
??
+
???
??
??
??
?
?
?
 (2.120) 
To simplify the integrand, the term 
 
1
1
cos
2sin
cos
2sin
n
B
n
B
??
?
??
?
+
?
??
??
?
??
??
??
?
 (2.121) 
is studied.  From the definition given in equation (2.39) and trigonometric identities, this 
term is 
 
()
()
()
1
1
cos
cos 1 cos
2sin2
cos
cos
sin 1 sin
2sin
2sin2
cos
cos 1 cos
cos
2sin2
2sin
sin
2
nn
xam y
bB
n
xam y
B
BB
nn
n
xam y
bB
B
n
xa
B
???
?
??
2
2
2
n
n
?
? ?? ?
?
?
???
??
?
?
?
+
?
??
+? +??
??
??
??
??
++?+??
??
??
??? ??
=
??? ??
+? +??
??
????
??
??
?+()
??
??
?
?
??
cos
1si
sin 2 2
nn
my
B
n
? ??
?
??
?+??
??
??
. (2.122) 
 37
 
If n is odd, the cosine terms are zero, leaving 
 
()
()
1
1
cos
cos
sin 1 sin
2sin
2sin2
cos
sin 1 sin
cos
2sin2
2sin
n
nn
xam y
B
BB
n
n
xam y
BB
B
??
2
2
n? ?? ?
?
?
? ?? ?
??
?
?
+
?
??
??
+? +??
??
??
??
??? ??
=
??? ??
?+?+??
????
??
??
??
?
?
 (2.123) 
which simplifies to 
 
1
1
cos
2sin
1
cos
2sin
n
B
n
B
??
?
??
?
+
?
??
??
??
=?
??
??
??
. (2.124) 
On the other hand, if n is even, the sine terms are zero, leaving 
 
()
()
1
1
cos
cos
cos 1 cos
2sin
2sin2
cos
cos 1 cos
cos
2sin2
2sin
n
nn
xam y
B
BB
n
n
xam y
BB
B
??
2
2
n? ?? ?
?
?
? ?? ?
??
?
?
+
?
??
??
+? +??
??
??
??
??? ??
=
??? ??
+? +??
????
??
??
??
?
?
 (2.125) 
which simplifies to 
 
1
1
cos
2sin
1
cos
2sin
n
B
n
B
??
?
??
?
+
?
??
??
??
=
??
??
??
. (2.126) 
Therefore, overall this function is equivalent to 
 ()
1
1
cos
2sin
1
cos
2sin
n
n
B
n
B
??
?
??
?
+
?
??
??
??
= ?
??
??
??
. (2.127) 
 38
 
So the equation (2.120) simplifies to 
 
()
() ()
() ()
2
,
2, 2,
2
1
1
2
,
11
2
0
0
1
0
11
2
2
0
1
4
sec
cos
sin 2 sin
sgn sgn
8
sec sin
2
sec
cos
sin 2 sin
sgn sgn .
sec sin
2
n
nm
jj
j
nm
A
RR
n
An B
d
n
B
k
B
n
B
d
n
k
B
?
?
?
???
??
? ?
?
??
???
??
?? ?
?
??
+?
=
?
+?
?
+?
?
??
??=
??
??
? ??
?
??
??
?
??
?
??
?
?
?
?
???
?
??
??
?
??
+?
??
?
+
?
?
?
?
?
? (2.128) 
Writing this in terms of Cheng and Wellicome functions produces 
 ()
(
2
,,
2, 2, 1,2 2,2
2
1
1
48
n
nm nm
jj
j
AAn
)
R Ref
B?
+?
=
??
??= +
??
??
?
ef (2.129) 
where 
 
() ()
()
() ()
()
1
2
1,2 1 1
2
0
0
1
0
2,2 1 1
2
2
0
sec
cos
sin 2 sin
sgn sgn ; 1
sec sin 1
2
sec
cos
sin 2 sin
sgn sgn ; 2.
sec sin 1
2
j
j
n
B
ef d j
n
k
B
n
B
ef d j
n
k
B
?
?
???
??
?? ?
?
??
???
??
?? ?
?
??
?
+?
?
+?
?
??
??
??
??
=?
??
+?
??
??
??
??
=?
??
+?
?
?
=
=
 (2.130) 
 39
 
To summarize, the total leading edge contribution is 
 
() () ()
() () ()
() ()
,
, , 1,1 1,1 1,2 1,2 1,3 1,3
2
2,1 2,1 2,2 2,2 2,3 2,3
,0
1,1 1,1 2,1 2,1
,
1,2 2,2
11 1
4
11
11
4
8
nn n
nm
nmL
nn n
nn
nm
nm
A
el el el el el el
el el el el el el
Ak
ef ef ef ef
An
ef ef
B
?
?
?
+ ?+?+
+? +? +?
+? +?
?
=??+???
?
?
+? ? +? ? ?? +
?
??
+??+??
??
??++
??
?
 (2.131) 
using the following integrals 
 
()
( ) ( )
()
()
32
2
101
,1 0
2
2
0
1
1
2
,2
2
0
0
1
1
,3
2
0
sgn sec sec
1
sec sin ( 1)
2
sec
sgn
sin 2 sin
(1)
2
sec sin ( 1)
2
sec
sgn
sin 2 sin
(1)
2
sec sin (
j
j
j
j
j
j
j
j
fk
el k d
n
k
B
n
f
B
n
el d
n
B
k
B
n
f
B
n
el
B
k
?
?
?
?? ??
?
?
??
???
?
??
?
?
?
??
???
?
??
?
??
??
?
?
?
?
?
?
?
?
=? ?
+?
??
??
??
=?
+?
??
??
??
=?
+
?
?
() ()
()
() ()
()
() ()
0
2
32
2
01
,1 1
2
2
0
1
2
1,2 1 1
2
0
0
2,2 1 1
1)
2
sec cos sec
11sgn
sec sin ( 1)
2
sec
cos
sin 2 sin
sgn sgn ; 1
sec sin 1
2
sec
cos
sin
sgn sgn
j
j
j
j
j
d
n
B
k
ef d
n
k
B
n
B
ef d j
n
k
B
n
ef
?
?
?
?
?
?
???
??
?
??
???
??
?? ?
?
??
?
?
??
?
?
??
?
?
+?
+?
?
??
=? ? ?
??
+?
??
??
??
??
=?
??
+?
??
=?
??
?
?
?
=
()
1
0
2
2
0
2sin
;2
sec sin 1
2
j
B
dj
n
k
B
?
??
?
?
?
??
?
?
??
??
??
=
+?
?
.
 (2.132) 
These integrals do not have an analytical solution; they must be numerically 
integrated.  Several of these integrals contain the even auxiliary function from equation 
 40
 
(2.91).  This function also does not have an analytical solution and either a series 
expansion or numerical integration must be employed. 
2.3.4. Evaluation Grid 
As was mentioned in equations (2.45) and (2.46), the wave height is given by 
 
() () ( )
() ()
,, ,,
11
1
,,
sin .
2
NM
nmL nmL
nm
N
n
n
gxy g xy xay
n
Ax yB
B
?? ? ? ?
?
==
=
,? ?=??
? ?
??
?+
??
??
??
?
 (2.133) 
In the previous section it was determined that this equation cannot be solved analytically; 
therefore, a numerical solution is employed.  To approximate this function numerically, a 
grid of solution points is created.  Two indices are already in use; m corresponds to the 
longitudinal pressure strip in question and n corresponds to the frequency of the sine 
wave in the pressure.  To define where on the surface the height is being solved for, two 
more indices are used: i corresponds to the longitudinal (x) direction and j corresponds to 
the transverse (y) direction.  Using these indices the x and y coordinates become 
 
1
2
1
.
2
i
j
L
x i
I
B
yj
J
??
=?
??
??
??
=?
??
??
 (2.134) 
I is the number of longitudinal points in the wetted area and J is the number of transverse 
points. 
 41
 
 
Fig. 7 Sinusoidal Pressure Hull Discretization 
i = 1
x = x
1
i = 2
x = x
2
    x = -L 
    x = 0 
    x = -a 
m = 1 
m = 2 
m = M
j = J
y = y
J
j = 2
y = y
2
y = B 
j = 1
y = y
1
y = 0
i = I
x = x
I
y,? 
x,? 
a 
Similarly, the pressure coefficient from the last term of the wave height (2.133) is written 
in terms of the longitudinal index: 
 ( )
,ni n
Ax A=
i
. (2.135) 
At this point a tensor containing the wave height from the leading edge contribution of 
each pressure strip and pressure frequency for each of the grid points is created: 
 
( )
,,
,,,
,
,
nmL i j
nmi j
nm
gxy
E
A
??
= . (2.136) 
It has been noted that the trailing edge contribution is expressed by shifting the leading 
edge equation by a patch length (2.43), i,e, 
 
1
2
i
x aa i a
??
+ =?+
??
??
. (2.137) 
This is equivalent to shifting the index of the longitudinal coordinate: 
 42
 
 43
i
 
1i
x ax
?
+ = . (2.138) 
The same logic is used to express each of the subsequent pressure strips as the first strip 
offset: 
 
() ()
()
1
1
11
2
1.
i
iim
xam a i am
xam x
+?
??
+?=?+?
??
??
+?=
 (2.139) 
These simplifications are then substituted back into the wave height equation (2.133): 
 
()(
,,,1,,1 ,
11 1
sin
2
NM N
ij nm n i mj n i mj ni j
nm n
n
gAEEAy
B
?
??
+? ?
== =
)
B
? ?
=? +
? ?
? ?
?? ?
. (2.140) 
This equation is further simplified by combining the n summations.  Additionally, the 
case where the transverse position is outside the half breadth of the craft (2.26) is now 
added back in, resulting in 
 
()
()
()
,,1,1, ,1,
11
,
,
,,1,1, ,1,
11
;
sin
2
;
NM
nm ni mj nimj
nm
j
ni j
ij
NM
j
nm n i m j n i m j
nm
j
AE E
B yB
n
AyB
g
B
yB
AE E
By
?
??
+? ?
==
+? ?
==
?
?
?
? ?
?
?
? ??
?
?
??
?
?+
=
?
??
??
?
?
?
??
? ?
?
?
?
??
??
?
. (2.141) 
 
The m sum is combined with the last term by including a condition when m = i, 
 
()
()()
()
,,1,1, ,1,
11
,
,,1, ,10, ,
,,1,1, ,1,
11
;
sin
2
;
NM
nm n i m j n i m j
nm
mi
j
ij
ni n j n j ni j
NM
j
nm n i mj n i mj
nm
j
AE E
B yB
ng
AE E A yB
B
yB
AE E
By
???
+? ?
==
?
+? ?
==
?
?
?
?
?
?
?
? ?
? ???
?
?
?
??=
??
?
+?? +
???
??
?
??
??
?
?
???
?
?
?
?
?
??
??
. (2.142) 
According to the discretized longitudinal position equation (2.134), the term E
n,1,0,j
 
corresponds to a location ahead of the craft.  Therefore, the wave height due this term is 
obviously zero.  A new wave height basis function tensor is generated that incorporates 
each of these changes. 
 
()
,1, 1 , ,1, ,
*
,,,
,1,1,
;  or  or 
sin ;  and -
2
ni mj nimj j j
nmi j
nj j j
EE imyBy
E
n
EyBi
B
?
+? ?
B? ?<?
?
?
=
? ??
?+=??
? ??
??
?
>
,
. (2.143) 
Substituting back into the total wave height (2.142) produces 
 . (2.144) 
*
,
11
NM
i j nm nmi j
nm
gAE??
==
=
??
This equation is linear in the pressure coefficients.  It can be written in matrix form by 
introducing two new indices: p corresponds to the pressure strip and frequency and k 
corresponds to the location on the surface, i.e. 
 
( )
()
1
1.
p mN
ki Jj
n= ?+
=? +
 (2.145) 
 44
 
The wave height equation then becomes 
 45
pk
  (2.146) 
*
,
1
MN
k
p
gAE??
=
=
?
which is written in matrix form as 
 
*
g??= AE . (2.147) 
In summary, the integrals in equation (2.132) are numerically integrated over a grid 
of points for the leading edge of the first pressure strip.  The grid is then shifted to 
generate the entire basis function tensor.  Using the basis function tensor and pressure 
coefficients the wave height is determined. 
2.4. CONSTANT PRESSURE PATCHES 
A second method to define the pressure patches is to assume a constant pressure over 
the entire patch [Scullen].  Multiple pressure patches with varying pressures are then 
created over the wetted area of the craft.  The wave height is linear in each of the 
contributing heights, so the heights from the individual pressure patches are added 
together to generate the total wave height, i.e. 
 () (
,
11
,
NM
nm
nm
)x y??
==
=
??
xy. (2.148) 
Scullen and Tuck present the following wave height  
 () ()
,
,
,
,
nm
nm
nm z
S
P
0x yGxyd
g
? ???
?
=??
??
? (2.149) 
 created by such a pressure patch, where 
 ()
( )
2 cos sin
20
020
,, Re sec
2se
ik x y kz
k e
Gxyz dkd
kk
? ??
?
2
c
? ?
??
? ?++
?
=
?
??
. (2.150) 
 
The partial derivative of this equation with respect to height, evaluated on the undisturbed 
free surface (z = 0), is 
 ()
( )
2 cos sin
20
22
020
,,0 Re sec
2s
ik x y
z
k ke
Gxy dkd
kk
? ??
?
ec
? ?
??
? ?+
?
=
?
??
. (2.151) 
 Therefore, the wave height due to a single constant pressure patch is 
 
()
() ()
,
2
,0 2
, 22
020
cos sin
,Resec
2
.
nm
nm
nm
ik x y
S
Pk
k
xy
gk
e dddkd
?
?
?? ??
??
sec? ??
? ??
?
?
?? +???
??
=
?
??
??
g
 (2.152) 
 
Fig. 8 Constant Pressure Hull Discretization 
    x = -L 
    x = 0 
    x = -a 
m = 1 
m = 2 
m = M
b
y,? 
y = 0y = B 
i = 1
x = x
1
i = 2
x = x
2
i = I
x = x
I
j = 1
y = y
1
j = 2
y = y
2
j = J
y = y
J
n = N n = 2 n = 1
y = b
x,? 
a 
As with the sinusoidal pressure method, a double integral over the wetted area of the 
craft must be evaluated to determine the surface height. 
 46
 
2.4.1. Surface Integral 
From Fig. 8 the surface integration is 
 
() () () ()
()
( )
,
1
cos sin cos sin
1
nm
am
bn
ikxy ikxy
Sambn
edde
?? ?? ?? ??
d? ??
??
?? +? ?? +??? ??
?? ??
??
=
?? ? ?
?. (2.153) 
To evaluate this integral two variable substitutions are made: 
 
*
*
.
x
y
? ?
? ?
= ?
= ?
 (2.154) 
The surface integral evaluates to 
 
( ) ( )
()() ()()
()() ()( )
,
cos sin
cos sin cos sin
22
cos sin cos sin
22
sin cos sin cos
.
sin cos sin cos
nm
ik x y
S
ik x am a y bn ik x am a y bn b
ik xam ybn ik xam ybnb
edd
ee
kk
?? ??
? ??
?? ? ?
??
?? ??
?
?
?? +???
??
?+? +? ?+? +?+???
???
?+ +? ?+ +?+???
???
=
+?
??
?
?
?
?
 (2.155) 
The single patch wave height (2.152) becomes 
 
()
()
()() ()()
()() ()( )
2
2
,
0
, 2
2
200
cos sin cos sin
cos sin cos sin
sec1
,Re
2sincossec
.
nm
nm
ik x am a y bn ik x am a y bn b
ik xam ybn ik xam ybnb
P
k
xy
g kk k
ee
?
?
dk
? ??
?? ? ?
?
?
?? ? ?
?
?
?
?
?
?+? +? ?+? +?+???
???
?+ +? ?+ +?+??? ?
??? ?
=?
?
?
?
?
?
?
??
g
?
?
 (2.156) 
The surface integration is accomplished.  Two integrations remain in this method, the 
wave angle and wave number integrations.  The first one to evaluate is the wave number 
integral. 
 47
 
2.4.2. Wave Number Integral 
To simplify the single patch wave height, a new function is created based on the four 
terms of the surface integral.  Essentially, a height component must be found for each of 
the four corners of the pressure patch.  The wave height equation becomes 
 
() ()(
()( )
,
,0 0
00
,,,
,,.
nm
nm
P
)x y x am a y bn x am a y bn b
g
xamybn xamybnb
?? ?
?
??
= +??? +??+?
?
?+ ?++ ?+?
?
 (2.157) 
where 
 ()
()
()
2
2
cos sin
0
0
2 2
200
sec11
,Re
2sincos sec
ik x y
k
x ye
kk k
?
??
?
?
dk? ?
??? ?
?
?+
?
=?
?
??
. (2.158) 
Evaluating the wave number integral produces 
 ()
( ) ( )
2
0 2
2
glog2Hsin
1
,
2sincos
TT TT
x yd
?
?
?
? ?
???
?
+?
=?
?
 (2.159) 
with 
 
2
00
sec sec sinTkx ky? ?=+ ?, (2.160) 
g the even auxiliary function defined in equation (2.91), and H the Heaviside step 
function, defined as 
 
()
0; 0
H
1; 0
z
z
z
<
?
=
?
>
?
. (2.161) 
2.4.3. Wave Angle Integral 
As with the sinusoidal pressure patches, the wave angle integral must be evaluated 
numerically.  To simplify this integration the corner integral (2.159) is divided into two 
parts, a local and a far field. 
 48
 
 ( ) ( ) ( )
0
,,
LF
,x yxyx???=+y (2.162) 
The local field corresponds to the area directly under the craft and does not contribute to 
the wake created by the craft; it does have an effect near the craft, however.  This term is 
 ()
( ) ( ) ( )
2
2
2
glog2HHsin
1
,
2sincos
L
TT TxT
x yd
?
?
?
? ?
???
?
+? ???
??
=?
?
. (2.163) 
The far field on the other hand contains the craft wake, but the integrand is much simpler: 
 ()
( )
2
2
H
sin
,
sin cos
F
x
T
x y
?
?
d? ?
? ??
?
=
?
. (2.164) 
The total wave height is found by summing the far field component at each corner of 
the pressure patch and accounting for the local field contribution which will be discussed 
in section 2.5.2. 
2.4.4. Evaluation Grid 
As with the sinusoidal pressure patches, the wave angle integral is solved 
numerically.  Therefore, a grid over which to solve the integral is created.  The same grid 
as the sinusoidal pressure method is used with one difference.  Previously, n 
corresponded to the frequency of the pressure; now it corresponds to the transverse 
pressure patch.  The location is discretized in the same way as in the previous method 
(2.134).  Similar to the sinusoidal pressure method, a basis tensor consisting of the 
contributions of each pressure point at each index location is found: 
 
()( )
)
,,, 0 0
00
,,
nmi j i j i j
ij ij
E x am a y bn x am a y bn b
x amy bn x amy bn b
??=+???+??+
?+ ?++ ?+.
 (2.165) 
This method requires a wave height contribution from each of the corners of the pressure 
patch.  Each of these contributions is identical except for the offset.  This offset is 
 49
 
 
()
()
1
2
1
2
1
1.
i
i
j
j
xama i am
x am a a i am a
ybnbj bn
ybnbbj bnb
??
+= ?+
??
??
??
+ ?= ?+ ?
??
??
?= ??
?+= ??+
 (2.166) 
These can also be written in terms of the index as 
 
1
1
.
iim
i
jjn
xamx
xamax
ybny
ybnby
?
i? +
?
? +
+=
+?=
?=
?+=
 (2.167) 
The basis tensor (2.165) then becomes 
 
()( ) ( )(
,,, 0 1 0 1 1 0 0 1
,,,,
nmij im jn im jn im jn im jn
Exyxyxyxy?? ??
?+ ? ?+ ?+ ? ? ? ?+
=? +
)
. (2.168) 
As with the previous method, only the first pressure patch contribution needs to be 
calculated.  The other patches are identical but offset.  Substituting this basis tensor back 
into the single pressure patch wave height equation (2.157) produces 
 
( )
,,
,
nm i j nm nmi j
gxyPE?? =
,
,
. (2.169) 
The total wave height (2.148) therefore becomes 
 . (2.170) 
,,
11
NM
i j nm nmi j
nm
gPE??
==
=
??
The system is linear in the pressure coefficients and is written in matrix form similar the 
sinusoidal pressure method, equations (2.145), (2.146), and (2.147). 
In this method equation (2.164) must be numerically integrated.  The integration is 
performed over a grid of points to produce the contribution of one corner of a pressure 
patch.  The total patch contribution is found by shifting the first corner to the other three 
 50
 
 51
corners and summing the results.  This first patch is then shifted to produce the entire 
basis function tensor.  The pressure amplitudes together with this tensor determine the 
total wave height. 
2.5. NUMERICAL ISSUES 
Both the sinusoidal and the constant pressure methods involve a numerical integration 
of the wave angle integral.  These integrals all contain features such as singularities and 
rapid oscillations which make the numerical integration difficult to perform. 
 
2.5.1. Sinusoidal Pressure Method 
The integrals for the sinusoidal pressure method (2.132) are 
 
()
( ) ( )
()
()
32
2
01 0 1
,1
2
2
0
1
1
2
,2
2
0
0
1
1
,3
2
0
sgn sec sec
1
sec sin ( 1)
2
sec
sgn
2 sin 2 sin
(1)
sec sin ( 1)
2
sec
sgn
2 sin 2 sin
(1)
sec sin (
j
j
j
j
j
j
j
j
kfk
el d
n
k
B
nn
f
BB
el d
n
k
B
nn
f
BB
el
k
?
?
?
?? ??
?
?
??
????
?
??
?
?
??
????
?
??
??
??
?
?
?
?
?
?
?
?
=? ?
+?
??
??
??
=?
+?
??
??
??
=?
+
?
?
()
() ( )
() ()
()
() ()
0
2
32
2
101
,1
2
2
0
1
112
1,2
2
0
0
11
2,2
1)
2
1 sgn sec cos sec
1
sec sin ( 1)
2
sec
sgn sgn cos
sin 2 sin
;1
sec sin 1
2
sec
sgn sgn cos
sin
j
j
j
j
j
d
n
B
k
ef d
n
k
B
n
B
ef d j
n
k
B
n
ef
?
?
?
?
?
?
?? ??
?
?
??
???
??
??
?
?
??
?
??
?
?
??
?
?
?
+?
+?
?
??
?
??
=? ?
+?
??
??
?
??
??
??
==
+?
??
?
??
=
?
?
?
()
1
0
2
2
0
2sin
;2
sec sin 1
2
j
B
dj
n
k
B
?
??
?
?
?
??
?
?
??
??
??
=
+?
?
.
 (2.171) 
It should be noted that the denominator is identical in each of these integrands: 
 
2
0
sec sin ( 1)
2
j
j
n
Dk
B
?
??=+?. (2.172) 
 52
 
The integrands also contain only a few common numerators: 
 
( ) ( )
()
() ( )
() ()
32
01 0 1
1
1
32
101
1
11
1sgnsecsec
sec
2sgn
2 sin 2 sin
1 1 sgn sec cos sec
sec
2 sgn sgn cos .
sin 2 sin
LN k f k
nn
LN f
BB
FN k
n
FN
B
?? ??
????
?
??
?? ??
???
??
? ?
?? ?
?
??
?? ?
?
+?
=
??
=
??
??
??
=?
??
??
=?
??
??
 (2.173) 
To cut down on the number of calculations performed during the numerical integration, 
the leading edge wave height equation (2.131) is divided into three terms: 
 () () (
,,0,
,, 2
44 8
nm nm nm
nmL
AAkAn
)I II III
B
?
??
=+ + . (2.174) 
The first term in this equation is 
 
() ( ) ( )
() () ()
1,1 1,1 1,2 1,2 1,3 1,3
2,1 2,1 2,2 2,2 2,3 2,3
11 1
nn n
Ielel elel elel
el el el el el el
+? +? +?
+? +? +?
=? ? +? ? ?? +
+? ? +? ? ?? + .
 (2.175) 
This is written in terms of the numerators and denominator above as 
 
() ()
() ()
() ()
() ()
0
12
2
2
10
2
11 1 12 2
11 1 12 2
11 1 12 2
11 1 12 2
.
nn
nn
nn
nn
LN LN L N L N
I
D
LN LN L N L N
d
D
LN LN L N L N
D
LN LN L N L N
d
D
?
?
?
?
+? + ?
?
+? + ?
+? + ?
+? + ?
?
??+??
= ?
?
?
?
?? + ?? +
+ ?
?
?
?
???
+ ?
?
?
?
?? + +? ?
+ ?
?
?
?
?
 (2.176) 
 53
 
The second term is 
 () ()
1,1 1,1 2,1 2,1
1
nn
IIefef efef
+ ?+
=? ? +? ?
?
; (2.177) 
or in terms of the common numerators and denominators 
 
() ()
2
122
11 1 11 1
nn
FN FN FN FN
IId
DD
?
?
?
+? +?
?
??
???
=+??
??
?
. (2.178) 
The third term is 
 
1,2 2,2
III ef ef= + ; (2.179) 
or in the common numerators and denominators 
 
2 0
1202
2FN FN
III d d
DD
?
?
? ?
?
=+
??
. (2.180) 
Combining the integrals in this way cuts down on the number of times the integration 
routine calculates common terms.  It also allows the integration routine to only be more 
accurate where needed and reduce the number of integrating points thereby speeding up 
the calculation.  Typical combined integrands are shown below. 
 54
 
 
-pi/2 -pi/4 0 pi/4 p i/2
-30
-20
-10
0
10
20
30
I
?
?? ??
0
? 2?
 
 
-pi/2 -pi/4 0 pi/4 p i/2
-30
-20
-10
0
10
20
30
II
?
??
 
??
0
 ? 2?
 
-pi/2 -pi/4 0 pi/4 p i/2
-30
-20
-10
0
10
20
30
III
?
?? ??
0
?
 
2?
Fig. 9 Sinusoidal Pressure Combined Integrands  
k
0
 = 3.08, x = 0.25, y = 0.14, m = 1, n = 1, a = 0.1, B = 0.5 
There are still a few issues preventing accurate numerical integration of these 
functions.  The first is the rapid oscillation in II as it approaches the integration limits.  
To aid in the integration of this oscillation a variable substitution is made: 
 tant ?= . (2.181)
This substitution applied to the II integral (2.178) leads to 
 
() ()
** **
12
11 1 11 1
nn
FN FN FN FN
IId
DD
+? +??
??
??
???
=+??
??
?
t (2.182) 
 55
 
with the numerator and denominator being 
 
( ) ( )
()()
*2
10
2
0
1
2
11sgn 1cos1
1(1)
2
1
.
1
j
j
FN t k t
n
Dkt t
B
xam yBt
t
2
1
? ?
?
?
??
?
???
=? + +
???
=++?
+?+
=
+
m
?
?
?
 (2.183) 
This new integrand is shown below. 
-5 0 5
-5
0
5
II
t
 
Fig. 10 Sinusoidal Pressure II Integrand with Slow Oscillations 
k
0
 = 3.08, x = 0.25, y = 0.14, m = 1, n = 1, a = 0.1, B = 0.5 
The oscillations are much more reasonable.  There is a new problem, however, the 
integration limits are now positive and negative infinity.  The integrand rapidly 
approaches zero as t approaches positive or negative infinity.  The integration is safely 
truncated with minimal inaccuracies in the result. 
Another problem exists; the integrand of each of the functions becomes infinite at 
some points.  To find where these singularities are, the value of the wave angle that sets 
the denominator equal to zero needs to be found: 
 
2* *
0
sec sin ( 1) 0.
2
j
j
n
Dk
B
?
??=+?= (2.184) 
 56
 
Instead of solving this equation directly, it is easier to solve it in terms of the transformed 
variable used to stretch the II integrand (2.181): 
 
**2
0
1(1)
2
j
j
n
Dkt t
B
0
?
= ++? =. (2.185) 
The solution where this singularity occurs is 
 
2
*
0
11
1
22
n
t
Bk
?
??
=? ? + +
??
??
. (2.186) 
The wave angle of the singularity is therefore 
 
2
*1
0
11
tan 1
22
n
Bk
?
?
?
? ?
??
? ?
=? ? + +
??
? ?
??
? ?
? ?
. (2.187) 
By inspection, when j = 1 in the denominator the sign of the wave angle must be positive 
and when j = 2 the sign must be negative to make the total denominator zero. 
The location of the singularity has been found, now the singularity must be dealt with.  
To remove the singularity from the integration, the integrands are modified.  In general, 
 
()
22
**
**
11
*
2
**
1
ln
NN
NN
dd
DD DD
??
?? ??
????
????
? ?
?
? ???
==
==
??
??
?
??=? +
?
??
??
??
??
?
. (2.188) 
The second term in the integrand effectively cancels out the singularity of the first term.  
To not add any net change to the equation, the integral of the second term is added back 
to the total equation.  This technique is used to remove the singularity from each of the 
integrals in both the original (wave angle) and transformed variable.  To apply this 
technique the derivative of the denominator must be computed.  In the original variable 
the derivative is 
 57
 
 
( )
23
0
2cos secDk
?
? ?=? ; (2.189) 
in the transformed variable the derivative is 
 
( )
2
0
2
12
1
kt
D
t
?
+
=
+
. (2.190) 
The modified integrands with the singularities removed are shown below. 
 
-pi/2 -pi/4 0 pi/4 p i/2
-10
-5
0
5
I
?
?? ??
0
? 2?
 
-10 -5 0 5 10
-4
-3
-2
-1
0
1
II
t
 
-pi/2 -pi/4 0 pi/4 p i/2
-2
0
2
4
6
III
?
?? 4??
0  
? 2?
Fig. 11 Sinusoidal Pressure Final Integrands 
k
0
 = 3.08, x = 0.25, y = 0.14, m = 1, n = 1, a = 0.1, B = 0.5 
The integrands still contain discontinuities, but they are now either due to different 
integration ranges, i.e. I and III contain two integrals one for negative wave angles and 
the other for positive wave angles, or to the sign function contained in the integrands.  
 58
 
The first discontinuity type is not an issue because over each integration range the 
function is continuous.  The sign discontinuity could be removed by splitting the 
integration into two separate integrals at this point.  This introduces several coding issues 
though; locating the discontinuity and determining which segment the singularity is in to 
name a couple.  Because of these issues the sign discontinuity is resolved by simply using 
many integration points.  While this is not the most elegant method, it does not introduce 
significant integration errors. 
2.5.2. Constant Pressure Method 
The constant pressure method has two terms to evaluate, equations (2.163) and 
(2.164); they are 
 
()
( ) ( ) ( )
()
()
2
2
2
2
2
glog2HHsin
1
,
2sincos
H
sin
,
sin cos
L
F
TT TxT
x yd
x
T
xy d
?
?
?
?
?
? ?
???
??
???
?
?
+? ???
??
=?
=
?
?
 (2.191) 
with 
 
2
00
sec sec sinTkx ky? ?=+ ?. (2.192) 
The local field integral is treated specially.  As Scullen and Tuck show, the local field 
contribution to the surface height is mainly directly under the hull.  It appears as a 
discontinuity in the far field contribution that is constant in the longitudinal direction.  It 
also contains the even auxiliary function (2.91) which must be computed either from an 
interpolation or a series expansion.  Therefore, instead of taking the computational time 
and effort, the local field contribution is approximated as the value of the far field at the 
leading edge of the wetted area.  The total wave height at this location is effectively zero.  
 59
 
Therefore, the local field must cancel the far field out.  The local field is assumed 
constant in the longitudinal direction for the length of the wetted area. 
All of the actual wave height in the constant pressure method comes from the far field 
contribution.  This integral must still be computed.  The far field function is much 
simpler than the previous integrals.  It does not have any complex functions in the 
integrand and only contains a single integral to evaluate. It does have the same issues as 
the sinusoidal pressure integrals.  The integrand oscillates as the wave angle approaches 
the integration limits and it also contains a singularity.  These problems are solved using 
the same methods developed for the sinusoidal pressure method. 
 
-pi/2 -p i/4 0 pi/4 p i/2
-20
0
20
40
?
F
?
?? 4??
0
? 2?
 
Fig. 12 Constant Pressure Original Far Field Integrand 
k
0
 = 3.08, x = 0.25, y = 0.14 
The integrand oscillation is countered by a variable transformation (2.181).  The far 
field integral then becomes 
 ()
( )H sin
,
F
x
T
x y
t
?
?
?
??
=
?
d (2.193) 
with 
 60
 
 
(
2
0
1Tk txyt=++). (2.194) 
This integrand is shown below. 
-10 -5 0 5 10
-10
-5
0
5
10
?
F
t
 
Fig. 13 Constant Pressure Far Field Integrand with Slow Oscillations 
k
0
 = 3.08, x = 0.25, y = 0.14 
This integrand still has a singularity at t = 0.  The singularity is removed using the 
method from the sinusoidal pressure patches (2.188).  Using this method the far field 
integral becomes 
 ()
( ) ( )
0
Hsinsin
,
F
x Tkx
x y
t
?
?
?
??
?
=
?
dt. (2.195) 
The second term in the integrand is odd so its integral is zero and nothing needs to be 
added to the integral to cancel it out.  This integrand is shown below. 
 61
 
-100 -50 0 50 100
-1
-0.5
0
0.5
1
?
F
t
 
Fig. 14 Constant Pressure Far Field Final Integrand 
k
0
 = 3.08, x = 0.25, y = 0.14 
The singularity has been removed; due to the scale of the integration limits the oscillation 
is more pronounced.  Many integration points must be used, but this function can be 
accurately numerically integrated. 
 62
 
3. METHOD VALIDATION
To validate the methods presented in the previous section several tests are performed.  
These tests compare the sinusoidal and constant pressure methods derived above and 
where possible include or reference outside results.  
3.1. SINUSOIDAL PRESSURE BASIS FUNCTION 
The first test is taken from Cheng and Wellicome?s paper presenting the sinusoidal 
pressure method.  The test utilizes the simplest pressure function available to this method, 
a single sine wave with unit amplitude, 
 
1
sin
2
P? y
? ?
?
=+
?
?
?? ?
??
? ?
. (3.1) 
The beam and length of this ?hull? are both one.  The pressure is constant in the 
longitudinal direction, so only one pressure strip is used, as shown in figure 1. 
 63
 
0 0.2 0.4 0.6 0.8 1
-0.5
0
0.5
x
y
0.2
0.4
0.6
0.8
Fig. 15 Sinusoidal Pressure Basis Function Test Pressure 
The fundamental wave number, representing craft speed, is 3.0779.  The density of 
the water and acceleration due to gravity are both set to one.  This test is shown in figure 
4 of Cheng and Wellicome?s paper.  Both the sinusoidal and constant pressure methods 
are implemented to attempt to reproduce this figure. 
-2 0 2 4 6 8 10
-5
0
5
x
y
Constant Pressure M ethod
Sinusoidal Pressure M ethod
-1
-0.5
0
0.5
1
 
Fig. 16 Sinusoidal Pressure Basis Function Test Wake 
The gross aspects of the two methods match each other quite well and also match the 
results from Cheng and Wellicome.  The results are very similar in the ?wake? region of 
the surface.  The peaks and valleys of the oscillations occur in the same places and are 
approximately the same amplitudes. On the other hand, there are significant differences 
 64
 
in the two methods.  It is expected that the sinusoidal pressure method would perform 
better in this test because the input pressure is exactly one of its basis functions.  
However, that method has significantly more noise both ahead of the applied pressure 
and between the pressure and the limits of the wake.  Ahead of the craft the surface 
height should obviously be zero because the craft?s speed carries the disturbance 
downstream.  Similarly, the double rows of spikes spreading out at nearly 45 degrees 
behind the pressure disturbance occur ahead of the predicted wake and are noise created 
by this method.  This noise does not create a problem using this method though because it 
is outside of the area of interest and does not corrupt the key results.  The constant 
pressure method has no problem with noise in its results. 
A much larger problem exists with the sinusoidal method directly under the applied 
pressure, where the ?hull? is. The sinusoidal pressure method predicts that the water 
immediately rises as a pressure is applied to the surface as seen in Fig. 17. 
0 0.2 0.4 0.6 0.8 1
-0.5
0
0.5
x
y
Constant Pressure M ethod
Sinusoidal Pressure M ethod
-1
-0.5
0
0.5
1
 
Fig. 17 Sinusoidal Pressure Basis Function Test Hull 
This rise is quite dramatic when viewed in the total wake surface, Fig. 16.  This rise 
cannot happen physically.  Approximately half way along the length of the pressure the 
water surface drops to match the surface predicted by the constant pressure method.  This 
 65
 
 66
result is not predicted in the Cheng and Wellicome paper and must be a numerical 
anomaly in the calculations, but no reason has been found for it.  This result is much 
more significant than the noise of the sinusoidal method.  The goal of this research is to 
match a hull shape to a predicted wake.  If the hull the sinusoidal method predicts is 
incorrect in this simple example, then the method is not a valid model for predicting more 
complex hull shapes. 
Another point that should be mentioned is the time to compute the results using each 
of these methods.  The sinusoidal pressure method takes orders of magnitude longer to 
complete the calculations than the constant pressure method.  This timing makes sense 
because the sinusoidal pressure method computes many more integrals than the constant 
pressure method.  Additionally, the integrals in the sinusoidal method often require an 
additional integration at each point in the integrand.  Even though this interior integral is 
accomplished with an interpolation, it still slows the integration down. 
Overall, both methods predict very similar wake shapes.  The sinusoidal pressure 
method has more numerical integration noise, but the noise is in parts of the surface that 
do not interfere with the hull and wake surfaces.  The sinusoidal pressure method also 
takes significantly longer to compute.  The major downfall of the sinusoidal pressure 
method, though, is the rise in the wave height near the forward section of the hull.  This 
rise does not match any physical phenomenon (such as a bow wave) in either size or 
shape; additionally it does not match the results in Cheng and Wellicome.  The rise 
makes this method unreliable for predicting a hull shape given a pressure definition. 
 
3.2. SIMPLIFIED REALISTIC HULL PRESSURE 
Scullen and Tuck also present a test case for a pressure distribution.  This test case 
represents a simplified function for the pressure a planing hull creates.  The pressure is 
given by 
 
()
2
2
141P
x
??
=??
??
??
y. (3.2) 
This ?hull? has a length of two and a beam of one. 
0 0.5 1 1.5 2
-0.5
0
0.5
x
y
1
2
3
4
 
Fig. 18 Simplified Realistic Hull Test Pressure 
For this test the fundamental wave number is one.  As with the previous test 
gravitational acceleration and the density of the water are both set to one. 
 67
 
-2 0 2 4 6 8 10
-5
0
5
x
y
Constant Pressure M ethod
Sinusoidal Pressure M ethod
-10
-5
0
5
 
Fig. 19 Simplified Realistic Hull Test Wake 
Once again, the major aspects of the results from both the constant and sinusoidal 
pressure methods match.  They both show a series of peaks and valleys in the 
approximate shape of an actual boat wake.  The area inside the major wake has smaller 
disturbances.  These results are very promising for the goal of using these methods to 
predict an actual wake.  The sinusoidal method has several of the same issues that 
plagued the sinusoidal pressure basis function test.  The noise ahead of the boat is not as 
apparent, but the double row of spikes behind the applied pressure is present as before, 
Fig. 16, and in this test intersects the actual wake making the issue more problematic.  
The rise in the surface under the applied pressure is also still present.  This rise matches 
the characteristics of the previous test, a rise in the front half of the pressure becoming a 
depression further along the pressure. 
 68
 
0 0.5 1 1.5 2
-0.5
0
0.5
x
y
Constant Pressure M ethod
Sinusoidal Pressure M ethod
-10
-5
0
5
 
Fig. 20 Simplified Realistic Hull Test Hull 
The area directly under the applied pressure using the constant pressure method 
matches the results obtained by Scullen and Tuck, figure 9.  As with the sinusoidal 
pressure basis function test, the constant pressure method completed the calculations 
many orders of magnitude faster than the sinusoidal pressure method. 
The results of this test are essentially the same as the previous test.  The constant 
pressure method performs quite well and the sinusoidal pressure method has several 
problems that make it an unreliable candidate to predict a hull and wake shape.  An 
additional promising result is the wake behind this pressure is beginning to resemble a 
boat?s wake. 
3.3. SINUSOIDAL PRESSURE CENTERLINES 
The previous tests explored the general shape of the wakes generated using the two 
pressure methods.  Cheng and Wellicome also present centerlines of the surfaces 
produced by the sinusoidal pressure basis function test run at different beam to length 
ratios.  The pressure for these tests is the same as in the sinusoidal pressure basis function 
test (3.1) except the beam is varied in each test, i.e.  
 69
 
 
1
sin
2
y
P
B
?
? ?
?
=+
?
?
?? ?
??
? ?
(3.3) 
where B is the beam.  The length and speed in this test are also the same as the sinusoidal 
pressure basis function test.  Cheng and Wellicome present the results of these tests in 
figure 3.  Their results are also presented in the Fig. 21.  The most obvious feature of 
each of these plots is the pressure rise in the sinusoidal pressure method at the leading 
edge of the pressure.  In each test, the surface this method predicts rises dramatically 
before dropping to a more reasonable value halfway along the pressure, x = -0.5.  Other 
details are also different in the three methods.  The first test, B = 0.4, is inconclusive.  
Neither method matches Cheng and Wellicome?s results.  The basic form of the three 
results is similar; a depression under the ?hull? followed by a rise and second depression 
aft of the boat.  The details of the three are quite different, however.  The second test,  
B = 1, provides more insight.  Again, the basic forms of the results are similar.  The 
amplitudes, though, of both the sinusoidal and constant pressure methods are not as high 
as Cheng and Wellicome?s results predict; they are approximately equal to each other, 
however.  This test corresponds to the sinusoidal pressure basis function test presented 
above.  In the sinusoidal pressure basis function test, the amplitude of the waves is 
around one.  This is also true in Cheng and Wellicome?s results for this test.  Their 
centerline, on the other hand, has an amplitude of over two.  This implies the amplitude 
of their centerline tests may not be accurate.  The other aspects of this test match Cheng 
and Wellicome?s results well.  The location of the second peak does not exactly match 
the peak of Cheng and Wellicome, but it is close in each method.  The last test, B = 10, 
has similar results.  The Cheng and Wellicome amplitude is higher, but the peaks line up 
 70
 
better than the second test.  Once again, the sinusoidal and constant pressure methods 
results match quite well. 
From Cheng and Wellicome
Sinusoidal Pressure
Constant Pressure
 
-3 -2 -1 0
-1
0
1
2
x
z
 
B = 0.4 
-3 -2 -1 0
-2
-1
0
1
2
3
4
x
z
B = 1 
-3 -2 -1 0
-2
0
2
4
6
x
z
B = 10 
Fig. 21 Sinusoidal Pressure Basis Function Centerlines 
Overall, the results from the methods tested do not match Cheng and Wellicome?s 
results exactly.  It is expected that the sinusoidal pressure method would.  Some 
difference is anticipated and acceptable in the constant pressure method.  In each test, the 
 71
 
 72
amplitudes do not match those predicted by Cheng and Wellicome.  The sinusoidal and 
constant pressure method amplitudes do match each other reasonably well, however.  The 
problem may be in the amplitude predicted by Cheng and Wellicome.  Each method 
predicts approximately the same frequency for the waves; wave peaks are close enough 
to be acceptable.  The surface rise in the sinusoidal pressure method, as in the previous 
tests, is a fatal problem.  The constant pressure method is a valid method to predict hull 
and wake shapes given a pressure distribution. 
 
 73
4. METHOD ISSUES
In addition to the test case inaccuracies discussed in the previous chapter, there are 
several other issues which may complicate using each of the methods to compute a boat 
wake.  Each of these issues deals with the different pressures the method is based around 
and involves symmetry and oscillation frequencies. 
4.1. CONSTANT PRESSURE METHOD SYMMETRY 
The first issue involves the fact that boats are generally symmetric.  The symmetry 
assumes that the port and starboard sides of the hull are identical and also that the center 
of gravity is directly over the centerline of the hull.  These two assumptions lead to a 
symmetric pressure distribution.  Because the pressure is symmetric it is safe to only 
analyze one side of the wake response and assume the other side is identical.  The 
constant pressure method utilizes constant pressure elements as the basis of the applied 
pressure.  Multiple pressure basis functions (at different locations) are added together to 
produce the total pressure.  A problem arises when only computing the response of half 
of the pressure patches, however.  The response of each pressure patch is symmetric 
about the center of the pressure patch, but the pressure patch is offset from the centerline 
of the hull making the pressure (and the response) non-symmetric about the centerline of 
the hull (Fig. 22).  Because this response is not symmetric it is no longer accurate to only 
examine one side of the wake. 
 
0 2 4 6 8 10
-6
-4
-2
0
2
4
6
x
y
 
Fig. 22 Single Constant Pressure Patch Wake 
One solution to this problem is to not take advantage of the symmetry and compute 
responses from pressure patches over the entire hull.  This method must be employed if 
the hull is not symmetric for any reason.  On the other hand, if the hull is symmetric there 
are corresponding pressure patches on either side of the centerline.  These two patches 
can be analyzed to produce a combined response. 
0 2 4 6 8 10
-6
-4
-2
0
2
4
6
x
y
 
Fig. 23 Corresponding Constant Pressure Patch Wakes 
This response is now symmetric; however, the advantage in time saving from only 
computing responses for one half of the pressure patches has been lost.  The response 
needs to be calculated for only one half of the wake, but for each of the pressure patches. 
 74
 
Another solution is to only use the pressure patch on one side, and reflect the wake 
created across the centerline back into the side being calculated.  This reflection actually 
comes from the corresponding symmetric pressure patch.  The response from the actual 
pressure patch that is lost across the symmetry line is identical to the response from the 
corresponding symmetric pressure patch.  This modified height basis function is now 
symmetric about the centerline and can be safely used while still only using the pressure 
patches on one side of the hull and computing the wake over the same half of the 
centerline. 
0 2 4 6 8 10
0
1
2
3
4
5
6
x
y
 
Fig. 24 Total Constant Pressure Patch Basis Function 
4.2. SINUSOIDAL PRESSURE METHOD SYMMETRY 
A similar symmetry condition exists for the sinusoidal pressure method; the 
symmetry is easier in this case though.  The pressure basis functions in this method are 
defined as existing over the entire width of the hull.  This means they can be inherently 
symmetric without reflecting the response of another pressure patch.  From (2.25) the 
pressure is defined as a series of sine waves with increasing frequency.  Essentially the 
pressure is approximated with a simplified Fourier series.  This pressure is symmetric if 
 75
 
only odd terms are used for n.  To verify this symmetry condition a sinusoidal pressure 
patch is simulated without the symmetry assumption.  As can be seen in Fig. 25, the 
resulting surface disturbance is indeed symmetric.  It is valid to only compute half of the 
wake. 
x
y
0 2 4 6 8 10
-5
0
5
 
Fig. 25 Sinusoidal Pressure Patch Wakes 
To strictly satisfy the derivation all Fourier terms are used to define the pressure, but 
because of symmetry if the hull (and thus pressure disturbance) is symmetric the 
coefficients corresponding to even n are zero.  Computation time can be saved, however, 
by simply discarding these terms and only computing the response from the odd terms of 
n.  To take this into account the index for the pressure terms is modified. 
 
*
2nn1= ?  (4.1) 
where .  This modified index is always odd.  It is used in place of the old 
index in each of the sinusoidal pressure method basis functions.  Now the pressure basis 
function is guaranteed to be symmetric and the resulting wake will be as well.  This 
means that once again only half the wake has to be simulated.  Also, only half the basis 
*
1, 2, 3,n = L
 76
 
functions need to be determined because each of the asymmetric pressure basis functions 
is ignored. 
4.3. SINUSOIDAL PRESSURE METHOD OSCILLATION 
The fact that the frequency is increasing in the sinusoidal pressure method brings up 
an interesting facet of this method.  To accurately capture the characteristics of a sine 
wave a minimum of two points per period of the sine wave must be known.  This is the 
minimum frequency (Nyquist frequency) at which a signal must be sampled in order to 
reconstruct it using a Fourier series [Franklin].  In practice, it is desirable to sample at 
more points per period than the Nyquist frequency suggests because a Fourier series is 
not generally used to reconstruct the signal; a simpler method (e.g. linear interpolation) is 
used in reconstruction. These simpler methods perform better with more than two 
samples per period. 
The number of points this method is allowed is limited, however.  In order for there to 
be as many pressure coefficients as there are calculated wave heights under the hull the 
number of points from the centerline to the beam must equal the number of pressure 
terms.  The number of samples per period is obviously lowest with the highest frequency.  
The worst case number of samples per period can therefore be calculated by comparing 
the number of points per half-beam to how many periods there are in a half-beam at the 
highest frequency.  This ratio is 
 
worst
case
#
21
samples N
period N
=
4
?
 (4.2) 
 77
where N is the number of pressure terms.  In the limit as more pressure terms are used, 
the number of samples per period approaches the Nyquist limit. 
 
0 5 10 15 20
2
2.5
3
3.5
4
Number of Pressure Terms
M
i
ni
m
u
m
 N
u
m
b
e
r
 of
 S
a
m
p
l
e
s
 
/
 
P
e
r
i
od
 
Fig. 26 Loss of Data as More Fourier Terms are Used 
It does not take many pressure terms before there are essentially two samples per period 
in the highest frequency pressure basis function.  This means the higher frequencies are 
in danger of loosing data due to not sampling often enough.  One method to examine the 
impact of this problem is to oversample the system and compare the results to the results 
produced by the actual sampling. 
The first function to study is the pressure basis function.  The lowest and highest 
frequency pressures are shown along with where they are actually sampled and a linear 
interpolation (Fig. 27).  The maximum number of pressure terms is 10. 
 78
 
0 0.075 0.175 0.275 0.375 0.475
-1
-0.5
0
0.5
1
y
P
*
 
n = 1 - oversampled
n = 1 - actual
n = 10 - oversampled
n = 10 - actual
 
Fig. 27 Oversampled Pressure Basis Function 
It is obvious in this plot that the sampling captures the character of the low frequency 
pressure, but does not capture the high frequency oscillations.  The amplitude of the high 
frequency oscillations remains constant, but the amplitude of the sampled signal seems to 
increase moving away from the centerline. 
It seems likely that similar behavior will be observed in the wake basis function that 
results from this pressure.  The model does not guarantee that the oscillations carry over 
to the wake, but physically it makes sense.  The plot only shows one slice of the wake, 
but the behavior is similar throughout the entire wake (Fig. 28).  Once again the low 
frequency component is captured well.  A blip at y = 0.175 is not captured, but it is most 
likely noise and should be ignored.  As with the pressure basis function the high 
frequency component, on the other hand, is not captured.  The amplitude actually 
decreases slightly, but the sampled function has a peak amplitude in the middle of the 
hull then it decreases again. 
 79
 
0 0.075 0.175 0.275 0.375 0.475
-0.5
0
0.5
1
y
z
 
n = 1 - oversampled
n = 1 - actual
n = 10 - oversampled
n = 10 - actual
 
Fig. 28 Oversampled Wake Basis Function 
m = 1, x = -1.1 
The data loss can also be examined by looking at the characteristics of the total wake 
basis function compared to an oversampled basis function (Fig. 29).  The oversampled 
function shows the oscillations are reasonably consistent in both directions; the peaks and 
valleys create a regular grid.  The sampled basis function, however, seems to have a 
series of nearly plane waves with peaks at a slight angle from the centerline.  The two 
behaviors are quite different.  It is also important to note that the data loss issue discussed 
above is in the transverse direction.  A similar behavior is observed in the longitudinal 
direction. 
 80
 
Actual
Oversampled
 
Fig. 29 Oversampled Wake Basis Function Surface 
m = 1, n = 10 
This problem becomes particularly pressing when the hull shape is known and the 
pressure coefficients are unknown.  Because of the data loss the higher frequency basis 
functions become closer to linearly dependant and the matrix inversion harder to 
accurately compute.  In practice, a high frequency term in a forward pressure strip is 
erroneously given a large amplitude.  This creates large high frequency oscillations 
further aft along the hull which are fixed by assigning an even larger amplitude to the 
high frequency terms in these pressure strips.  The inversion results in an estimated hull 
that when oversampled is nothing but an exponentially growing high frequency 
oscillation.  The wake this hull creates is obviously not accurate. 
In conclusion, it is possible to take advantage of the symmetry of boat hulls to reduce 
the number of calculations needed.  With the constant pressure method, this requires 
slightly modifying the basis function.  For the sinusoidal pressure method all that is 
needed is to discard the pressure terms that are asymmetric.  These procedures effectively 
reduce the computation time for each method.  The fact that the sinusoidal pressure 
method is based on a Fourier series also raises issues related to the number of points used 
 81
 
to capture the system behavior.  In general, these issues along with all the inadequacies 
discussed in chapter 3 make the sinusoidal pressure method unacceptable for computing 
the shape of a boat wake.  Throughout the remainder of the work the constant pressure 
method is used exclusively. 
4.4. ISSUES WITH INVERSION  
As was mentioned in chapter 2, the wave height over a grid of points is determined by 
summing a series of basis functions multiplied by pressure coefficients at each point 
(2.170).  In other words, the wave heights are a linear combination of basis functions: 
 g??= AE . (4.3) 
The ideal method to determine a boat?s wake is to define the surface of the hull and 
compute the basis functions over the wetted area.  The number of points defining the hull 
should be the same as the number of unknown pressure coefficients (
hull
E  is square).  
The pressure coefficients are then determined using 
 
()
1
hull hull
Ag??
?
= E . (4.4) 
A new set of basis functions is then computed over the wake area and the pressure 
coefficients determined with the hull are used to solve for the height of the wake: 
 .
wake
wake
A
g
?
?
=
E
 (4.5) 
The problem with this method is that it involves inverting the basis function matrix.  
Because of the uncertainties that go into creating this matrix the inversion is not a stable 
numerical operation. 
 82
 
While the basis function matrix is nonsingular (theoretically invertible) it is very 
poorly conditioned and thus sensitive to noise.  The condition number of a matrix is the 
ratio of the largest and smallest singular values of the matrix.  For the ideal case (an 
orthogonal matrix) the condition number is one.  As the condition number grows the 
matrix inversion becomes less stable and the accuracy of the inverted matrix decreases.  
Chapter 3 presents two cases to validate the solution methods, a sinusoidal pressure and a 
simplified hull pressure.  In these cases the condition numbers of the hull basis functions 
are 7.5?10
9
 and 1.2?10
11
 respectively.  In these tests the pressure (as opposed to the hull 
shape) was defined; however it would not be safe to invert the hull basis function 
matrices. 
To demonstrate the instability the basis functions from the simplified hull pressure 
will be used with two hull shapes and the resulting pressures compared.  The first hull 
shape is the one computed using constant pressure patches (Fig. 30). 
 
Fig. 30 Original Test Surface 
When the basis function matrix is inverted the pressure (Fig. 31) is identical (to machine 
precision) to the actual pressure used to generate the surface (Fig. 18). 
 83
 
 
Fig. 31 Pressure Resulting From Original Test Surface 
The surface is then rotated by 0.00001? about the y ? axis (adjusting the trim by a 
miniscule amount).  This small change in the hull should produce virtually no change in 
pressure; however, the pressure solved for (Fig. 32) is unrealistic.  The pressure rise at the 
front of the surface displacement is still visible.  However, further aft the pressure 
oscillates uncontrollably.  As the surface is rotated more (or other perturbations are 
applied) the amplitude of the oscillation becomes much larger. 
 84
 
 
Fig. 32 Pressure Resulting From Rotated Test Surface 
To better understand the reason for these oscillations the basis function matrix must 
be examined.  The hull shape equation (4.3) can be written in block form as a series of 
smaller matrices due to its formation from a multidimensional tensor equation, i.e. 
 
1,1 1 ,1,1,1 1 ,2,1,1 1 , ,1,1 1 ,1
2,1 1 ,1,2,1 1 ,2,2,1 1 , ,2,1 1 ,2
,1 1,1,,1 1,2,,1 1,,,1 1,
hull
JNJNJ NMJ
JNJNJ NMJN
IJ NIJ NIJ NMIJ NM
gA
EE E A
g
EE E A
??
?
?
?
?
????? ???
????? ???
????? ???
=
??? ??
??? ??
??? ??
=
??? ??
??? ??
??? ??
E
L
L
MMMO
L
?
?
N
?
?
?
?
M
 (4.6) 
where 
,ij
? , , and  are as in (2.170)and ?? ? represents the whole range of 
values (i.e. a vector/matrix).  The basis matrix can be simplified by noting that the 
contribution of a pressure patch only affects the flow aft of the patch: 
,,,nmi j
E
,nm
A
 
,,,
0 ; 1
nmi j
Eim= ??. (4.7) 
 85
 
This leads to: 
 86
?
?
?
?
?
?
 . (4.8) 
1,1,1
1,1 1,1
1 ,1,2,1 1 ,2,2,1
2,1 1,2
,1 1,
1 ,1, ,1 1 ,2, ,1 1 , , ,1
00
00
0
NJ
J N
NJ N J
J N
IJ NM
NIJ NIJ NMIJ
E
A
EE
A
g
A
EE E
?
?
?
?
??
? ?
?? ??
? ?
? ?
?? ?? ? ?
??
?? ?
??
?? ?
?? ?
=
?? ?
?? ?
?? ?
L
L
MMOM
L
The first row of hull heights (
1,1 J
?
?
) can be determined independent of any pressures 
except for the first row ( ).  Conversely, the first row pressure vector can be found 
by inverting the first row basis function matrix ( ) to match the first row hull 
height vector.  This pressure creates a disturbance down stream.  The second row of 
pressures must cancel this disturbance as well as match the second row height to the hull.  
Due to noise in the basis function calculations, the down stream height disturbances 
contain small oscillations.  To cancel the oscillations, the pressure in the next inversion 
contains the same oscillation with a slightly larger amplitude.  As the solution propagates 
aft the oscillation in the pressure continues to grow causing the instability. 
1,N
A
? 1
J1,1,1N
E
??
 
0.1 0.2 0.3 0.4
-15
-10
-5
0
5
10
y
P
x = 1.5
x = 1.7
x = 1.9
 
Fig. 33 Increasing Pressure Oscillation 
This instability and sensitivity to small hull shape changes require a change in 
solution procedure.  Instead of defining the hull, determining the pressure that produces 
that hull, and finally determining the wake the new procedure is to define a pressure 
shape with unknown parameters; vary those parameters over a reasonable range; and then 
determine both the hull and wake shape the pressure produces.  One potential problem 
with this method is that the hull shapes produced may not be realistic.  The hull shape 
needs to be monitored to guarantee that it is practical.  This may also be a benefit, 
however; since the hull is not being varied directly, some shapes may be discovered that 
are possible to manufacture but would not otherwise be investigated. 
 87
 
 88
5. HULL OPTIMIZATION
The goal of this research is to study methodology for designing hulls for different 
types of professional and recreational tow boats, specifically for towing a skier or a 
wakeboarder.  Algorithms for predicting the wake (and hull shape) given a pressure 
disturbance have been developed.  The next step is to describe what is ?good? for a wake 
and what parameters are defined in each event.  The wake shape equations depend on the 
speed the boat is traveling; the speed is in turn dependent on what and who is being 
towed.  Additionally, it does not make sense to examine the shape of the entire wake; the 
rider will only be in a portion of the wake defined by the tow rope length which depends 
on the event and the rider.  Different events require different parameters and different 
riders have different preferences. 
5.1. EVENT DEFINITION 
Skiing has existed for the longest time and its events are the most formalized.  In 
particular two governing bodies oversee most of the professional skiing tournaments, the 
International Water Ski Federation (IWSF) and the American Water Ski Association 
(AWSA).  These two bodies have very similar rules concerning boat speed and rope 
length for events.  Two events will be considered for skiing: jumping and slaloming. 
 
Jumping is defined as the skier using a ramp to jump and get the maximum distance 
before landing.  The skier can choose a speed between 45 and 57 kph (28 and 35 mph) 
and the tow rope length is 23 m (75 ft) [AWSA]. 
Fig. 34 Ski Jumping Example and Course Layout 
Slaloming is defined as the skier going around a series of buoys to either side of the 
boat?s path.  The speed is set at 58 kph (36 mph); the rope length varies between 11.25 
and 18.25 m (37 and 60 ft) [AWSA].  The rope gets shorter with each run making the 
event more difficult. 
 89
 
Fig. 35 Slaloming Example and Course Layout 
The conditions for recreational (amateur) waterskiing vary much more based on the 
skier?s preference.  Typically the speed is less than for professionals and the tow rope 
length is towards the longer end of the professional range.  One factor ties each of these 
activities together: the activity is not based on the wake.  The boat must be towing the 
skier, but fewer disturbances from the wake are better. 
Wakeboarding is a newer sport; two bodies also govern its professional events, the 
World Wakeboard Association (WWA) and the World Wakeboard Council (WWC).  
Tournaments sponsored by these two bodies have different events that mainly consist of 
various freeform jumps, flips, etc.  Both agree, however, that boat speed and rope length 
are at the discretion of the rider [WWC]; in general the riders prefer boat speeds much 
slower than skiing, 12 to 20 mph (19 to 32 kph), and tow rope length in the middle of the 
slalom lengths [Favret, Solomon].  This is also true for recreational wakeboarding.  One 
factor that is consistent for most wakeboarding activities is that the wake itself provides 
the launching platform for many of the tricks.  Therefore, the wake should be larger and 
provide a ramp for the wakeboard. 
 90
 
Fig. 36 Wakeboarding Examples 
In summary, besides the actual hull shape there are two parameters that are varied for 
the different activities: the boat speed and the tow rope length, or distance behind the boat 
the wake will be examined.  Two basic types of wake are also desired depending on the 
activity.  For skiing a flat wake as small as possible is desired.  For wakeboarding a larger 
wake with ramp characteristics is ideal. 
To characterize these wake shapes two parameters will be examined: the maximum 
height of the wake and the slope of the main disturbance.  These parameters are 
calculated along the arc the skier makes behind the boat and are defined as 
 
() ()
1
tan .
max min
max min
max min
height z z
zz
slope
Rz Rz??
?
= ?
?? ?
=
? ?
?
? ?
 (5.1) 
 91
 
z 
?
z
min
z
max
( )
z
max
?
( )
z
min
?
R
?
y 
x
Fig. 37 Wake Parameters 
For each speed and tow length under consideration the pressure distribution that creates 
the hull will be varied to attempt to either minimize or maximize (depending on the 
activity) these parameters.  The hull shape that created the wake will then be examined 
for reasonability and shape. 
5.2. TYPICAL HULL SHAPE 
Because the hull shape is being determined from the surface deformation made by the 
pressure instead of being defined directly, the shape must be examined to guarantee that 
it is reasonable.  The gross shape of most planing hulls has not changed drastically in 
 92
 
many years, although minor improvements have been made.  The shape is characterized 
by a ?V?-bottom that is sharper near the bow with decreasing deadrise angle towards the 
stern.  This basic shape is shown below [Comstock]. 
 
L
C
L
B
0 
? 
1? 
1 
2 
3 
4 5 6 
7 
8 
9 
10
Fig. 38 Basic Planing Hull Shape 
Two critical parameters characterize this shape.  The chine is the sharp break in the 
transverse slope of the hull (the edge of the ?V?-bottom).  The slope of the ?V? is known 
as the deadrise; as was previously mentioned it varies along the length of the hull. 
A third parameter needed to characterize the hull?s shape in the water is the trim 
angle.  This angle is how parallel the keel is to the free surface.  This is not a parameter of 
the hull itself; the trim angle is maintained to balance the pitch moments acting on the 
hull. 
Modern planing hulls are more complicated than this example.  For instance most 
have strakes, longitudinal ridges running along the bottom of the hull; however, the 
chine, deadrise, and trim angle characterize the overall free surface deformation created 
by the hull well enough for the purposes of this research. 
 93
 
 94
5.3. TEST CASES 
As mentioned in section 5.1, the conditions that represent each event include speed 
and tow rope length.  Six cases are studied to find optimal hull shapes.  To represent ski 
jumping the rope length is defined as 23 m and the speed can vary between 45 and 57 
kph; these limits are tested.  In slaloming the speed is set at 58 kph and the rope length 
can vary between 11.25 and 18.25 m; 11.25 and 14.25 m are evaluated.  Wakeboarding is 
not regulated but two typical cases, 14.75 m rope length at 19 kph and 32 kph, are tested.  
These parameters are summarized in Table 1. 
Table 1 Test Parameters 
 Speed (kph) Tow Rope Length (m) 
Jump 1 57 23 
Jump 2 45 23 
Slalom 1 58 11.25 
Slalom 2 58 14.25 
Wakeboard 1 19 14.75 
Wakeboard 2 32 14.75 
 
Three other parameters must be set for each test: the displacement, length, and beam 
of the boat.  When planing the volume of water displaced does not balance the weight of 
the boat due to the lift created by planing.  Instead of measuring displacement the lift 
generated by the pressure is adjusted to match the weight of the boat.  The length and 
beam are more difficult.  As more lift is created the boat rises out of the water.  
Therefore, the wetted area will not necessarily stay the same as parameters change.  In 
 
order to minimize variations and concentrate on how hull shape (as opposed to size) 
affects the wake for this study the length and beam remain constant for each test. 
From section 5.2, planing hulls normally have a chine where the shape drastically 
changes; above the chine the hull is nearly vertical.  When planing the waterline is very 
near the chine (see Fig. 39).  Therefore, the wetted beam is assumed to be 90% of the 
overall beam.  Similarly, approximately two thirds of the length of the hull is in the water 
when planing (see Fig. 39).  For the tests a typical tow boat with parameters in Table 2 is 
used. 
Fig. 39 Examples of Planing Boats 
 
 95
 
Table 2 Typical Tow Boat Parameters 
 Overall Value Wetted Value 
Mass (kg) 1445 1445 
Length (m) 6.45 4.3 
Beam (m) 2.31 2 
 
The pressure distribution is varied for each test.  The pressure distribution used in 
section 3.2 represents a simplified pressure produced by a planing hull.  In particular it 
has characteristics that are present in planing craft such as a singularity at the leading 
edge and zero value at the other edges [Scullen].  These characteristics can be maintained 
and the pressure allowed to vary by modifying the pressure equation to 
 
11
c d
L y
P
x B
??? ?
?? ??
=
??
??? ?????
????
? ???
. (5.2) 
In this equation c effectively controls how long the peak of the pressure is and d controls 
how wide the peak is.  For the skiing tests (jumping and slaloming) the goal is to 
minimize the wake height and slope; for the wakeboarding tests the goal is opposite, to 
maximize the wake height and slope. 
In each test using the conditions listed above (Table 1 and Table 2) the pressure is 
varied according to equation (5.2) and then modified so that lift matches weight.  The 
wake generated by this pressure is computed and examined in an arc defined by the tow 
rope.  The maximum wake height and wake slope along this arc are determined using 
equation (5.1).  These two parameters are then compared over the range of pressure 
parameters so that for each test the slope and height can either be maximized or 
minimized as appropriate. 
 96
 
 97
5.4. JUMP 1 TEST RESULTS 
The first test is the higher speed ski jump test.  The surfaces shown below represent 
the maximum wave height and slope as a function of the two pressure parameters (c and 
d).  The goal of this test is to minimize both the maximum height and the slope.  The 
maximum height is minimized when either c or d is at the minimum value tested; the 
absolute minimum occurs when both c and d are at the minimum tested value.  The 
minimum value of c tested is 0.01.  Values below this do not appreciably modify the 
pressure from this value and so are not tested.  Similarly for d testing any lower does not 
change the results. 
The minimum slope occurs where d is approximately 4 and c is greater than 5.  It 
stays near this minimum along the line c = 3.5.  To compromise between these two 
minimizations values of c = 2.3 and d = 2 are chosen for the optimal wake. 
 
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
0.4
0.45
0.5
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
28
29
30
31
32
 
Fig. 40 Jump 1 Minimization 
5.4.1. Resulting Pressure and Hull Shape 
These pressure parameters result in the pressure shown below (Fig. 41).  This 
pressure produces the wake also shown below.  In general this hull shape looks 
reasonable.  The trim angle is 5.7 deg; at the stern of the boat the deadrise is 6 deg.  
Unlike the typical hull presented in section 5.2 the deadrise decreases toward the front of 
the boat.  This is due to two reasons.  First, the wetted area is defined as a rectangle 
instead of the triangle that a more typical hull would have.  Without knowing the hull 
shape beforehand it is not possible to know the shape of the wetted area triangle so a 
 98
 
 99
rectangle must be assumed.  Second, near the bow the pressure function used (5.2) has 
value for the width of the boat instead of only near the centerline (again a rectangular 
wetted area instead of a triangle).  A different pressure function might produce a more 
typical hull shape.  One interesting feature of the hull shape is the positive height on the 
outside rear corner.  At the transom approximately 0.8 m from the centerline the hull 
begins to rise quickly.  This is characteristic of the chine line.  The chine widens as it 
approaches the transom; this is not typical, but has the same explanation as the atypical 
deadrise.  Also, outside the chine the height becomes positive.  It is common for the 
water to spray up on the sides of the boat (see Fig. 39).  The positive height is physically 
reasonable in this light. 
 
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
2
3
4
5
6
x 10
4
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
-0.4
-0.2
0
0.2
 
Fig. 41 Jump 1 Pressure and Hull Shape 
5.4.2. Resulting Wake 
The wake this pressure produces is shown below in Fig. 42.  The black line represents 
the arc a skier would follow.  The height the skier encounters is also plotted vs. the 
distance along the arc.  The square indicates the highest point on the arc and the circle the 
lowest.  A typical wake behind a boat is reasonably flat directly behind the boat with a 
crest spreading in a ?V? outside this flat area.  Outside the crest is a valley which levels 
off to the undisturbed free surface height.  The shape shown below has several of these 
characteristics.  In particular the size and location of the crest and valley are the 
 100
 
 101
parameters used to characterize the wake.  The area inside the wake crest (inside the ?V? 
behind the boat) is not accurate.  This area has longitudinal oscillations that are not 
present in physical wakes.  These oscillations narrow further behind the boat until the 
surface inside the wake becomes essentially noise (around x = 25 m in this test).  These 
oscillations can also be seen in the centerline validation tests (section 3.3).  However, in 
this area of the wake the extra disturbance from the boat?s propeller affects the fluid flow; 
the disturbance from the propeller is not included in this simulation.  Therefore, the area 
inside the ?V? behind the boat can safely be ignored when analyzing the wake shape. 
Ignoring the area directly behind the boat, the height profile along the skier?s arc 
appears reasonable.  The valley outside the wake is not typically that much more extreme 
than the surface height inside the wake crest, but this is most likely due to a combination 
of the atypical hull and the no propeller assumption.  The height of the wake is 39.8 cm 
and the slope is 28.4 deg. 
 
 
 x  (m)
 y 
 (m
)
5 10 15 20 25
1
2
3
4
-0.4
-0.2
0
0.2
 
1 2 3 4 5
-0.3
-0.2
-0.1
0
0.1
0.2
Distance Along Arc (m)
 z 
 (m
)
 
Fig. 42 Jump 1 Wake 
5.5. JUMP 2 TEST RESULTS 
The next test case is the slower ski jump.  As before, the surfaces to minimize (wake 
height and slope) are shown below.  The height shows a strong trend: as c is decreased 
the height decreases.  The slope is not as clear.  There is a rough minimum along the line 
c = 2.1 and the line d = 2 is also a minimum for c greater than 4.  The slope is also near 
minimum at c = 0.01 and d = 14.6; this is the condition chosen as the optimum 
compromise. 
 102
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
0.56
0.58
0.6
0.62
0.64
0.66
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
25
25.5
26
26.5
27
 
Fig. 43 Jump 2 Minimization 
5.5.1. Resulting Pressure and Hull Shape 
These parameters result in a pressure that is nearly linear while still maintaining the 
required boundary conditions.  The hull shape is also interesting.  The trim is normal at 
5.2 deg, and as before the hull gets wider forward.  The deadrise, however, is not 
apparent.  In fact, the rear outside edge is lower than the centerline.  This shape is known 
as a catamaran and while not typically used in towboats is common in other power boat 
designs. 
 103
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1000
2000
3000
4000
5000
6000
7000
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
-0.4
-0.2
0
0.2
 
Fig. 44 Jump 2 Pressure and Hull Shape 
Although this hull shape is not common for towboats, some manufacturers are 
beginning to produce boats with similar features.  The boat shown below is the 
MasterCraft X-Star.  It is not a true catamaran because the ?V? hull is still present; 
however, outriggers are present similar to a tri-hull.  The X-Star is a wakeboarding tow 
boat; it is not meant to imply that the hull shape presented above is necessarily the best 
ski tow boat.  It does, however, show that similar designs are feasible and being used in 
industry. 
 104
 
 
Fig. 45 MasterCraft X-Star 
5.5.2. Resulting Wake 
As before, the wake looks reasonable except for the area immediately behind the 
boat.  The maximum wake height is 56.2 cm and the slope is 25.9 deg.  In general, this 
wake is not as preferable as the previous test case.  As boat speed increases typically the 
wake becomes smaller and flatter because less of the hull is disturbing the free surface to 
achieve the same lift.  The comparison of the two speeds qualitatively matches the 
physical expectations.  The magnitude of the height for both this and the previous test 
seems high.  More evaluation of this technique is warranted to guarantee its accuracy. 
 105
 
 
 x  (m)
 y 
 (m
)
5 10 15 20 25
1
2
3
4
-0.5
0
0.5
 
1 2 3 4 5
-0.2
-0.1
0
0.1
0.2
Distance Along Arc (m)
 z 
 (m
)
 
Fig. 46 Jump 2 Wake 
5.6. SLALOM 1 TEST RESULTS 
The next two tests are the slalom runs.  The first slalom test is with the shorter rope.  
The trends in both the height and slope are strong.  The height decreases as either c or d 
decrease; the absolute minimum occurs at c = 0.01 and d = 2.  The slope has similar 
trends except the minimum in c occurs at 2; for c < 2 the slope begins to rise again.  The 
gradient in the d ? direction is stronger, however (d has more affect on slope than c).  
Because c = 0.01 and d = 2 minimizes the height and is near the minimum slope this 
point is chosen as the optimal compromise. 
 106
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
0.4
0.45
0.5
0.55
0.6
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
28
29
30
31
32
 
Fig. 47 Slalom 1 Minimization 
5.6.1. Resulting Pressure and Hull Shape 
The pressure this condition produces is fairly linear longitudinally and curved 
transversely.  The hull, however, does not exhibit the catamaran shape of the previous 
linear hull; it is a more traditional ?V? shape.  The trim angle is 4.9 deg and the deadrise 
at the transom is 2.6 deg. 
 107
 
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
2000
4000
6000
8000
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
-0.3
-0.2
-0.1
0
0.1
0.2
 
Fig. 48 Slalom 1 Pressure and Hull Shape 
5.6.2. Resulting Wake 
The wake this pressure produces is slightly different from the previous two test cases.  
The lowest height of the wake occurs inside the wake crest as opposed to the valley 
outside the crest.  In fact, no valley outside the wake crest is present.  This difference is 
due to the fact that the skier is so close to the boat; further from the boat the wake crest 
for this peak dies and the wake crest the previous tests used is formed.  This close to the 
boat the skier is ahead of the formation of the main wake.  The height the skier sees is 
38.4 cm and the slope is 28.2 deg. 
 108
 
 x  (m)
 y 
 (m
)
5 10 15 20 25
1
2
3
4
-0.4
-0.2
0
0.2
 
1 2 3 4 5
-0.4
-0.3
-0.2
-0.1
0
0.1
Distance Along Arc (m)
 z 
 (m
)
 
Fig. 49 Slalom 1 Wake 
5.7. SLALOM 2 TEST RESULTS 
The longer rope slalom test results are interesting because for many pressure 
distributions the wake is not valid; in particular the main wake crest is below the 
undisturbed free surface which is not realistic.  These obviously incorrect wakes are not 
considered in the optimization.  Of the wakes that are reasonable the heights and slopes 
follow the trends seen previously in the minimization.  As c and d decrease the height and 
slope also decrease; d has a bigger effect on slope than c does.  The optimal combination 
for this test is c = 2.3 and d = 17.4. 
 109
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
0.5
0.52
0.54
0.56
0.58
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
39
39.5
40
40.5
 
Fig. 50 Slalom 2 Minimization 
5.7.1. Resulting Pressure and Hull Shape 
These pressure coefficients result in a distribution that is wide, but concentrated at the 
front of the wetted area.  The hull has the catamaran shape seen in a previous test.  The 
trim angle is 3.8 deg; the deadrise angle is meaningless in a catamaran hull. 
 110
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
2
3
4
x 10
4
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
-0.4
-0.2
0
0.2
0.4
 
Fig. 51 Slalom 2 Pressure and Hull Shape 
5.7.2. Resulting Wake 
The longer rope in this test means the skier is back in the main wake so the valley 
outside the wake crest is apparent.  The wake height is 48.5 cm and the slope is 39.2 deg.  
As with the ski jumping tests, the wake heights are excessive compared to physical 
wakes. 
 111
 
 x  (m)
 y 
 (m
)
5 10 15 20 25
1
2
3
4
-0.5
0
0.5
 
1 2 3 4 5
-0.4
-0.3
-0.2
-0.1
0
0.1
Distance Along Arc (m)
 z 
 (m
)
 
Fig. 52 Slalom 2 Wake 
5.8. WAKEBOARD 1 TEST RESULTS 
The last two tests are for wakeboarding.  The main difference between these tests and 
the skiing tests is the lower speeds.  Additionally, instead of minimizing the wake height 
and slope the goal of the tests is to maximize them.  The first wakeboard test is the slower 
of the two.  The height and slope show similar trends as the other tests except c has a 
much larger effect on both the slope and the height than d; in fact unlike before as d 
decreases the height and slope may increase.  The maximum height and slope both occur 
at c = 10 and d = 30. 
 112
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
1.2
1.4
1.6
1.8
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
24
26
28
30
32
 
Fig. 53 Wakeboard 1 Maximization 
5.8.1. Resulting Pressure and Hull Shape 
This pressure is virtually constant over the width of the boat and is concentrated 
exclusively at the leading edge of the wetted area.  The hull shape this produces is 
problematic.  Before the transom of the boat the hull is rising implying the hull is bowl 
shaped, which is not a practical planing hull design.  Considering only the centerline 
forward of the minimum hull height the trim angle is 35.2 degrees, very steep for a 
planing hull.  As boat speed increases the boat transitions from operating in a 
displacement mode (lift comes from displaced water) to a planing mode.  During this 
transition the boat trim is very large.  The hull in this test with these pressure parameters 
 113
 
is going slow enough that it has not made the transition to full planing.  The wake shape 
equations are only valid for full planing.  This is the most likely cause for the unrealistic 
hull shape. 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
2
4
6
8
x 10
4
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
-1.5
-1
-0.5
0
 
Fig. 54 Wakeboard 1 Pressure and Hull Shape 
5.8.2. Resulting Non-Optimal Pressure and Hull Shape 
To find a reasonable hull the parameters which minimize the wake height and slope  
(c = 0.01, d = 2) are examined.  The hull this produces still curves upward at the stern, 
but not much. 
 114
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
2000
4000
6000
8000
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
-1.5
-1
-0.5
0
 
Fig. 55 Wakeboard 1 Non-Optimal Pressure and Hull Shape 
5.8.3. Resulting Non-Optimal Wake 
The wake the non-optimal pressure coefficients produce is obviously not the best 
wake, but it does look reasonable.  The crest ?V? is very wide which is typical of slower 
boats, particularly when not fully transitioned to planing mode.  The lowest point of the 
wake is inside the crest which is not unreasonable.  The wake height is 110.6 cm and the 
slope is 28.5 deg.  As with all the other tests this height seems excessively large. 
 115
 
 x  (m)
 y 
 (m
)
5 10 15 20 25
1
2
3
4
-1.5
-1
-0.5
0
0.5
 
1 2 3 4 5
0
0.2
0.4
0.6
0.8
Distance Along Arc (m)
 z 
 (m
)
 
Fig. 56 Wakeboard 1 Non-Optimal Wake 
5.9. WAKEBOARD 2 TEST RESULTS 
The final test is the faster wakeboard test.  The height and slope profiles have similar 
trends as the slower wakeboard test.  The maximum of each occurs at c = 10 and d = 30. 
 116
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
0.65
0.7
0.75
0.8
0.85
0.9
0.95
 
 c
 d
2 4 6 8 10
5
10
15
20
25
30
31.5
32
32.5
33
33.5
34
 
Fig. 57 Wakeboard 2 Maximization 
5.9.1. Resulting Pressure and Hull Shape 
This pressure is identical to the optimal wake in the previous wakeboard test, but this 
hull is more reasonable.  One interesting feature of the hull is a depression near the 
forward outside corner of the wetted area.  This is similar to a catamaran, but vanishes at 
the stern of the boat.  No tow boats are currently manufactured in this shape, but it is not 
infeasible.  The trim angle is 11 deg and there is no deadrise. 
 117
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
2
4
6
8
x 10
4
 
 x  (m)
 y 
 (m
)
1 2 3 4
0.2
0.4
0.6
0.8
1
-1
-0.5
0
0.5
 
Fig. 58 Wakeboard 2 Resulting Pressure and Hull Shape 
5.9.2. Resulting Wake 
This test has a high enough speed that the wake looks more like the skiing wakes.  
The valley outside the wake crest is the lowest point although the valley inside the crest is 
not much higher.  The wake height is 95.3 cm and the slope is 34.3 deg.  It does not seem 
reasonable for a boat to create a wake nearly one meter high. 
 118
 
 x  (m)
 y 
 (m
)
5 10 15 20 25
1
2
3
4
-1
-0.5
0
0.5
1
 
1 2 3 4 5
-0.6
-0.4
-0.2
0
0.2
0.4
Distance Along Arc (m)
 z 
 (m
)
 
Fig. 59 Wakeboard 2 Wake 
5.10. RESULTS SUMMARY 
In each test the wake the boat produced appears reasonable.  A ?V? shaped crest is 
visible with a valley inside it and outside either a valley or drop to the undisturbed free 
surface.  The wake is wide when the boat is slow and narrows as speed increases 
matching reality.  The area directly behind the boat is not realistic, but this could be due 
to ignoring propeller effects.  The wake shape also shows a strong correlation to the 
pressure distribution, and the hull shapes while not always typical are usually reasonable. 
There is however a problem with the wake height.  In every case tested the height is 
larger than can be expected in a real wake.  The shape of the wetted area also needs to be 
 119
 
 120
examined.  In particular, each hull produced is flat at the leading edge of the wetted area 
instead of showing the characteristic deadrise present in tow boats.  Without the ability to 
directly define the hull shape defining the shape of the wetted area becomes problematic. 
 
 121
6. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH
 
6.1. CONCLUSIONS 
In conclusion, this work attempts to predict the wake shape behind a given boat 
moving at a given speed.  To do this two methods for determining the free surface 
deformation from a pressure distribution are presented.  Both methods derive from the 
same velocity potential function; the difference is the form of the pressure distribution 
and the resulting integrals that must be evaluated numerically.  Each method is compared 
to previously published results to verify its accuracy.  The sinusoidal pressure basis 
function method does not yield results matching the original authors.  Furthermore, this 
method has potential instabilities beyond the numerical inaccuracies; in order to produce 
a square linear system the pressure oscillates too quickly for the solution grid to capture.  
The constant pressure basis function method yields results that closely match the 
published results.  The numeric algorithm for the constant pressure basis function also 
runs much faster than the sinusoidal pressure basis function.  Because of the sinusoidal 
pressure basis function method?s shortfalls, the constant pressure basis function method is 
used to predict the wake shape behind the boat.  Even the constant pressure basis function 
method is not stable enough to invert and determine a pressure distribution from a boat 
hull shape.  Instead a variety of pressures are examined for wake shape and hull shape. 
 
 122
Tests are performed for a range of scenarios encountered in real life.  In particular, ski 
jumping and slaloming, and wakeboarding conditions are simulated.  To find the optimal 
hull shape for each of these activities the wake shapes produced by the various pressure 
distributions are examined for characteristics important to the activities: wake height and 
slope.  The hulls produced by the tested pressures are feasible for the most part.  
However, because the hull is not directly defined some characteristics are not typical.  
Specifically, every hull produced is flat at the leading edge of the wetted area; this is not 
typical of modern towboats.  The flat hull shape is due to the fact that a rectangular 
wetted area must be assumed.  Additionally, several hulls had depressions along the 
outside edge.  This shape is common in power boats, but not in tow boats.  The wakes 
observed in the simulations are also qualitatively reasonable.  A definite ?V? crest is 
visible behind the boat.  This ?V? widens as the boat moves slower matching physical 
intuition.  The wake height and slope also show a strong correlation to pressure 
distribution.  Oscillations appear inside the ?V? that do not represent reality, but these are 
most likely due to ignoring the effects of the propeller.  The major discrepancy between 
the simulated wake and a physical wake is the difference in height.  The simulated wakes 
are much larger than is typical of tow boat wakes.  This discrepancy warrants further 
research. 
One final important note is that the hull optimization performed in this research only 
considers wake parameters.  Boat hulls must be designed with many other considerations 
such as handling, drag (engine size), and ride comfort in mind.  For example, most of the 
hulls studied in this research have nearly flat bottoms.  Flat bottom boats are common for 
small one or two person, low power boats for use in calm waters.  Larger boats (such as 
 
 123
tow boats) never have flat bottoms because of considerations completely outside wake 
shape.  Flat bottom boats are not as maneuverable as ?V? hulls; they slide over the water 
instead of cutting through it.  This sliding behavior also makes their ride in rough water 
very violent; the boat goes up over each wave instead of cutting through them.  Any 
designer who wished to make use of the procedures developed in this research would 
need to combine the wake considerations with other boat performance considerations.  
Until a method to invert the surface deformation procedure (define a hull shape instead of 
a pressure distribution) is developed it will be very difficult to use this method in 
conjunction with other hull considerations. 
6.2. SUGGESTIONS FOR FUTURE RESEARCH 
There are several areas that could be investigated to improve the method developed in 
this research.  These areas generally fall into two categories: numerical algorithms used 
and range of tests to optimize hull shape.  The most complex algorithms in this research 
involve the numerical integration of functions.  These functions have features making 
them difficult to integrate, singularities and rapid oscillations.  Transformations are used 
to remove the singularity and slow the oscillations, but these change the integration range 
to negative and positive infinity introducing inaccuracies when the integration is 
truncated.  Better analytical transformations may be possible to improve the numerical 
integration accuracy.  The particular pressure basis functions used can also improve the 
numerical algorithms.  The sinusoidal pressure basis function method has the problem of 
data loss from not capturing higher frequency oscillations.  The method does satisfy the 
Nyquist criterion for number of points per period necessary to uniquely define a sine 
wave; however, this absolute minimum is not enough to accurately capture the sine wave.  
 
 124
This apparent discrepancy is due to the fact that the Nyquist criterion is for fitting the 
data with a Fourier series instead of the piecewise linear interpolation used in the method 
developed for this research.  If the wake surface were described by a Fourier series 
instead of being determined by inverting the basis function matrix it is possible the higher 
frequency oscillations could be captured thereby stabilizing the sinusoidal pressure basis 
function method.  Another improvement that could be made with the pressure basis 
functions is in the constant pressure basis function method.  This method approximates 
the pressure distribution by piecewise constant rectangular patches.  This creates 
discontinuities in the applied pressure disturbance.  The free surface disturbance may be 
smoother if these pressure discontinuities were not present.  Specifically if the pressure 
distribution could be approximated by either two dimensional piecewise linear or cubic 
spline rectangles a smoother wake may result.  For this method to succeed the velocity 
potential function would need to be derived for this new pressure type.  All of the 
numerical improvements have the same goal in mind: to allow the free surface 
deformation problem to be inverted.  For this research to move from an analysis tool to a 
design tool, a way to define the hull shape, and thereby determine the wake shape, must 
be developed.  In particular this means lowering the condition number of the surface 
basis function matrix.  The condition number may be lowered either by more accurate 
numerical integration or by using better pressure basis functions. 
The second category for recommended future research is in the range of tests to 
optimize the hull.  If a method to directly define the hull shape is developed many more 
tests become available.  One of the more interesting scenarios to investigate is the mixed 
design tow boat, one that is designed to tow both skiers and wakeboarders.  These 
 
 125
activities have different wake requirements, but are also performed at different speeds.  
Could one hull be designed to produce an ideal wakeboard wake at low speeds and an 
ideal ski wake at high speeds?  If the pressure must be defined instead of the hull shape 
then a wider range of pressure distributions should be examined.  Specifically the 
knowledge that a ?V? hull produces a triangular wetted area should be taken into account.  
Regardless of whether the hull shape or pressure distribution is defined the results need to 
be compared to physical experiments.  This work highlighted results that do not seem 
physically reasonable.  These results need to be compared to a real wake. 
 
 126
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