A SECOND-ORDER DIFFRACTION THEORY FOR WAVE RUNUP AND WAVE
FORCES ON A VERTICAL CIRCULAR CYLINDER
By
DAVID LANE KRIEBEL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ................................
LIST OF TABLES ....................................
LIST OF FIGURES ....................................
LIST OF SYMBOLS ...................................
ABSTRACT ........................................
CHAPTERS
1 INTRODUCTION ...................................
1.1 Problem Statement. ...... ..........................
1.2 Problem Formulation ...............................
1.3 First-Order Solution ...............................
1.4 Second-Order Problem Formulation .......................
2 REVIEW OF THE LITERATURE .........................
2.1 Introduction ... ................. .... ..........
2.2 Solutions Prior to 1979 ...............................
2.3 Asymptotic Solutions...............................
2.4 Green's Function Solutions ...........................
2.5 Fourier-Bessel Integral Solutions .......................
2.6 Other Recent Solutions .............................
2.7 Irregularity of Second-Order Solution ................ ....
2.8 Discussion of Literature
3 SOLUTION FOR SECOND-ORDER VELOCITY POTENTIAL
3.1 Introduction .
vii
xiii
xviii
. . . . . .
3.2 General Solution Procedure ........................... 27
3.3 Far-Field Boundary Condition . . . . . ... 30
3.4 Second-Order Solution ... ...... ................. 35
3.4.1 Solution for Scattered Potential . . . . 35
3.4.2 Solution for Incident Potential p . . . . 37
3.4.3 Solutions for O, and 40, .................... .. .. 38
3.4.4 Summary of Forced Wave Motion. . . . . 47
3.5 Complete Solution Satisfying the No-Flow Condition . . . .. 47
3.6 Discussion of Solution for (2 ................... ...... 49
3.7 Numerical Computation of 2 . . . . . ... 52
4 FREE SURFACE ELEVATIONS AND WAVE RUNUP .. ............ 57
4.1 Introduction .. .................................. 57
4.2 First-Order Water Surface ............................ 58
4.3 Quadratic Components of the Second-Order Water Surface . ... 63
4.3.1 Mean Water Levels ............................ 64
4.3.2 Oscillatory Quadratic Free Surface . . . . 67
4.4 Free Surface from Second-Order Velocity Potential . . ... 69
4.5 Free Surface Complete to Second-Order . . . . . 73
4.6 Summary of Nonlinear free surface . . . . . ... 79
5 WAVE FORCES ................... ............... 89
5.1 Introduction ............... ................... 89
5.2 First-Order Wave Forces ............................. ..... 91
5.3 Quadratic Components of Second-Order Force . . . ... 94
5.3.1 Mean Drift Forces ........................... 95
5.3.2 Oscillatory Quadratic Forces ... . . . . 97
5.4 Forces from Second-Order Velocity Potential . . . . 99
5.5 Wave Forces Complete to Second-Order . . . ... 102
5.6 Summary of Nonlinear Forces . . . .. . 104
6 EXPERIMENTAL VERIFICATION OF SECOND-ORDER THEORY ..... 111
6.1 Description of Laboratory Experiments ................... 111
6.2 Literature Review of Previous Experiments ....... ............ 117
6.3 Results of Wave Runup/Rundown Experiments ................... 122
6.4 Results for Photopole Experiments ........... .. ... ..... 129
6.5 Results of Wave Force Experiments ................. ..... 131
7 SUMMARY AND CONCLUSIONS ..... ......... ....... ...... 140
APPENDICES
A DERIVATION OF QUADRATIC TERMS ..................... 154
A.1 Notation ................. .................... .. 154
A.2 Quadratic Forcing in Combined Free Surface Boundary Condition ..... 155
A.3 Representation of Quadratic Terms by Fourier Series ............ 158
A.4 Exact Form of fii Quadratic Forcing Terms ................... 159
A.5 Asymptotic Form of the Free Surface Forcing Terms ............ 160
A.6 Quadratic Terms For Evaluation of Second-Order Free Surface ....... 161.
A.7 Quadratic Terms For Evaluation of Second-Order Wave Forces ...... 163
B DERIVATION OF COEFFICIENTS FOR SCATTERED WAVE SOLUTION 169
C USEFUL INTEGRALS ............................... 171
D RESULTS OF WAVE RUNUP EXPERIMENTS ................... 173
E RESULTS OF PHOTOPOLE EXPERIMENTS ................... 196
F RESULTS FOR WAVE FORCE EXPERIMENTS .................. 209
BIBLIOGRAPHY .................... ................. 232
BIOGRAPHICAL SKETCH ................................ 236
ACKNOWLEDGEMENTS
I am most appreciative of the advice and friendship of Dr. Robert G. Dean and I thank
him for his guidance not only during this study but during the six years that I have worked
with him. I am grateful to Dr. James Kirby, Dr. Robert Dalrymple, and Marc Perlin for
numerous useful discussions, to Claudio Neves for customizing the word processing soft-
ware, and to my committee members, Drs. Thomas Bowman, Joseph Hammack, Lawrence
Malvern, Michel Ochi, and Hsiang Wang for their input. I also thank Dr. Wang and the
Coastal and Oceanographic Engineering Department for financial support, as well as the
staff of the Coastal Engineering Laboratory for help in the experimental and numerical
phases of the study.
Numerous friends and fellow students provided daily encouragement and made this a
great experience, especially Mike Barnett, Kevin Bodge, Danny Brown, Bill Dally, Claudio
Neves, Marc and Terri Perlin, and Harley Winer.
Most of all, however, I would like to express my gratitude to my best friend, my wife
Debbie. She has endured more than her share of lonely nights and weekends but her faith,
her optimism, her patience, and her love, have enabled us both to realize this goal.
LIST OF TABLES
6.1 Wave Conditions for Laboratory Experiments .............. 113
6.2 Nondimensional Parameters From Laboratory Experiments ....... 113
6.3 Normalized Fourier Amplitudes for Wave and Force Data ........ 132
LIST OF FIGURES
1.1 Definition sketch ....... .... .. ............... 4
3.1 Amplitude of f"/ quadratic forcing applied to free surface, (a) side view,
(b) front view ............... ................ 33
3.2 Amplitude of f" quadratic forcing applied to free surface, (a) side view,
(b) front view .................. ........... .... 34
3.3 Definition sketch for point-source of applied surface pressure . 39
3.4 Amplitude of f}* forcing with Gibb's phenomenon after one iteration 55
3.5 Amplitude of fj* forcing with reduced Gibb's phenomenon after 50 iter-
ations .................. ............. ..... 55
4.1 Runup profiles from linear theory for various values of ka. .. . 60
4.2 Maximum wave runup as a function of diffraction parameter, ka. ... 60
4.3 Envelope of first-order free surface along axis of wave propagation, ka =
1.0 .......... .............. .................. 61
4.4 Contours of first-order wave crest envelope, ka = 1.0 . . 62
4.5 Second-order mean water levels along z axis for reference case, ka = 1.0,
kd = 1.57, and kH = 0.5 ............... ........... 65
4.6 Contours of second-order mean water level for reference case, ka = 1.0,
kd = 1.57, and kH = 0.5 .................. ........ 66
4.7 Envelope of second-order oscillatory quadratic components, ka = 1.0,
kd = 1.57, and kH = 0.5 ............................. 68
4.8 Envelope from second-order velocity potential and effect of second-order
potential on free surface, ka = 1.0, kd = 1.57, and kH = 0.5 . 72
4.9 Total wave envelope to second-order for reference case ka = 1.0, kd =
1.57, and kH = 0.5 .............................. .. 74
4.10 Contours of total wave crest amplitude to second-order for reference case
ka = 1.0, kd = 1.57, and kH = 0.5 ..................... .77
4.11 Oblique view of wave field near cylinder to second- order, phase of max-
imum runup-r/2, for reference case ka = 1.0, kd = 1.57, and kH = 0.57 78
4.12 Oblique view of wave field near cylinder to second- order, phase of max-
imum runup, for reference case ka = 1.0, kd = 1.57, and kH = 0.5 79
4.13 Example of second-order wave envelope for large cylinder, ka = 1.57,
kd= 1.57, kH =0.5 ..... ...................... 81
4.14 Example of second-order wave envelope for small cylinder, ka = 0.2,
kd= 1.57, kH = 0.5 ............................ 82
4.15 Example of second-order wave envelope for deep water, ka = 1.0, kd =
3.14, kH = 0.5 .................. ........ . 83
4.16 Example of second-order wave envelope for near-shallow water, ka = 1.0,
kd = 0.628, kH = 0.1 .......................... 84
4.17 Comparison of first and second-order wave runup at 0 = x as a function
of the diffraction parameter and wave steepness . . ..... 86
4.18 Percent difference between second and first order maximum runup 87
5.1 Maximum linear wave force as a function of the diffraction parameter ka 93
5.2 Oscillatory quadratic force compared to linear force for reference condi-
tions ................... ........... ...... .. 99
5.3 Second-order forces from the second-order velocity potential for reference
condition ......................................... 102
5.4 Total second-order wave forces for reference conditions . . 103
5.5 Example of second-order force for large cylinder, ka = 1.57, kd = 1.57,
and kH = 0.5 ................... ....... .... 104
5.6 Example of second-order force for small cylinder, ka = 0.2, kd = 1.57,
and kH = 0.5 ................... ............. 105
5.7 Example of second-order force for deep water, ka = 1.0, kd = 3.14, and
kH = 0.5 ... .................. ................. 106
5.8 Example of second-order force for near-shallow water, ka = 1.0, kd =
0.628, and kH = 0.1 ............................. 107
5.9 Comparison of first and second-order maximum forces as a function of
the diffraction parameter and wave steepness . . . 109
5.10 Percent difference between second and first-order maximum force 110
6.1 Model basin and experimental setup . . . .... 112
6.2 Test conditions compared to approximate limits of validity for Stokes
theories ................... ................. 114
6.3 Instrumented cylinder for measuring total wave forces . .. 116
6.4 Side view of cylinder with photopole array . . ... 117
6.5 Comparison of linear theory to available force data in the diffraction
regime ......... ........ .. ............. ... 120
6.6 Comparison of measured runup to linear and second-order theories 126
6.7 Percentage difference between measured maximum runup and linear the-
ory ....... .............................. 127
6.8 Percentage difference between measured maximum runup and second-
order theory ............. ..................... 128
6.9 Comparison of measured maximum forces to linear and second-order
theories .................. .. ..................... 135
6.10 Percentage difference between measured maximum forces and linear the-
ory ....................... ...................... 137
6.11 Percentage difference between measured maximum forces and second-
order theory .................. .............. 138
6.12 Comparison of measured drift forces to second-order theory ..... .139
D.1 Runup/rundown envelope for ka = 0.271, kd = 0.750, and kH = 0.132 174
D.2 Runup/rundown envelope for ka = 0.271, kd = 0.750, and kH = 0.178 175
D.3 Runup/rundown envelope for ka = 0.271, kd = 0.750, and kH = 0.215 176
D.4 Runup/rundown envelope for ka = 0.308, kd = 0.853, and kH = 0.085 177
D.5 Runup/rundown envelope for ka = 0.308, kd = 0.853, and kH = 0.137 178
D.6 Runup/rundown envelope for ka = 0.308, kd = 0.853, and kH = 0.182 179
D.7 Runup/rundown envelope for ka = 0.308, kd = 0.853, and kH = 0.250 180
D.8 Runup/rundown envelope for ka = 0.308, kd = 0.853, and kH = 0.296 181
D.9 Runup/rundown envelope for ka = 0.374, kd = 1.036, and kH = 0.122 182
D.10 Runup/rundown envelope for ka = 0.374, kd = 1.036, and kH = 0.205 183
D.11 Runup/rundown envelope for ka = 0.374, kd = 1.036, and kH = 0.286 184
D.12 Runup/rundown envelope for ka = 0.374, kd = 1.036, and kH = 0.385 185
D.13 Runup/rundown envelope for ka = 0.374, kd = 1.036, and kH = 0.402 186
D.14 Runup/rundown envelope for ka = 0.481, kd = 1.332, and kH = 0.186 187
D.15 Runup/rundown envelope for ka = 0.481, kd= 1.332, and kH = 0.317 188
D.16 Runup/rundown envelope for ka = 0.481, kd = 1.332, and kH = 0.438 189
D.17 Runup/rundown envelope for ka = 0.481, kd = 1.332, and kH = 0.530 190
D.18 Runup/rundown envelope for ka = 0.631, kd = 1.745, and kH = 0.683 191
D.19 Runup/rundown envelope for ka = 0.684, kd = 1.894, and kH = 0.391 192
D.20 Runup/rundown envelope for ka = 0.684, kd = 1.894, and kH = 0.572 193
D.21 Runup/rundown envelope for ka = 0.917, kd = 2.536, and kH = 0.631 194
D.22 Runup/rundown envelope for ka = 0.917, kd = 2.536, and kH = 0.806 195
E.1 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.308, kd = 0.853, and kH = 0.085 ...... ............ 197
E.2 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.308, kd = 0.853, and kH = 0.137 . . . ..... 198
E.3 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.308, kd = 0.853, and kH = 0.182 .................. 199
E.4 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.374, kd = 1.036, and kH = 0.122 . . . ..... 200
E.5 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.374, kd = 1.036, and kH = 0.205 . . . ..... 201
E.6 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.374, kd = 1.036, and kH = 0.286 .................. 202
E.7 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.481, kd = 1.332, and kH = 0.186 . . . ... 203
E.8 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.481, kd = 1.332, and kH = 0.317 . . . ... 204
E.9 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.481, kd = 1.332, and kH = 0.438 . . . ..... 205
E.10 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.684, kd = 1.894, and kH = 0.391 . . ..... 206
E.11 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.684, kd = 1.894, and kH = 0.572 . . . ..... 207
E.12 Photopole data compared to theoretical wave envelope along x-axis for
ka = 0.917, kd = 2.536, and kH = 0.631 .. . . 208
F.1 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.271, kd = 0.750, and
kH = 0.132 ........ ..... .............. ...... 210
F.2 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.271, kd = 0.750, and
kH = 0.178 ................... ............ ... 211
F.3 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = w/2 and (2) depth-integrated force, for ka = 0.271, kd = 0.750, and
kH = 0.215 .......................... ... .. .. 212
F.4 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.308, kd = 0.853, and
kH = 0.085 ............... ..... ............ .. 213
F.5 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.308, kd = 0.853, and
kH = 0.137 .................................. .... .. 214
F.6 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = s/2 and (2) depth-integrated force, for ka = 0.308, kd = 0.853, and
kH = 0.182 ................... ............ 215
F.7 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.308, kd = 0.853, and
kH = 0.250 ................... ............ 216
F.8 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.308, kd = 0.853, and
kH = 0.296 ........... ....................... 217
F.9 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.374, kd = 1.036, and
kH = 0.122 .................. ............. 218
F.10 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.374, kd = 1.036, and
kH = 0.205 ................... ............... 219
F.11 Theoretical and experimental results for (1) wave profile at r = 10a and
9 = r/2 and (2) depth-integrated force, for ka = 0.374, kd = 1.036, and
kH = 0.286 ................... ...... ......... 220
F.12 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.374, kd = 1.036, and
kH = 0.385 ........ ... ............... ....... 221
F.13 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.374, kd = 1.036, and
kH = 0.402 ................. ................ 222
F.14 Theoretical and experimental results for (1) wave profile at r = 10a and
8 = r/2 and (2) depth-integrated force, for ka = 0.481, kd = 1.332, and
kH = 0.186 .................................. 223
F.15 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.481, kd = 1.332, and
kH = 0.317 .................................. 224
F.16 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.481, kd = 1.332, and
kH = 0.438 ................... ............... 225
F.17 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.481, kd = 1.332, and
kH = 0.530 .................................. 226
F.18 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.631, kd = 1.745, and
kH = 0.683 ................. ............. 227
F.19 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = x/2 and (2) depth-integrated force, for ka = 0.684, kd = 1.894, and
kH = 0.391 .... .................... ....... ... ..228
F.20 Theoretical and experimental results for (1) wave profile at r = 10a and
9 = X/2 and (2) depth-integrated force, for ka = 0.684, kd = 1.894, and
kH = 0.572 ................... .............. 229
F.21 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.917, kd = 2.536, and
kH = 0.631 ............ ... ................ .. 230
F.22 Theoretical and experimental results for (1) wave profile at r = 10a and
0 = r/2 and (2) depth-integrated force, for ka = 0.917, kd = 2.536, and
kH = 0.806 ................. ............... 231
LIST OF SYMBOLS
a cylinder radius
an coefficient in scattered wave solutions
an coefficient in scattered wave solutions
An radial function for the first order velocity potential
c.c. complex conjugate quantity
C general constant coefficient
CFSBC combined free surface boundary condition
C(t) Bernoulli function
d water depth
DFSBC dynamic free surface boundary condition
Dn wavenumber spectrum
f quadratic forcing terms in second-order CFSBC
f" forcing due to self-interaction of first-order incident wave
f" forcing due to cross-interaction of first-order incident and scattered waves
f" forcing due to self-interaction of first-order scattered wave
fn quadratic forcing in series form of second-order CFSBC; function of r only
fii forcing due to self-interaction of first-order incident wave; function of r only
f* forcing due to cross-interaction of first-order incident and scattered waves;
function of r only
fn" forcing due to self-interaction of first-order scattered wave; function of r only
F total depth-integrated force on cylinder in z direction
FI force to first-order
Fz force to second-order
F21 quadratic second-order force, total
Fzt2 quadratic second-order force, from first-order free surface
F21, quadratic second-order force, from velocity-squared terms
2 oscillatory quadratic second-order force
F2i steady quadratic second-order force or drift force
F2!f steady quadratic second-order force, from first-order free surface
F,21 oscillatory quadratic second-order force, from first-order free surface
F21 oscillatory quadratic second-order force, from velocity-squared terms
F3I, steady quadratic second-order force, from velocity-squared terms
F22 second-order force from second-order velocity potential
g gravitational acceleration constant
9,,, nondimensional function of r in series expansions in Appendix A
g(e) general complex function of 0
G Green's function
GI radial function for oscillatory quadratic free surface; defined in Appendix A.6
G< radial function for steady quadratic free surface; defined in Appendix A.6
h,,n nondimensional function of r in series expansions in Appendix A
h(O) general function of 0
H wave height
H, Hankel function of the first kind of order n
HI, radial derivative of Hankel function with respect to argument
J. Bessel function of the first kind of order n
J' radial derivative of Bessel function with respect to argument
k first-order free wavenumber from o2 = gk tanh kd
ka relative cylinder radius
kd relative depth
kH wave steepness
k2 second-order free wavenumber from 4o2 = gk2 tanh k2d
KBBC kinematic bottom boundary condition
KFSBC kinematic free surface boundary condition
KSBC kinematic structural boundary condition
K. modified Bessel function of second kind of order n
K'h radial derivative of modified Bessel function with respect to argument
P pressure, either on free surface or in fluid
PV Cauchy principal value integral
r radial distance, center of cylinder to field point
r' radial distance, center of cylinder to source point
R radial distance, source point to field point
t time
T wave period
u radial water particle velocity
v tangential water particle velocity
w vertical water particle velocity
z cartesian coordinate in direction of wave propagation
y cartesian coordinate transverse to direction of wave propagation
Yn Bessel function of the second kind of order n
Yn radial derivative of Bessel function with respect to argument
z cartesian coordinate in vertical direction
a horizontal angle from x axis in local coordinate system relative to point source
?n complex coefficient defined as (2 Sno)in
f y coordinate of source point
6 Dirac delta function
5no Kronecker delta function, nonzero when n = 0
C perturbation parameter equal to wave steepness
S z coordinate of source point
q free surface elevation
ql first-order free surface elevation
qrt second-order free surface elevation
2I second-order quadratic free surface components derived from first-order
velocity potential
)21t oscillatory second-order quadratic free surface components
t72 steady second-order quadratic free surface components
z22 second-order free surface component derived from second-order velocity
potential
0 horizontal angle to field point in global coordinate system
9' horizontal angle to source point in global coordinate system
I general wavenumber
Vy first-order wavenumber for evanescent modes; from Cr2 = -gri tan Kjd
KCi second-order wavenumber for evanescent modes; from 4r2 = -g~2i tan I2id
p z component of wavenumber vector
v y component of wavenumber vector
z x coordinate of source point
p density of water
a wave frequency equal to 2x/T
Velocity potential with time dependence removed; same definition applies with
any subscript or superscript
F velocity potential for forced wave due to point source of pressure
OR velocity potential for radiated wave due to point source of pressure
4 velocity potential, including time dependence
'1 first-order velocity potential
'<2 second-order velocity potential
first-order incident velocity potential
<& first-order scattered velocity potential
Q- second-order incident velocity potential
'2 second-order scattered velocity potential
f second-order velocity potential for homogeneous solutions
Ssecond-order velocity potential for particular solutions
e second-order homogeneous solution for scattering of forced wave due to fi
forcing
QiH second-order homogeneous solution for scattering of forced wave due to fio
forcing
V21 second-order homogeneous solution for scattering of forced wave due to f/
forcing
t2sp second-order particular solution for forced wave due to fi forcing
2,p second-order particular solution for forced wave due to f" forcing
P, second-order particular solution for forced wave due to f" forcing
,n phase angle defined as tan-I(Y,(ka)/J' (ka))
V gradient operator
3R real part of complex quantity
used over another symbol to denote nondimensional quantity
used over another symbol to denote complex conjugate quantity
xvii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
A SECOND-ORDER DIFFRACTION THEORY FOR WAVE RUNUP AND WAVE
FORCES ON A VERTICAL CIRCULAR CYLINDER
By
DAVID LANE KRIEBEL
August 1987
Chairman: Robert G. Dean
Major Department: Engineering Sciences
A solution is developed for the interaction of a second-order Stokes wave with a large
vertical circular cylinder. At first-order, the solution is the usual linear diffraction theory,
consisting of an incident plane wave and an outwardly radiating scattered wave. At second-
order, quadratic forcing terms, due to nonlinear interactions of the first-order incident and
scattered waves, appear in the free surface boundary condition and behave like an applied
surface pressure to generate additional waves at twice the frequency of the linear waves.
The second-order solution consists of (1) particular solutions representing forced wave
motions due to the quadratic forcing and (2) complementary or homogeneous solutions
representing scattered wave motions due to the interaction of the forced waves with the
cylinder. One forced wave component is the usual Stokes second-order plane wave. Other
forced wave motions are found in closed integral form by applying a source distribution
method in which the source strengths and phases are given by the quadratic forcing on the
free surface. The scattered wave solutions consist of outwardly radiating second-order free
waves plus local evanescent modes which satisfy the homogeneous form of the second-order
boundary value problem.
Theoretical results are compared to 22 laboratory experiments in which measurements
were obtained for (1) wave runup and rundown around the circumference of the cylinder, (2)
xviii
wave crest and trough envelopes in front and behind the cylinder, and (3) total wave forces.
The second-order theory explains a significant portion of the measured wave runup and
scattered/diffracted wave envelope. Measured maximum runup exceeds linear theory by 44
percent on average but exceeds the nonlinear theory by only 11 percent; in some cases both
measured runup and the second-order theory exceed the linear prediction by more than 50
percent. The second-order forces also agree with measured data and exceed linear forces
by as much as 20 percent for certain conditions. The nonlinear diffraction theory is found
to be valid for the same relative depth and wave steepness conditions applicable to Stokes
plane wave theory.
CHAPTER 1
INTRODUCTION
1.1 Problem Statement
At present, there is no consensus on a complete and consistent second-order theory
for the scattering and diffraction of water waves about a large surface-piercing vertical
circular cylinder. For small piles, second and higher-order estimates of wave loadings are
obtained through the semi-empirical Morison equation, e.g. Morison et al. (1950), with wave
kinematics described by one of the higher-order wave theories, based on the assumption that
the incident wave is not distorted by the cylinder For large piles, this assumption is no
longer reasonable. Higher-order solutions for wave flow past a large cylinder must include
(1) wave-structure interactions, such as scattering of the incident plane wave, and (2) wave-
wave interactions between the incident and scattered waves. In theory, inertial terms are
dominant in the equations of motion for flow around a large cylinder and the resulting
boundary value problem is "simplified" to the potential flow limit. In practice, however,
the resulting problem has proved difficult to solve when extended to second-order due to
the combination of imposed boundary conditions, which include quadratic forcing in the
free surface boundary condition, the kinematic requirement for no-flow through the cylinder
boundary, and a poorly specified form of the radiation condition far from the cylinder.
The boundary value problem for the first perturbation order, corresponding to linear
diffraction theory, was solved by Havelock (1940) for the. deep water case and by MacCamy
and Fuchs (1954) for finite water depth. Since the mid-1970s, several solutions for the
second-order diffraction theory have been proposed, yet none has achieved acceptance as
a "correct" solution. The earlier solutions, i.e. before 1979, either failed to satisfy the
nonhomogeneous free surface boundary condition imposed at second-order, or failed to
2
recognize that both free and forced wave motion must exist at second order. Forced waves
must exist due to the applied quadratic forcing in the second-order free surface boundary
condition while dispersive free waves must exist due to the scattering of the forced waves
from the cylinder. Since 1979, proposed solutions have seemingly satisfied the required free
surface boundary conditions and have included free and forced wave components. However,
due to the mathematical form of these solutions, and due to continued lack of agreement
on a proper far-field radiation condition, it is not clear that any of these proposed solutions
are complete such that all physical second-order wave motions are represented.
The ultimate goal of all proposed second-order solutions has been to determine second-
order wave forces for the design of large offshore structures. Given the lack of agreement on
a complete solution for the second-order velocity potential, there is similarly no agreement
on a proper formulation for second-order forces. The importance of second-order effects on
wave loadings is not known with certainty and no conclusive design criteria are available.
Based on a comparison of laboratory wave force data to linear diffraction theory, it has
been speculated that second-order effects may account for as much as a 20 percent increase
in wave forces. However, the comparison of theoretical second-order force solutions to
laboratory data has been incomplete and none of the proposed solutions is well verified.
Surprisingly, no effort has been made in any previous solution to consider the behavior
of the wave field around the cylinder or to compare the resulting scattering or diffraction
effects with phenomena observed in other wave-structure interaction problems. Only two of
the proposed second-order solutions consider the resulting second-order wave runup around
the cylinder, and neither of these have developed general design guidelines for wave runup on
cylindrical members. In general, little physical interpretation of the second-order free and
forced wave motions is offered in any of the proposed methods; this has made it especially
difficult to evaluate the completeness of the proposed solutions.
The basic objective of this study is to obtain a complete solution, consistent to the
second perturbation order, for the interaction of regular Stoke's waves with a large surface
3
piercing vertical circular cylinder. Second-order effects are investigated, both through devel-
opment of a unified second-order theory and through comparison of the theory to laboratory
data. Specific goals, addressed in subsequent chapters, are to
1. Review the previously proposed solutions for the second-order velocity potential in
order to assess the completeness and validity of each method.
2. Develop a complete solution for the second-order velocity potential which rigorously
accounts for all free and forced wave motions.
3. Determine the second-order free surface displacements around the pile, including wave
runup, scattering and diffraction effects, and second-order mean water levels.
4. Determine the second-order wave forces on the cylinder, including the mean drift
forces and the time-dependent dynamic forces.
5. Compare the theoretical potential flow solution for the dynamic forces, mean drift
forces, and wave runup to laboratory data.
6. Assess the relative importance of second-order effects over the range of relevant nondi-
mensional parameters to determine the range of validity of the second-order solution.
1.2 Problem Formulation
A fixed, vertical, surface-piercing cylinder, of radius, a, in water of uniform depth, d,
is subjected to regular periodic waves of height, H, propagating in the positive x direction
as shown in Figure 1.1. It is assumed that the fluid is irrotational and incompressible such
that the velocity vector may be represented by the positive gradient of a scalar velocity
potential. In cylindrical coordinates, the velocity potential is given as Q(r, 0, z, t) such that
the physical fluid velocity components are
aU
u = _e (1.1)
= a- = 7 -- (1.2)
r a80 r
W =(1.3)
z
y
--- (x,y) or (r.e)
r
Incident
Wave
e=x 0 =0
"upwave" "downwave"
zx-d
Figure 1.1: Definition sketch
The governing partial differential equation is the equation for conservation of fluid mass
which, for an incompressible fluid, is given by the Laplace equation
V'2. = 0 (1.4)
In cylindrical coordinates this is
1 1
r,+ 4r + r--I + P0,. = 0 (1.5)
In addition, appropriate boundary conditions must be satisfied. These include kine-
matic conditions governing the flow field along the bottom, the cylinder, and the free sur-
face, as well as dynamic conditions which require the free surface to deform in order to
maintain a uniform pressure. As a consequence of the deformable free surface, the position
of the upper fluid boundary is an additional unknown and is given by the equation
z = r(r, t)
~
(1.6)
The derivations of the boundary conditions are well known, e.g. Stoker (1957), Mei (1983),
or Dean and Dalrymple (1984). Kinematic boundary conditions are
Kinematic Bottom Boundary Condition (KBBC) for a flat bottom
9, = 0 on z = -d (1.7)
Kinematic Structural Boundary Condition (KSBC) for a circular cylinder
0, = 0 on r =a (1.8)
Kinematic Free Surface Boundary Condition (KFSBC)
~Ie + 'rI7r, + -74et 4), = 0 on z = r (1.9)
The dynamic boundary condition is
Dynamic Free Surface Boundary Condition (DFSBC)
)t + 9g + [(,r)2 + (1 )2 + (',) = C(t) on z = 7 (1.10)
where the external atmospheric pressure along z = 1r, is taken to be zero. The Bernoulli
function, C(t), may be specified to be zero such that the solution is referenced to the mean
water level datum in deep water which is identical to the still water level.
An additional boundary condition which must be specified is a proper form of the
so-called radiation condition. This condition ensures that mathematical solutions for cylin-
drical waves physically represent waves that radiate outward away from the cylinder rather
than waves that radiate inward toward the cylinder from sources at infinity. Sommerfeld
(1949, p. 193) gives the radiation condition as
lim r2 (4 iic')= 0 (1.11)
where n is the scattered wavenumber which satisfies a dispersion relationship relating the
wavenumber, the water depth, and the frequency of the oscillation.
j
6
The nonlinearity of the boundary value problem, imposed by the free surface boundary
conditions, effectively prevents an exact solution from being obtained. In order to obtain an
approximate solution, a perturbation expansion is employed which is identical to the usual
perturbation solution for two-dimensional Stokes waves, e.g. Dean and Dalrymple (1984,
pp. 296-305). The velocity potential and the free surface elevation are approximated in
power series form as
00
4 = E", (1.12)
n=l
= = E "t (1.13)
n=1
where e is equal to the nondimensional wave steepness; i.e. c = kH. In addition, since the
location of the water surface, z = r, in the free surface boundary conditions is unknown, the
KFSBC and DFSBC are expanded in a Taylor series about the known reference datum, z =
0. Substitution of the power series representations for 4 and tq into the governing equation
and the boundary conditions, including the expanded free surface conditions, results in
separate boundary value problems for each order c term in the power series. In this study,
only solutions through the second order are considered.
At order c, the first-order boundary value problem is
V'l = 0 (1.14)
1,i = 0 on z = -d (1.15)
lut =s = 0 on z = 0 (1.16)
Oit + 9gl = 0 on z = 0 (1.17)
'I, = 0 on r = a (1.18)
lim r1'/ (cf, ici') = 0 (1.19)
along with symmetry conditions on 0. Elimination of ri from Equations 1.16 and 1.17 gives
a single Combined Free Surface Boundary Condition (CFSBC) for tZ
itt + g41 = 0 on z = 0 (1.20)
At order J<, the second-order boundary value problem is
V2' = 0 (1.21)
"o = 0 on z = -d (1.22)
nlt + r?1,lr + -i10910 2s 1,r71 = 0 on z = 0 (1.23)
9t +g + (r) + ( ie)2 + (i,) + 'it, =0 on z = 0 (1.24)
2r = 0 on r = a (1.25)
and where T2 satisfies symmetry conditions on 0. While the first-order radiation condition
may be written in closed form, the second-order radiation condition is not well-posed; addi-
tional conditions on the far-field behavior of 2 are required. Elimination of 172 from Equa-
tions 1.23 and 1.24 yields the second-order Combined Free Surface Boundary Condition
(CFSBC)
2tt + 92, =
Il [tl + .I] [(Qir)2 + (1,)' + (.s)2]t on z = 0 (1.26)
At first-order, the right-hand side of the CFSBC is identically equal to zero and the
CFSBC may be termed homogeneous. At second-order, non-zero quadratic forcing terms
appear on the right side of the CFSBC which render the problem nonhomogeneous. These
forcing terms, mathematically, contain nonlinear products of derivatives of the first-order
solution. Physically, they may be interpreted as an oscillatory pressure applied to the free
surface which generates second-order forced waves in order to more closely satisfy the exact
nonlinear free surface boundary conditions, e.g. Lighthill (1979).
As discussed by Stoker (1957), the linear water wave problem, with a homogeneous
CFSBC, is analogous to a spring-mass system in which the oscillation of a mass, m, attached
to a spring with a force constant, c, is given by
mrne + cz = 0 (1.27)
The resulting motions are simple harmonic oscillations that are "free" in the sense that
the modes of oscillation are determined only by the characteristics of the system. For the
mechanical system, the natural period of oscillation is determined by the ratio c/m, while
for the linear water waves, the free or natural wavelength is related to the wave period and
water depth through the linear dispersion relationship
ao2= gktanhkd (1.28)
In contrast, the second-order water wave problem is analogous to a forced spring-mass
system described by
mxt + cx = f (1.29)
where f is an externally applied oscillatory force. For forced systems, the resulting motion
is composed of two types of oscillations: (1) a unique forced oscillation imposed by the
applied forcing and (2) free oscillations that satisfy the homogeneous free surface bound-
ary condition, and which correspond to the natural modes of oscillation of the system. At
second-order, the free wave modes oscillate at frequency 2r but the wavelength is gov-
erned by the homogeneous form of the second-order CFSBC which gives the second-order
dispersion relationship
4a2 = gk2 tanh k2d (1.30)
where k2 is the wavenumber of the,free second harmonic. In deep water, k2 approaches 4k
while in shallow water, k2 approaches 2k.
1.3 First-Order Solution
The solution for the order c boundary value problem in Equations 1.14 to 1.19 has been
given by Havelock (1940) for deep water and by MacCamy and Fuchs (1954) for finite water
depth. The solution is obtained by a separation of the potential, '1, into the sum of the
specified incident wave and a scattered wave
-1 = Vi + 0' (1.31)
This separation is typical in diffraction problems and leads directly to the solution of the
total first order potential. At second-order, however, such a separation does not simplify
the problem in the same way since additional wave motions are generated by the quadratic
forcing in the free surface boundary condition.
The first-order incident potential, representing a plane wave propagating in the +z
direction independent of the presence of the cylinder, must satisfy the usual boundary
value problem for linear wave theory
V20 = 0 (1.32)
D, = 0 on z = -d (1.33)
4itu + St, = 0 on z = 0 (1.34)
The solution is given by
igH cosh k(d + z) e0(k-a')
2 cosh kd 2 + c. (
where the "c.c." denotes the complex conjugate quantity. The wavenumber, k, and the
wave frequency, a, satisfy the linear dispersion relationship
2 = gktanhkd (1.36)
obtained by substituting e1 into the CFSBC, Equation 1.34.
In cylindrical coordinates, the incident plane wave may be represented by a partial wave
expansion in terms of cylindrical standing waves, eg. Mei (1983), as
igHcoshk(d+ z) 'o i.8n(kr) cos O c.c. (1.37)
= krconO- + c.c. 37)
n=0
based on the identity, e.g. Abramowitz and Stegun (1972)
ehk = eseco= = nJn(kr) cos nO (1.38)
n=O
where
a6. = (2 6ro)i" (1.39)
and S.o is the Kronecker delta function.
The boundary value problem for the scattered wave potential, O, is given by
V24P = 0 (1.40)
o, = 0 on z = -d (1.41)
~1t + gQ1 =0 on = 0 (1.42)
!r -- -1, onr = a (1.43)
lim r2 (P,. icc) = 0 (1.44)
r-.co I
where the scattered wavenumber, K, is so far not specified. The scattered wave motions are
"generated" by the nonhomogeneous boundary condition on the cylinder, in which there
is an apparent velocity, -~i, at r = a as a result of the interaction of the incident plane
wave with the fixed cylinder. In this way, the scattered wave problem is like a cylindrical
wavemaker problem; this analogy may be applied equally at first and second-order.
The solution to the Laplace equation by separation of variables yields the following
complete set of eigensolutions which satisfy the kinematic bottom boundary condition, the
radiation condition, and symmetry conditions on 0, e.g. Mei (1983) or Dean and Dalrym-
ple (1984)
0o e-it
S= a O (B,,n(r) oscos ncosh(d + z) 2 +c.c.
n=0O
oo oo -iat
+ E E an,Kn(ijr)cosnOcoscj(d+ z)-- +c.c. (1.45)
n=Oj=l
where H,,(ir) is understood to be the Hankel function of the first kind that represents an
outwardly propagating wave while Kn,(ir) is the modified Bessel function of the second
kind that represents local standing wave motion. Substitution of [' into the CFSBC in
Equation 1.42 yields two dispersion relationships: (1) for real eigenvalues
S2=ga tanh cd (1.46)
which has one positive root, k, identical to the incident free wave number, and (2) for
imaginary eigenvalues
o1 = -gij tan ijd (1.47)
which has many roots Jci, with
('- ). < ic < jr (1.48)
Applying the boundary condition on the cylinder, Equation 1.43, yields the following
condition relating the unknown coefficients in the scattered wave solution, an and an, to
the incident wave solution
00
akH'(ka) cosh k(d+ z) + a,,njiK'(sja) cos rj,(d + z) =
i=1
igH cosh k(d + z) .k (ka) (1.4)
tkJ,(ka^) (1.49)
2a cosh kd
where J.(ka) is understood to be the derivative of J,,(kr) with respect to the argument
kr, evaluated at r = a. Since cosh k(d + z) and cos ki(d + z) form an orthogonal set, the
unknown coefficients in Equation 1.49, are found as
i gH 1 pnkJ,'(ka)
a, =(1.50)
2a cosh kd kH,(ka)
ani = 0 (1.51)
Substituting Equation 1.50 into Equation 1.45 gives the linear scattered wave solution as
ig H cosh k(d + z) J',(ka) e-iot
igH cosh Hn(kr) cos n + c.c. (1.52)
2o cosh kd = H-(ka) 2
which is the result of MacCamy and Fuchs (1954). The first-order scattered wave solution
is represented exactly by an outwardly propagating wave since aj equals zero. This result,
that the local standing modes vanish identically for the linear problem, is well known; the
same result does not apply to the second-order scattered potential.
By adding the incident and scattered solutions from Equations 1.37 and 1.52, the total
first order potential may now be written as
igH cosh k(d + z) r J' (ka) 1 e-io
#1 =Pn [Jn(kr) H,(kr)I cos n9
2o cosh kd H'(ka) 2" cos 2
+ c.c. (1.53)
By redefining the radial dependence as
Ji(ka)
An(kr) = Jn(kr) ka) H,(kr) (1.54)
the first-order solution becomes
igH cosh k(d +z) e-ot
1 = H coshkd z.,A,,kr) cos nO + c.c. (1.55)
n=O
1.4 Second-Order Problem Formulation
Since the solution for 01 includes the sum of the incident plus scattered waves, the
quadratic terms in the second-order CFSBC, Equation 1.26, have the following general
form
'2tt + g 2z, = + f/ + f" (1.56)
where each forcing term on the right hand side may be defined physically as
fi = due to nonlinear self-interaction of the first-order incident plane wave
fi* = due to nonlinear cross-interaction of the incident and scattered waves
f*a = due to nonlinear self-interaction of the first-order scattered wave
Solutions for the second-order potential, %2, must conceptually include three particular
solutions which represent the forced wave motions due to the nonhomogeneous forcing on
the free surface as
2f = ', + #'2p + '02P (1.57)
These forced waves may be thought of as the ambient or "incident" waves at second-order,
although only the 90'p term is really incident in the sense that it is a second-order plane
13
wave propagating in the z direction with uniform amplitude everywhere. These forced wave
motions exist only because of nonlinear wave-wave interactions. They also exist without
further regard for the cylinder; therefore, they do not, by themselves, satisfy the KSBC
requiring no-flow through the cylinder boundary.
In order to satisfy the no-flow condition at the cylinder boundary, a second set of
solutions must exist which, when added to the particular solutions, compensates for the
apparent velocity of the forced waves through the cylinder. These additional solutions are
the complementary or homogeneous solutions of the corresponding boundary value problem
with a homogeneous free surface boundary condition
S+ =0 (1.58)
These honogeneous solutions are the second-order scattered waves, and, conceptually,
one homogeneous component is associated with each component of the forced wave motion
as
O = + + + + (1.59)
While the forced waves arise because of nonlinear wave-wave interactions, the scattered
waves arise because of wave-structure interactions. Just as the first-order scattered wave
potential, Vi, was "generated" by an apparent motion of the cylinder boundary, -'ir, so
too are the second-order scattered wave motions "generated" by an apparent motion of the
cylinder, -O F. Furthermore, the second-order scattered wave solutions may be determined
by exactly the same methods used to obtain the first-order scattered wave solution.
CHAPTER 2
REVIEW OF THE LITERATURE
2.1 Introduction
Since 1972, numerous solutions for the second-order velocity potential have been pro-
posed; however, after 15 years of research, it is not clear whether any solutions correctly
represent the physics of the nonlinear wave-structure interaction problem. The first group
of solutions, developed between 1972 and 1979, generally failed to (1) satisfy the nonhomo-
geneous combined free surface boundary condition, or (2) recognize that the solution must
,contain both forced wave motions, due to the quadratic terms in the CFSBC, as well as scat-
tered free wave motions, due to the interaction of the forced waves with the cylinder. The
solutions proposed since 1979 have seemingly corrected these deficiencies; however, while
they mathematically satisfy the CFSBC and the KSBC, the solutions differ substantially in
the way in which these boundary conditions are satisfied. Likewise, the proposed solutions
incorporate the radiation condition in different ways and it is not certain that any of the
methods contain a unique description of the second-order wave motions.
2.2 Solutions Prior to 1979
Chakrabarti (1972) appears to be the first to explicitly propose a solution for a higher-
order velocity potential for the problem of nonlinear wave interaction with a large vertical
cylinder. Using a Stokes 5th order incident wave, a series solution was assumed for the total
potential which, at each order, m, had a form similar to that of the first-order solution as
42 == )mcoshmk(d +z)(, J ) [(mkr) Jmk) (mkrcos nOe-r
m= n= Hmka)
(2.1)
15
The coefficients, A,, are those associated with the mth order component of the incident
Stokes 5th order wave. While this solution satisfies the kinematic structural boundary
condition, it does not satisfy the nonhomogeneous free surface boundary condition since it
does not include the effects of all nonlinear self-and cross-interaction terms.
Chakrabarti (1975) reviews the solution of Yamaguchi and Tsuchiya (1974) in which
the second-order potential is given by
'0 = cosh2k(d+ z) C, [YV(2ka)J,(2kr) J(2ka)Y,(2kr)] cos nic-"0e
n=0
(2.2)
This solution also satisfies the KSBC automatically but it assumes that all wave motion
is forced with wavenumber 2k, neglecting any second-order free wave components. The
complex coefficients, Cn are determined by substituting 02 into the CFSBC and evaluating
at r = a; thus, the solution does not satisfy the required free surface conditions over the
full domain.
The form of the solution for second-order terms assumed by Chakrabarti (1972) and
Yamaguchi and Tsuchiya (1974) is based on a radial dependence like
J" (2ka)
J(2kr) HJ',(2ka) H,(2kr) (2.3)
This form is valid at first-order since the wavenumber of the free scattered wave is identical
to that of the incident wave. It is also useful mathematically since it automatically satisfies
the no-flow condition on the cylinder. At second and higher-orders, however, this form
of the solution is not valid since it neglects the change in wavenumber required after the
nonlinear forced wave components scatter from the pile.
A standard solution technique, identical to that employed at first-order, has been to
separate the second-order potential into incident and "scattered" components. Mathemati-
cally, the second-order incident wave is found from a separated boundary value problem in
which no structure is present
V'2 = 0 (2.4)
Qr2 =0 on z= -d (2.5)
Vi$2 + 9V2, = f' on z = O (2.6)
The solution is easily obtained to be the usual Stokes second-order plane wave component
since fi represents the familiar plane wave forcing.
The remaining problem for the "scattered" wave motion is
V2~ = 0 (2.7)
QI, = 0 on z = -d (2.8)
2tt + gQ;, = "i + f" on z = 0 (2.9)
r = -- 2r on r = a (2.10)
Note that the nonhomogeneity in the CFSBC is not removed, and a second nonhomogeneous
boundary condition is now present on the cylinder boundary. With this formulation the
so-called "scattered" wave potential, VQ, must not only represent the true scattering of
the incident component, t', but it must also represent the remaining forced wave motion
generated by fi' and f", as well as any interactions of these forced waves with the cylinder.
This separation into second-order incident and "scattered" waves is not wrong; however, all
components of wave motion must be carefully accounted for in 'Q.
Raman and Venkatanarasaiah (1976a, 1976b) employ the separation outlined above and
attempt to solve for the remaining "scattered"potential by taking the finite Fourier cosine
transforms of the governing boundary value problem for V2. The final solution is found to
have the form
t4 = C cosh k2(d + z) E hn(kr, k2r) cos nOe-i2 (2.11)
n=0
where the function h,(kr, k2r) contains combinations of Bessel functions, Hankel functions,
and their derivatives. The solution was the first to display the second-order free wavenum-
ber, k2, which satisfies the second-order dispersion relationship
4a2 = gk2 tanh k2d (2.12)
As pointed out by Chakrabarti (1977,1978) and Isaacson (1977c), however, the solution
with only the wavenumber, k2, satisfies only the homogeneous CFSBC and cannot satisfy
the remaining quadratic forcing in the CFSBC.
2.3 Asymptotic Solutions
Molin (1979) proposes a solution for the second-order wave forces based on Haskind's
reciprocal relationship, e.g. Mei (1983), in which only an asymptotic form of the second-
order velocity potential is required. Asymptotic forms of the "scattered" potential are
proposed based on the leading asymptotic form of the quadratic forcing terms. For large r,
the f" forcing is found to behave like
f' = --. Cei(kr(x+o.S)-}) (2.13)
Molin-proposes an asymptotic solution for -I'p in the form
P = F( [cosh(k(2 + 2 cos 0)(d + z))l e1i(r(1+ e-2) (2.14)
which represents forced wave motion, propagating away from the cylinder everywhere, but
which does not satisfy the Laplace equation. The f" forcing is found to behave asymptot-
ically like
f" = lei(2r-ot) (2.15)
r
but Molin assumes a form for C, like
h()= -cosh k(d+ z)ci(r-2't) (2.16)
Molin argues that since P~, is forced by an outwardly radiating motion that decays
rapidly, the wave motion far from the cylinder should be a second-order free wave satisfying
the radiation condition. On the other hand, Molin assumes that the solution for 4I'p
should not satisfy a radiation condition, even though it too results from an oscillatory and
outwardly propagating forcing applied to the free surface.
18
Lighthill (1979) proposes a solution for the second-order wave force which, like Molin's,
does not require a solution for the second-order velocity potential. Green's theorem is
applied to two potentials, the first, Ah, representing the second-order potential due to the
presence of the fixed cylinder in waves, the other, ', due to the unit oscillation of a cylinder
in otherwise calm water. Four surface integral are obtained; however, one is zero due to the
bottom boundary condition and another is zero since Lighthill assumes that the scatteredd"
potential satisfies the radiation condition.
Of the remaining two integrals, one along the cylinder boundary is shown to be identical
to the desired depth-integrated force on the cylinder, while the other may be reduced to
an integral of the quadratic forcing from the CFSBC multiplied by the vertical velocity
at z = 0 resulting from the unit oscillation of a cylinder in calm water. The method is
attractive since forces may be computed without a solution for iP2; however, the method is
also limited in that only the integrated force is obtained and other quantities, such as the
wave runup or wave kinematics, cannot be determined. Lighthill (1979) applies the method
for deep water and for small piles; general solutions for finite depth are given by Rahman
(1984), Demirbilek and Gaston (1985), and Eatock Taylor and Hung (1987).
2.4 Green's Function Solutions
Garrison (1979) proposes a numerical solution for the second-order velocity potential
based on a source distribution method using Green's functions. By introducing the Green's
function, G(C, ,' ), which represents a source of unit strength at z = C, y = Y, and z = f,
the solution for the "scattered" potential is found by applying Green's theorem to V2 and
G. Contributions to '2 are found from the surface integral over the cylinder at r = a and
from the surface integral over the free surface at z = 0; the radiation condition applied
to VI and G ensures no contribution from the surface integral around an imaginary outer
cylinder at some large value of r.
The resulting integral solution for A' is
(xy, z) = Y Zf f.(C,, )G(z, y, z; 7, &)dS
+ (f"(.Y7, 0)+ f "(,-y,0))G(z,y,z;,- y,O)dS (2.17)
where /,( 7, ) is an unknown source strength distribution function over the cylinder at
r = a, while the source strength distribution along the free surface is given exactly by the
quadratic terms in the CFSBC, f' and f". The solution for f.((,7y,) and D$ proceeds
by discretizing the cylinder boundary and the free surface into area elements and then nu-
merically approximating the cinematic structural boundary condition using Equation 2.17.
For later reference, the form of the Green's function used to represent the sources along the
free surface is
loo cosh n(d + z)
G(z,y,0;,0) = 2-PV cosh(+ z)Jo(nR)d
o cosh ocd(n tanh xd k2 tanh kzd)
k2 cosh k2d cosh k2(d + z) R)
+ 4x+ Jo(k-R) (2.18)
sinh 2kd + 2k2d
where the "PV" signifies the Cauchy Principal Value of the singular integral and R is the
distance from the individual source at (, 7, 0) to the field point of interest (z, y, 0).
Chen (1979) and Chen and Hudspeth (1982) propose a second-order diffraction solution
based on an eigenfunction expansion for Green's functions. The second-order problem is
first separated into the incident Stokes plane wave potential and the remaining "scattered"
potential, which is further separated into two potentials as
Q 2 = h + 2, (2.19)
The potential i^' satisfies the homogeneous CFSBC, the radiation condition, and a non-
homogeneous structural boundary condition; this represents the scattering of free waves
produced by the interaction of the incident plane wave component with the cylinder. This
solution is obtained from an integral equation with a Green's function defined to include
outwardly propagating free waves as well as local evanescent modes. This component is
analogous to the '0g, solution discussed in Chapter 1 and, for later reference, the solution
for 0h is
3ioaH2k cosh kd cosh 2kd 4or2
Ssinh 4 d (4k k)(sinh 2k2d + 2k2d) (2k tanh 2kd )
8 sinh4 kd (4k2 k)(sinh 2k2d + 2kzd) g
20
.cosh k2(d + z) (a) H,(kr) cos ne-i2 +c..
=o H' (k2a)
3+ H28 k cos 2id cosh2kd 4a
+ (2sinh4k (4k+ 22) (2k tanh2kd- -)
8sinh kd i=1 (4k + i4)(sin2X2id+2id) g
*cos X2i(d + z) J'(2k)Kn(scr) cos nke-a) + c.c. (2.20)
co(rs-,j( +2i a ,e +
where xsi denotes the second-order evanescent wavenumbers.
The potential 9'2 is assumed to satisfy the remaining non-homogeneous forcing terms
in the CFSBC, but is not required to satisfy a radiation condition. A Green's function
solution is obtained in the form of double summations of products of Bessel functions.
Unknown coefficients are determined by satisfying the CFSBC; however, eigenvalues, i.e.
wavenumbers, are found by satisfying o = 0 on r = a based on zeros of
J'(qia) = 0 (2.21)
In this way, a set of discrete wavenumbers, qi, are found, which satisfy the no-flow conditions
for an interior boundary value problem and correspond to the irregular frequencies that
occur in numerical Green's function solutions.
2.5 Fourier-Bessel Integral Solutions
Hunt and Baddour (1980a,1980b), Hunt and Baddour (1981), and Hunt and Williams
(1982) proposed solutions for the second order velocity potential based on a Fourier-Bessel
integral method for the conditions of deep water standing waves, deep water progressive
waves, and progressive waves in finite depth, respectively. Rahman and Heaps (1983) also
proposed a solution for progressive waves in finite depth that is identical in content to that
of Hunt and Williams (1982).
Upon substituting the first-order solution into the second-order CFSBC, the quadratic
forcing terms are ultimately represented by a single sum with separable harmonics in 0 as
-i24t
'2tt + 9g, = f(r) cos n -- + c.c. (2.22)
n=O
where f,(r) is a complicated function of r that includes sums and products of A,(kr) and
A',(kr), with An defined in Equation 1.54.
A solution for %2 is then sought in the form of an integral over a continuum of wavenum-
bers as
co = no [0 D,,(r.)An(Dcr) cosh rc(d + z)da + c.c. (2.23)
n=o0
This form satisfies the Laplace equation, the KBBC, and, since A,(rr) is defined as in
Equation 1.54 for each wavenumber i, it also automatically satisfies the KSBC. By sub-
stituting Equation 2.23 into the CFSBC and equating harmonics in 9, integral equations
are obtained for the spectrum of unknown coefficients D,,(c). Inversion of these integral
equations yields the final solution for %2 as
C os no f00 mcosh r(d + z)A,(rcr)
n= fo cosh icd(K tanh Pcd k2 tanh k2d)
r'f,(r')A,(ir')dr'd-- + c.c. (2.24)
Jo 2
The integral over all wavenumbers n is singular at the second-order free wavenumber, k2.
A singular integral of this type represents local standing waves which are forced wave mo-
tions concentrated where the value of the forcing, f,(r), is greatest. Farther away, however,
these forced wave modes die out and only one standing wave, with the free wavenumber,
k2, survives. The solution method requires that the prescribed forcing, f,,(r) be sufficiently
localized to permit a Fourier transform. Without explanation, the method is applied to all
forcing terms, including the incident plane wave term, f", which has uniform amplitude
everywhere and which should not be transformable. The question of appropriate far field
conditions is not fully addressed, and the solution given in Equation 2.24 does not satisfy
a radiation condition, since it represents both inward and outward radiating cylindrical
waves.
2.6 Other Recent Solutions
Chakrabarti (1983) presents a solution for the second-order potential based on a separa-
tion of the boundary value problem similar to that used by Chen and Hudspeth (1982). The
"scattered" potential is separated into a "structural" problem (homogeneous CFSBC with
22
nonhomogeneous structural boundary condition) and a "free surface" problem (nonhomoge-
neous CFSBC but with a homogeneous structural boundary condition). For the structural
problem, Chakrabarti adopts the solution of Chen and Hudspeth as given in Equation 2.20.
For the free surface problem, a solution similar to that of Yamaguchi and Tsuchiya (1974)
is assumed which automatically satisfies the homogeneous structural boundary condition
and which has wavenumber dependence 2k, therefore neglecting the scattered free wave
components. Unknown coefficients are determined by satisfying the free surface boundary
condition at r = a only.
Sabuncu and Goren (1985) derive a solution for the second-order potential based on
a direct eigenfunction expansion. The incident plane wave component was first separated;
however, the scattering of this component was assumed to also have a 2k wavenumber
dependence, unlike the more complete scattered solution given in Equation 2.20 by Chen
and Hudspeth (1982). The remaining forced wave motions were termed the "particular"
solutions and were assumed to have the form
h2P = cos n, o H,(2kr)coash 2k(d + z)
n=O I \ Ht( a)
+ E. cos2c(d + z) + c.c. (2.25)
j=1 K n(2rcij) 2
where a 2k wavenumber dependence is assumed but additional local evanescent motions are
also included. The unknown coefficients are determined by numerically fitting the CFSBC
through the method of collocation, in which Equation 2.25 is forced to satisfy the CFSBC at
a series of discrete points on the free surface in the vicinity of the cylinder. The scattering of
these forced wave motions is described by a series of homogeneous solutions which includes
outwardly radiating free waves and local evanescent modes in an eigenfunction expansion
form similar to that used at first-order.
2.7 Irregularity of Second-Order Solution
A controversy over the formal extension of the Stokes perturbation solution to second-
order was initiated by Isaacson (1977b) who asserted that the CFSBC and the KSBC are
23
"incompatible" in the overlap region at z = 0 and r = a. By differentiating the CFSBC
with respect to r and then evaluating at r = a, one obtains
(82 = a /32 /ai (2.26)
Isaacson found that, since 0, a 0 on r = a, the resulting irregular condition occurs
S (a= 0 (2.27)
In general, this statement is not true, indicating an inconsistency in the solution at the
juncture of the two boundaries. Isaacson (1977b, p. 169) concluded that the theory for
wave diffraction "may not be extended formally to a second approximation by the usual
perturbation procedure for Stokes wave."
A great deal of discussion subsequently occurred in the literature, with various authors
claiming that the Isaacson irregularity was either wrong or misinterpreted. The most salient
points seem to have been made by (1) Hunt and Baddour (1980b), who noted that in po-
tential flow, discontinuities are to be expected in the tangential velocity at solid boundaries
as found by Isaacson, and by (2) Wehausen (1980), who provided further proof that such an
irregularity exists along the circular water line contour and who showed that such an irreg-
ularity occurs at any order whenever a junction of nonhomogeneous boundary conditions
is present. Wehausen continues to state, however, that such a solution does not prevent
a useful solution from being attained. Miloh (1980, p. 281) examined the mathematical
form of the irregularity and concluded that "such an irregularity must be present any
purported solution that is not singular on the line of intersection must be incorrect."
In summary, it seems that a consensus has been reached that 42 must exhibit an
irregularity along the water line contour since the KSBC requires the radial velocity to be
zero there while the CFSBC is nonhomogeneous and contains nonzero tangential velocity
terms at r = a. The irregularity, however, does not mean that P2, is discontinuous or that
a solution does not exist at second-order, but only that through the limiting process, the
irregularity must be present in any second-order solution.
24
2.8 Discussion of Literature
Based on a review of the literature, various differences and inconsistencies between the
previous solutions may be identified. In general, it is difficult to identify a single "correct"
theory. However, Garrison's (1979) numerical solution, based on the method of Green's
functions, seems to rigorously incorporate all of the physical elements of (1) forced wave
motion due to the quadratic forcing applied to the free surface, as well as (2) the free wave
motion due to the scattering of the forced waves from the cylinder.
One major source of difficulty in obtaining a consistent and complete solution has been
the separation of I2 into the incident plane Stokes wave potential and the 'scattered" po-
tential, as presented in Equations 2.4 through 2.10. At first-order, this separation is natural
and clearly displays the origin of the scattered waves. At second-order, however, the same
separation does not reduce the complexity of the problem or aid in the interpretation of the
physics. As noted, the "scattered" potential must satisfy a combination of nonhomogeneous
boundary conditions, including (1) the remaining quadratic terms in the CFSBC as well as
(2) the apparent velocity at r = a due to the interaction of the second-order plane wave
component with the cylinder.
If the plane wave component, Q1, is separated as above, then the scattered wave motion
due to the interaction of 4'j (or 4'p as defined in Chapter 1) with the cylinder may be found
by methods identical to those employed at first-order. At both orders, the incident plane
wave components, *i' and -' (or ~0p), lead to apparent motions of the cylinder boundary,
-'i. and -9 (or -94 r). The scattered wave components, V, and tjI, are then free
waves required to compensate for this apparent motion and should consist of outwardly
propagating waves as well as possibly local evanescent wave modes. The correct solution to
this portion of the problem seems to have been determined in closed form by Chen (1979)
and Chen and Hudspeth (1982), and in integral form, with a numerical solution, by Garrison
(1979).
25
For the remaining wave motions, which include the forced waves generated by f'" and
f", as well as the free waves due to the interactions of these forced wave motions with
the cylinder, the previous solutions may be classified into two categories: (1) those that
represent forced wave motion by discrete wavenumber solutions and (2) those that represent
forced ware motions by the continuum of wavenumbers. In the first category, Chakrabarti
(1972,1983), Yamaguchi and Tsuchiya (1974), and Sabuncu and Goren (1985) all represent
forced wave motion with a 2k wavenumber dependence. Chen and Hudspeth (1982) repre-
sent the motion by a set of discrete wavenumbers corresponding to the eigenvalues of the
interior Newman boundary value problem. The second method, based on a Fourier-Bessel
integral incorporating the spectrum of wavenumbers, is used by Garrison (1979), Hunt and
Baddour (1981), Hunt and Williams (1982), and Rahman and Heaps (1983).
As will be discussed in later chapters, the asymptotic forms of the forcing terms fi and
f' seem to have have strong 2k wavenumber dependence; however, other spatial oscillations
and decay terms are also present, especially near the cylinder. It is therefore not possible
to represent the forcing by a simple 2k wavenumber solution; repeated numerical tests by
the author using the method of collocation and the method of least squares have confirmed
that satisfaction of the CFSBC requires more freedom than is obtained from a simple 2k
wavenumber solution. The Fourier-Bessel integral methods are required since they have
sufficient generality to fit the complicated free surface forcing. These methods are also
preferable based on knowledge of similar exterior boundary value problems in which the
domain extends to infinity. In these cases, no discrete eigenvalues are physically suggested
by the geometry and a continuum of wavenumbers is required, e.g. Sommerfeld (1949, p.
188) and Courant and Hilbert (1961, p. 339).
Several methods represent the velocity potential by a function that automatically sat-
isfies the no-flow condition in the KSBC while the CFSBC is satisfied as a last step. This
form of the solution is adopted by Chakrabarti (1972, 1983), Yamaguchi and Tsuchiya
(1974), Hunt and Baddour (1981), Hunt and Williams (1982), Rahman and Heaps (1983),
26
and Chen and Hudspeth (1982). In these cases, the scattered wave motions have the same
wavenumber dependence as the forced wave motions. However, the process of scattering of
nonlinear waves is expected to lead to a change of wavenumber unlike the result at first-
order. In addition, since the scattered waves are physically present only because of the
reflection of the forced waves from the cylinder, it seems that the KSBC should be the last
boundary condition to be mathematically satisfied. In this regard, the solutions of Garri-
son (1979) and Sabuncu and Goren (1985) clearly first satisfy the CFSBC then determine
the scattered wave solutions by satisfying the KSBC; these solutions also reflect a change
of wavenumber due to the scattering process with scattered solutions that are outwardly
propagating second-order free waves.
A final aspect that differentiates the proposed solutions is the degree to which the
Sommerfeld radiation condition is satisfied. For the wave motion generated directly by
the quadratic forcing in the CFSBC, the proper far field condition is not clearly specified.
The incident plane wave, generated by the f" forcing, certainly is not required to satisfy
a radiation condition. As noted by Molin (1979), the other quadratic forcing terms have
an outwardly propagating form but with rapid radial decay, such that waves are generated
that radiate outward. Most proposed solutions, including those of Chakrabarti (1972,1983),
Yamaguchi and Tsuchiya (1974), Raman and Venkatanarasaiah (1976a, 1976b), Hunt and
Baddour (1981), Hunt and Williams (1982), Chen and Hudspeth (1982), Rahman and Heaps
(1983), and Sabuncu and Goren (1985), do not require the forced wave motions to satisfy
a radiation condition. Molin (1979) does not require the wave motion generated by the f"
forcing to satisfy the radiation condition; however, the wave motion due to the f"' forcing
is assumed to satisfy the radiation condition. Garrison (1979) and Lighthill (1979), on the
other hand, require the wave motions generated by f" and f" to satisfy the radiation
condition.
CHAPTER 3
SOLUTION FOR SECOND-ORDER VELOCITY POTENTIAL
3.1 Introduction
The proposed solution for the second-order velocity potential is obtained in a complete
and physically consistent fashion with the final result expressed in closed integral form.
The solution is complicated and contains numerous terms; however, the physical origin of
each term in the solution, including the forced wave motions due to the quadratic forcing in
the CFSBC, and the free wave motions due to the scattering from the cylinder, are clearly
displayed. Although it is derived in an independent way, the solution for forced wave motion
is found to be identical to that proposed by Garrison (1979), through application of Green's
theorem, in that forced wave motions are found based on a distribution of wave sources
over the free surface with the source strengths and phases defined by the quadratic terms
in the CFSBC. Garrison then obtains the scattered wave solutions in terms of an integral
equation of a Green's function and an unknown source distribution function on the cylinder,
which must be found numerically. The scattered wave motions in the proposed solution are
obtained in closed-form based on the complete set of eigenfunction solutions which satisfy
the exterior boundary value problem with a homogeneous free surface boundary condition.
3.2 General Solution Procedure
A complete statement of the second-order boundary value problem is
V'2( = 0 (3.1)
'2tt + 9 2, = /" + f" + f/ on z = 0 (3.2)
2. = 0 on z = -d (3.3)
n2, = 0 on r =a (3.4)
along with symmetry conditions on 42 plus a form of the radiation condition that may be
applied to all terms except the incident plane wave.
The quadratic forcing terms in the CFSBC in Equation 3.2 are
/" + f' + / = -iet {(4I.z + 9,Z}
9
2 f{r 'lirt + zleSt + lizt } (3.5)
In Appendix A, the complete first-order solution is substituted into Equation 3.5 to obtain
expressions for fi, /fi, and f". One general feature of the quadratic forcing is that all
forcing is periodic in time with frequency 2oa. The first-order potential has frequency a;
therefore, the quadratic forcing is expected to contain sum frequencies at 2a, and possi-
bly difference frequencies that are independent of time. As shown in Appendix A.2, all
terms with difference frequencies vanish identically such that there are no steady or time-
independent forcing terms. As a consequence, the time dependence of the second-order
forcing may be separated; and, as shown in Appendix A.3, the quadratic forcing may be
separated into a series form with harmonics of cos nO as
-=i2ot
/" + /" + f" = > [I(r) + f/(r) + f"(r)] cos nO 2 + c.c. (3.6)
n=0
The time-dependence of the second-order potential may also be removed as
-i2ot
02(r, 0, z, t) = 42(r, 0, z)-- + c.c. (3.7)
The nonhomogeneous combined free surface boundary condition from Equation 3.2 can then
be rewritten as
=2 E [f + fi + f] cos n0 (3.8)
9 9 n=o
Following conceptual developments presented in Chapter 1, the general solution for 02
can be obtained in terms of particular solutions,
solutions, O2f as
2 = 2P + 04 (3.9)
The particular solutions ,0, represent the second-order ambient forced wave motions
due to the nonlinear wave-wave interactions of the first-order incident and scattered waves.
These solutions must satisfy the Laplace equation, the bottom boundary condition, the
nonhomogeneous free surface boundary condition, and some components must satisfy a
form of the radiation condition. The particular solutions do not, by themselves, satisfy
the no-flow condition on the cylinder boundary. As discussed in Chapter 1, the particular
solution, 2 may be further separated as
O= '2p + # OA + (3.10)
The component, O*p, represents the forced wave motion generated by the f" forcing; this
is the usual Stokes second-order plane wave component whose solution is well known. In
the same way, two other forced wave components exist in the ambient or "incident" wave
field due to the remaining fi and f" forcing terms.
The homogeneous solutions, ", represent free wave motions that exist due to the wave-
structure interaction of the second-order ambient forced waves with the circular cylinder.
These solutions must satisfy the Laplace equation, the bottom boundary condition, the
radiation condition, and the homogeneous free surface boundary condition. The homoge-
neous solutions may be determined to within a set of unknown constant coefficients. Like
the particular solutions, the homogeneous solutions do not satisfy the kinematic boundary
condition on the cylinder.
Once the complete solution is specified, i.e. Of + 4f, the unknown coefficients in
the homogeneous solutions are determined by requiring the total solution to satisfy the
kinematic condition of no-flow normal to the cylinder boundary. Determination of these
unknown coefficients as a final step in the solution is correct mathematically and physically,
as the homogeneous free wave motions exist at second-order only because of the reflection
or scattering of the forced waves from the cylinder.
30
3.3 Far-Field Boundary Condition
Mathematically, the second-order boundary value problem is well posed with the excep-
tion of the conditions to be applied in the far field. It is known that 02 must be bounded,
i.e. finite, in the far field; but, the correct form of the radiation condition is not known with
certainty. As noted in Chapter 2, previous solutions have variously invoked or ignored the
radiation condition with little discussion or physical reasoning as to its proper form. The
radiation condition, as posed by Sommerfeld (1949, p. 193), is a far field boundary condition
required to distinguish physically relevant cylindrical wave solutions from solutions which
are mathematically possible. In the context of linear diffraction theory, the incident wave is
prescribed as a plane wave propagating in the z direction, for which a radiation condition
is not required. For the scattered wave problem, finite solutions to the governing equations
and boundary conditions include inward radiating waves in addition to the physically ex-
pected outward radiating modes. The radiation condition is applied to the scattered wave
solution as
lim rn (V,, ik'7) = 0 (3.11)
r-co
which can only be satisfied by outwardly propagating waves. It requires the vanishing
of any inward radiating wave components and imposes a uniqueness to ensure that the
mathematical solution agrees with the physics.
For the second-order problem, the radiation condition must be considered for two types
of "disturbances" which generate waves at the fluid boundaries: (1) the direct quadratic
forcing applied to the free surface, and (2) the apparent forcing at r = a due to reflection of
the forced wave motions with the cylinder. In the latter case, it is clear by analogy to the
first-order problem, that a radiation condition is required to obtain proper homogeneous
solutions which represent outwardly radiating scattered free waves. The far field condition
lim rri ( ik2,") = 0 (3.12)
Stereore aot
is therefore adopted.
31
The correct far field behavior of the particular solutions, 4P', may be investigated
in part by considering the asymptotic form of the quadratic forcing terms. In Appendix
A.5, the asymptotic forms of the forcing terms are found as
fi a le2kr co e-i2a t (3.13)
i [.ikr(1+co0) + il(+cos)e-i2 (3.14)
tr3 r3 J
ft '4 ei2krL + aei2kr + i2kr -i2at (3.15)
r r2 r3
The leading coefficients are generally complex and are functions of the three dimensionless
parameters, kH, kh, and ka, as well as functions of 0. The forcing term f" is in the form
of a periodic plane wave propagating in the +x direction at wavenumber 2k with uniform
amplitude. This forcing is associated with the usual Stokes second harmonic plane wave
component for which the Sommerfeld radiation condition is certainly not required.
The general form of the f" forcing is that of an outwardly propagating oscillation;
however, the radial decay is quite rapid and the forcing is concentrated in the vicinity of the
cylinder. The wave motion generated by this forcing is expected to be composed of local
forced wave motion, to satisfy the rapid radial decay; but far from the cylinder, the forced
wave motions should decay such that only freely dispersive wave motions survive. The far
field wave motion should, therefore, consist of free waves radiating outward from the region
of localized forcing and, as Molin (1979) proposed, the particular solution due to the f'"
forcing should satisfy a form of the radiation condition.
The f'" forcing appears to have a radial decay like r-'/2 to leading order, but like
the f" forcing, also has higher order terms which give a more concentrated forcing near
the cylinder; the decay rate also varies with 0. The propagation characteristics are quite
complicated and, as noted by Mei (1983), range from an outward propagation with phase
speed c = a/k along 0 = 0 side to a simple radially-decaying oscillation along 0 = x side.
The forcing propagates away from the cylinder over all angular positions and it is expected
that the resulting wave motion will also be outwardly propagating but with significant local
standing wave components in the region of most intense forcing. Far from this region,
32
forced oscillations should again die out and the resulting wave potential is expected to
be composed of outwardly propagating free waves. While Molin (1979) assumed that the
asymptotic wave motions should have the same form as the forcing, it appears, instead,
that the wave motions generated by the nonuniform forcing fir should satisfy a form of the
radiation condition.
The physical arguments outlined above may be supported graphically by displaying the
amplitude envelope of the free surface forcing. In Figures 3.1 and 3.2, the amplitude of the
forcing is shown for the f" and f' forcing respectively, with a side view at the top of the
page, for 0 = 0, 9 = x, and the circumference of the cylinder, and a front view at the bottom
of each page, for 0 = x/2 and the circumference of the cylinder. For the f" forcing, the
amplitudes are clearly concentrated near the cylinder and are expected to generate waves
that radiate away from the cylinder.
For the f'* forcing, the behavior is quite different. Along 0 = r, the first-order scattered
wave opposes the incident wave to produce a standing wave system that, in turn, produces
large free surface forcing magnitudes. Along 9 = 0, however, the first-order scattered
wave propagates in the same direction as the incident wave with the same frequency and
wavenumber. The resulting quadratic forcing is small but also decays away from the cylinder
as the strength of the nonlinear interactions decreases. A general feature of the f" forcing
is that the amplitude of the forcing in the upwave region is larger than the forcing in the
downwave region. Around the cylinder, gradients are found in the forcing amplitudes, and
large forcing magnitudes are also found along the sides of the cylinder. Generally, waves
are generated near the front and sides of the cylinder, and then propagate both outward
and around to the rear. It is expected that due to the concentrated nature of the forcing,
and due to the spatial gradients of the forcing, free waves will radiate from the region of
most intense forcing.
The application of the radiation condition to the wave potentials generated by the
f* and f" forcing is supported implicitly by Lighthill (1979) and directly by Garrison
4.00
R/f
a a
ENVELOPE OF
9_. .F(SS) FORCING o
o
z.
LL.
8.00 6.00 '1.00 6.00 8.00_______.
8.00 6.00 .00 R
R/A
1.0000
1.5700
0.5000
Figure 3.1: Amplitude of f" quadratic forcing applied to free surface, (a) side view, (b)
front view
0.00
~1~
6.00 1.00
KA =
KD =
KH =
Ou7X
r -T T
- ENVELOPE OF
F IS) FORCING
86 0
a.oo 6.00 4.00 2.00 0.00 4.05 6.00 e.00
R/F /R/R
SI
S. .R 1.0000
a MRVE DIRECTION KO 1.S700
i KH = 0.5000 C
Figure 3.2: Amplitude of f"i quadratic forcing applied to free surface, (a) side view, (b)
front view
35
(1979). As noted, Garrison's solution is based on application of a Green's function, which
represents the wave motion generated by an individual source of unit strength and which
must satisfy a radiation condition. The associated source distribution function specifies the
strength, phasing, and spatial distribution of the unit sources; from Green's theorem the
source distribution function is given exactly by the quadratic forcing terms fi" and f".
From Green's theorem, the wave potential does not have the same form as the forcing, but
instead is composed of the contributions from infinitely many individual sources which may
reinforce or cancel depending on their spatial distribution and phasing but which lead to
outwardly radiating free wave motion in the far field. Since the radial decay of the forcing
varies over 0, the amplitude of the radiated free waves is expected to vary over 0 as well;
thus, the radiation condition cannot be applied in closed form to the total solution but,
rather, must be applied to each elemental point source.
3.4 Second-Order Solution
3.4.1 Solution for Scattered Potential #'
At first-order, the scattered wave solutions are obtained from the general set of eigen-
function solutions that satisfy the governing equation and the boundary conditions for the
region exterior to the cylinder and oscillating at frequency a. At second-order, the same
method will be employed to determine the homogeneous solutions; however, since all forcing
is found to oscillate at frequency 2<, the associated free wave motions must also oscillate
at frequency 2a as in Equation 3.7. The homogeneous form of the CFSBC in Equation 3.8,
which is satisfied by the second-order scattered free waves, is then
H- = (3.16)
As given by Mei (1983) or Dean and Dalrymple (1984), the solution to the Laplace
equation in cylindrical coordinates which satisfies the homogeneous free surface boundary
condition, the bottom boundary condition, the Sommerfeld radiation condition, and which
36
is symmetric in 0, is given by the complete set of eigenfunction solutions
4 = anocosh.K(d+ z)H,(cr)cosnO
n=0
+ anjcosrc1(d+ z)K,4(rcr)cosn0 (3.17)
n=0j=1
The second-order free wavenumbers associated with the homogeneous solutions are found
by substituting Equation 3.17 into the homogeneous CFSBC in Equation 3.16. Two second-
order dispersion relationships result: (1) for the propagating free waves
4o2 = gk2 tanh k2d (3.18)
which has a single positive root equal to the second-order free wavenumber, k2, and (2) for
the standing or evanescent wave modes
4o2 = -g9X2 tan K%2d (3.19)
which has infinitely many positive roots, x2i, given by
(j <) ,2ih _< jr (3.20)
The evanescent modes vanished identically at first-order since the depth dependence
of the scattered wave was identical to that of the incident waves; i.e. both incident and
scattered waves were free waves with wavenumber k. At second-order, the ambient or "in-
cident" forced waves have a more complicated vertical distribution of water particle motion
that differs from that of the freely propagating second-order scattered waves. This mismatch
along the cylinder boundary requires the existence of the evanescent modes at second-order,
in the same sense that the evanescent modes are required to satisfy the mismatch between
the motion of a vertical wavemaker paddle and the fluid motion of the waves which are
generated.
For later application, it will be useful to define a nondimensional form of the free wave
solutions. By normalizing by the amplitude of the first-order velocity potential as
S 2 (3.21)
37
and by defining normalized amplitude coefficients as
2a
n0 = -ano (3.22)
gH
2a
ni = ani (3.23)
the nondimensional homogeneous solution for the second-order scattered waves is given as
00
2 = Cnocosh k(d+ z)HH(k2r)cosn0
n=0
00 00
+ i Eaincos C2j(d+ z) K(1r2r)cos n (3.24)
n=Oj=1
3.4.2 Solution for Incident Potential 0i'pP
The solution for the forced wave motion generated by the f" forcing is well-known and
may be shown to be the usual second-order plane wave component obtained from the Stokes
perturbation expansion. From Equation A.60 in Appendix A.4, the free surface boundary
condition for the 4Pp component may be written as
42 423 gHk2 oo
p, p = g H (tanh kd 1) fJJn(2kr) cos nO (3.25)
9 -8 n=O
A solution is assumed for ',0p that satisfies the Laplace equation and the bottom boundary
condition as
00
p = A" cosh 2k(d+ z) fJ.n(2kr) cos n (3.26)
n=0
The amplitude A" is completely specified by the free surface boundary condition, and
substitution of Equation 3.26 into Equation 3.25 yields
A" = 3 gH2k2 (tanh2 kd 1)
8 a (2k tanh 2kd k2 tanh k2d) cosh 2kd (3
which is singular in shallow water when the free wave number, k2, equals 2k such that the
free surface forcing is resonant. The amplitude coefficient may also be written as
A" = -i 3 _H2 1 (3.28)
32 sinh4 kdh
38
The solution for 'iP, including the time dependence, may then be given as
.3 2cosh2k(d+ z) '0 e-ia
'p = -i1-H hk- / nJ.(2kr) cosnO- + c.. (3.29)
32 sinh4 kd "= 2
n=0
Using the partial wave expansion, e.g. Abramowitz and Stegun (1972)
00
E #Jn(2kr) cosn9 = ei2krco = ej2k' (3.30)
n=0
this becomes
Vi =3 2cosh2k(d+ z)
S= 32 hsik sin 2(kz ot) (3.31)
which is the familiar Stokes second-order incident plane wave component, e.g. Dean and
Dalrymple (1984, p. 301).
For later application, <'Zp is normalized by the amplitude of the first-order velocity
potential as
iP = 2 ,ip (3.32)
gH
If the form of the second-order amplitude A" given in Equation 3.27 is used, the nondimen-
sional form of O#p becomes
3 k(tanh2 kd 1) cosh 2k(d + z) T
qfp= -i-kH cash 2k(d Z Jn(2kr)cosn0 (3.33)
-4 (2k tanh 2kd k2 tanh kd) cosh 2kd sn (3.33)
3.4.3 Solutions for p and 4,*p
The solutions for #' and 0,p are based on a general solution for water waves due to a
pressure distribution applied to the free surface. First, the solution is obtained for a single
point source of pressure applied to the free surface. Next, the solution for an arbitrary
two-dimensional pressure distribution is obtained by summing the effects of elemental point
sources over the entire free surface. The quadratic forcing in the CFSBC is functionally
equivalent to an external pressure applied to the free surface; and, by specifying the form
of the pressure distribution, based on the derivation for the quadratic forcing in Appendix
A.2, several simplifications may be made to obtain a closed-form integral solution for i0~p
and ~j,.
Source Point
p(xy) 6(x-) 5 (y.-Y)
z=o ._T .. r -
Field Point
(x,y,O)
(r, 0 e,0)
x
z=-d
Figure 3.3: Definition sketch for point-source of applied surface pressure
Let the center of the vertical cylinder be at z = 0 and y = 0 in a global coordinate
system, where z = 0 defines the still water level. At some point on the free surface, x = (,
y = 7 and z = 0, let a point source of pressure oscillate at a frequency 2a with a magnitude
and spatial distribution defined by a Dirac delta function as P(z, y)S(x C)5(y 7), where
P(x, y) is a complex quantity. In global cylindrical coordinates, the point source is located
at (r', O',0). From the point of application of the source, define a local coordinate system,
(R,a,z), such that R is the radial distance from the point source to the field point (z,y,0)
as
R = (- )2 + (y 7)2) = (r2 + r12 2rr' cos(O 0')) 2 (3.34)
as defined in Figure 3.3.
40
The boundary value problem for the velocity potential resulting from this oscillating
point source is
V'2 = 0 (3.35)
= 0 on z = -d (3.36)
+,,+ = _a P(X,y)6(x-_)(y-7-) -2
+ c.c. on z = 0 (3.37)
lim Ri (PR ikA2) = 0 (3.38)
R-oo
The radiation condition may be easily written in closed form since it is applied with respect
to the point of application of the pressure, not the center of the cylinder.
A solution is sought in the form
(-i2)t -i2#t
4 (z, y, z, t) = OF (x, y, z)- + c.c. + OR(z, y, z) + c.c. (3.39)
where # represents forced wave motion while, as suggested by Wehausen and Laitone
(1960), #R is required to ensure that the total solution,
Substituting Equation 3.39 into the boundary value problem in Equations 3.35 to 3.38
gives two separate boundary value problems that are identical except for their free surface
boundary conditions
4O i2a
4 = 2P(z, y)5(x 0E)(y -Y) on z = 0 (3.40)
9 P9
and
RO- -4,2R 0 on z = 0 (3.41)
The radiation condition is applied to the total solution as
lim R2 [( + ) ik(F + f R)] = 0 (3.42)
R-oo
The solution for OF may be obtained by applying the two-dimensional Fourier transform
which is defined by the transform pair
41
F(~, v, z) = f f ,y, z)e-'+)dd (3.43)
F(Xy, z) = + + (00 ("V, z)e+i("+)dpdv (3.44)
Transforming the Laplace equation for OF gives a governing ordinary differential equation
as
S- ,F = 0 (3.45)
where
I2 = 02 + V2 (3.46)
Solutions to Equation 3.45 which also satisfy the transformed bottom boundary condi-
tion are
S(I Y, z) = A(, v) cosh n(d + z) (3.47)
Transforming the CFSBC, in Equation 3.40, gives
F 4 F = i2 P(,7) e-i(r.+) (3.48)
g9 pg 2r
Substitution of Equation 3.47, into Equation 3.48, determines the coefficient A(i, v) and,
after applying the inverse transform in Equation 3.44, the fundamental solution for OF is
obtained as
Fi2 P(, 7)
F= (2x)2
+00 +00 csh x(d + (349)
J-o J-o cosh icd(c tanh ,d k2 tanh k2d)
This solution may be expressed in simpler form, first by changing spatial variables to
local coordinates, then by changing the variables of integration as
S- = Rcosa 1 = K Cos
y 7 = Rsin a
V = Ksind
such that the solution is
~F(RIz; ) -L2u P(C, y)
pg (2r)2
+00oo +r cosh c(d + z) cinR co`(e-a)dedi (3.50)
o -i cosh #cd(c tanh ,'d k2 tanh k2d)
The identity, e.g Watson (1962)
Sei'CR.((-a)d$ = 2xJo(#cR) (3.51)
f4+r
may be used to obtain the simple result
F (R,a, z; ,,7) =
i2a P(,7)PV f+ cosh i(d + z)
pg 2ir cosh cd(c tanh rd k2 tanh k2d)
Because of the singularity of the integral at the free wavenumber, k2, the integral must be
interpreted in the Cauchy Principal Value sense, e.g. Wehausen and Laitone (1960).
The solution for the additional term, OR, required to satisfy the radiation condition
may be obtained from the general set of solutions which satisfy the Laplace equation, the
bottom boundary condition, the homogeneous free surface boundary condition, and which
are finite at R = 0. This solution is found as
SR(R, a,z) = Ccosh k2(d+ z)Jo(k2R) (3.53)
In order to establish the free coefficient, C, the entire solution, OF + OR, is required to
satisfy the radiation condition. To do this, the far field behavior of the forced wave motion,
OF, must be considered; this is accomplished by evaluating the singular integral for large
R using the Cauchy Integral Theorem. The desired integral may be obtained from the real
part of the complex integral with Jo(xR) replaced by the Hankel function, Ho(IR). In
addition, the singular function may be expanded about the singularity and leading order
terms retained as
x tanh Kd k2 tanh k2d (c k2) tanh k2d k2+ .c d +-
I cosh2 k2d]
x k2 [sinh 2k2d + 2k2d] + ... (3.54)
2 cosh2 k2d
~
43
Carrying out the contour integration, e.g. Wehausen and Laitone (1960, p. 477), the
asymptotic form for OF may be found as
F lim R [i2a P(_, 7) 2 cosh2 k2d 00 xccosh .(d + z) 1
R-.oo pg 2x (sinh2k2d+2k2d)Jo (K-k2)cosh dHo(R
i2a P(t,"t) f 2k2 cosh kd "
= PO 2- 1sinh 2kd + 2kcosh k(d + z)Yo(kR) (3.55)
In order to satisfy the radiation condition, the total solution, OJ + OR, must behave as an
outgoing free wave; thus, the value of the coefficient, C, in Equation 3.53 may be found as
= i2a P(t,7) 2k2coshkd (3.56)
pg 27 r sinh2kd+2kd (356)
The final solution for the velocity potential due to a single point source of strength,
P(z,y), oscillating at frequency 2a, is found by summing OF and OR to obtain
(R, az; ,,7) p= 2osinh 2k coshk2d cosh k(d+ z)Jo(k2R)
pg 29 sinh2k2d + 2k2d
Sscoshn(d + z) d O d e-i
+ f00 rK cJsh d(d+az) C2a
+ p cosh ~d( tanh xd kc tanh kzd) 2o(c
+ c.c. (3.57)
This elemental solution may be interpreted as follows. The singular integral represents
forced wave motion that is composed of contributions from all wavenumbers, X, and which
represents cylindrical standing waves. From the Cauchy Integral Theorem, the standing
waves have a strong local source-like behavior but, farther from the source, dispersion
selectively filters all wavenumbers such that only the contribution at k2, the second-order
free wavenumber, survives. This term is a cylindrical standing wave and, by itself, cannot
satisfy the radiation condition; therefore, a second standing wave solution, represented by
-OR, is required to ensure outward progressive wave motion. The solution is therefore a
combination of forced and radiated wave motions.
A mathematical description of the effects of a two-dimensional applied surface pressure
distribution is given by Wehausen and Laitone (1960, pp. 592-595). Application of the two-
dimensional Fourier transform to an arbitrary but transformable, i.e. sufficiently localized,
applied pressure distribution gives a two-term solution that represents local forced wave
44
motion, but with a selective radiation of an outwardly propagating free waves. As proved by
Hudimac (1958), the same result may be obtained by solving the more general problem of
an arbitrary distribution of wave sources anywhere in the fluid and then taking the limit as
the source distribution approaches the free surface. In this study, a more intuitive approach
is used, but the same result is obtained by summing an infinite number of point sources,
each with an elemental solution given in Equation 3.57, but with pressure amplitudes and
phases given by the quadratic forcing in the CFSBC. The result turns out to be equivalent
to that obtained by application of Green's theorem, as found by Garrison (1979). The
fundamental solution in Equation 3.57 is identical to the Green's function used by Garrison
(1979) to represent surface wave sources, in Equation 2.18, with a source strength of
i2o'
-P(,7) (3.58)
P9
The solution at a given field point due to a distribution of applied surface pressure is
obtained by a straightforward integration of the solution for q(R, a, z; C, 7) over the spatial
domain for all C and 7, as
f+OO +00
(R, a, z) = f, (R, a, z; 7)dd-y (3.59)
By introducing the change of variables of integration into Equation 3.57 as
z = rcos = r'cos 0'
y = rsin 0 7 = r' sin 0' (3.60)
and recognizing that the limits of the physical domain over which the forcing is applied are
a < r < oo and -x < 9 < 7, it is found that
1 i2a [. 2kz cosh kzd
S(r, ,z) = 2k2 d cosh k2(d + z)
2ir pg Pg sinh2ktd+2k2d
00 P(r', ')Jo(k2 r2 + 712 + 2rr' cos( O')r'dr'dO'
+ [0 J P(r',8O')PV cosh c(d + z)
-a -Jo cosh icd(; tanh Kd k2 tanh k2d)
SJo(oC/r2 + r,2 + 2rr'cos(O O'))r'dcdr'dO'] (3.61)
45
Simplifications may be achieved by first introducing the Addition Theorem for Bessel
functions, e.g. Watson (1962)
00
Jo(#/r + r2 + 2rr' cos(O 0')) = E c.J.(cr)J,,(Kr') cosn(0 0') (3.62)
n=0
where
Cn = 2 6.o (3.63)
Next, the applied pressure distribution may be related to the quadratic forcing terms in the
second-order CFSBC. In Appendix A.2, it is shown that the quadratic forcing may replace
the generic pressure such that
i2or 1 00
P(r, O') = f,(r') cos mO' (3.64)
P9 9 m=o
SThe solution may then be written as
1 r. 2k2 cosh kzd
(r,0, z) = --I cosh kz(d + z)
2;g smnh2k2d+2k2d
-" 0f1 00 00
r r E Z CcosmO' cos n(0 O')f,(r')Jn(k2r')Jn(k2r)r'dr'd0'
a n=0 m=0
+ E E cosEmnO'cos n(O 0') f, (r') do'
a -r n=O m=O
.PV c coshc(d+ z) Jn(.ir') J.(cr)r'dcdr' (3.65)
Scosh d(t tanh id k2 tanh kzd)
Carrying out the integration over 0', it is found that
f E en cosm0'cos n(O 0')dO' = 0 ifm n (3.66)
0=O{O 2n rEcos n if m = n
Finally, the solution #(r, 0,z), due to an arbitrarily distributed applied forcing in the CFSBC
is found to be
1 00 2k2 cosh k2dcosh k2(d + z)
0(r, 0, z) = cosn sinh 2kd + 2k Jn(k2r)
n= sinh 2k2d + 2kOd
f r'f,(r')Jn(k2r')dr'
+ loo ic cosh i(d + z)
o cosh Kd(r tanh id k2 tanh k2d)n
f r'f,(r') J.(nr')dr'di] (3.67)
*aloJ
46
The integrals over r' may be conveniently redefined as
D.(r) = L r f(r')J.(Pr')dr (3.68)
where Dn() represents a wavenumber spectrum for cylindrical wave motion. This term
is analogous to the usual spectrum obtained by the Fourier transform and is equal to the
Hankel transform of the applied quadratic forcing, e.g. Sneddon (1951). In order to obtain
the solutions 02p$ and b;,, corresponding to the wave potentials generated by the fi, and
f" forcing, the generic forcing in Equation 3.67 may be replaced by specific values of the
forcing. In this way the wavenumber spectra D*'( ) and DI'(K) may be defined individually
as
DZ'() = f r'f'(r')J.(ir')dr' (3.69)
D(ic) = r'f '(r')Jn(r')dr' (3.70)
Since the potentials O42, and q$p always have the same mathematical form, they will always
be taken together in this study with the single wavenumber spectrum D,(p), as
D,(x) = D"(c) + D,'(c) (3.71)
The solution for qp + j6, may be made nondimensional by dividing by the ampli-
tude of the first order potential. The wavenumber spectrum is normalized based on the
nondimensional forcing, e.g Appendix A.2, as
2ok 2ak *
2H Dn() = r'fn(r')Jn(or')dr' (3.72)
2 fk cr, (-ir9'k,.'H'
)= f -(r') J 1 .(r r')dr' (3.73)
9- a ( 80
-ikH= f k r' J(r')n(r')dr' (3.74)
4
.kH
= -i-b (x) (3.75)
The nondimensional particular solution for the forced-radiated wave motion ffP + 0q2 is
then given by
-, .kH r2 k2 cosh kzdcosh k(d+ z) ,(k)(k
'.p + = ---- icosn nh2O '^2r)kr2
An=o k sinh 2k2d + 2k2d
47
+ PV cosh x(d + z) J(cr)D,(K)d
Jo kcosh Kd( tanh d k2 tanh kd)
(3.76)
3.4.4 Summary of Forced Wave Motion
The solution for the portion of the second-order velocity potential generated by the
quadratic forcing in the CFSBC may now be summarized from Equations 3.33 and Equa-
tion 3.76. The nondimensional particular solutions are
00
ikH cos nO
n=0
[3 k(tanh' kd 1) cosh 2k(d + z) ,
4 (2k tanh 2kd k2 tanh kzd) cosh 2kd
is k2 cosh kzd cosh k2(d + z) D (k2)J,(kr)
+ 2k sinh2k2d+2k2d
1 /f00 _ecosh_(d+ z) (n r1
+ PV f 00 cosh P (. (it) J. (rT r) dr\
+4 J kcosh cd(n tanh d k2 tanh k2d)
(3.77)
3.5 Complete Solution Satisfying the No-Flow Condition
The solution to the complete second-order potential is determined by adding the homo-
geneous solution, in Equation 3.24, to the particular solutions for the forced wave motion,
in Equation 3.77. The unknown coefficients, Sno and ,,m in the homogeneous solution are
determined by requiring the total solution to satisfy the kinematic boundary condition on
the cylinder as
+r +2 = 0 (3.78)
Details of the derivation are presented in Appendix B, but the procedure is identical to
that at first order where the coefficients were determined by utilizing the orthogonality
properties of cosh k(d + z) and cos ic(d + z). With an and ai determined by Equations
B.4 and B.6 in Appendix B respectively, the homogeneous solutions are obtained as
00
SH = ikH Ecos ne
n=0
cosh kd coshk2d + z)Hkr) 6k(tanh2 kd- 1) J#(2ka)
sinh 2k2d + 2k2d (4k2 k) H',(k2a)
1 ro2 D2 (x)j '(a) i, k2 Jn(k2a)
+PV f ) i + 7j n(k2) (a
+kH,(k2a) o ( 2-k) 2 k H (k2a)
+os cK2d 2 +( (tanh2 kd 1) J'(2ka)
=1 + I 24k2 + Ij) Kn(rc-2a)
+ 1 # )' (nia) d 2 ) (3.79)
kK.f(x,,a) o ( 2'j+ dy)
The total solution for the second-order velocity potential may now be given in nondi-
mensional form, based on the sum of p +O Combining Equation 3.77 and Equation 3.79
and adding the time dependence gives the final solution for the second-order velocity po-
tential as
-i2at
S= -ikH-- cos nO
n=O
[{3 k(tanh2 kd- 1) cosh2k(d + z)J(2kr)
4 (2k tanh 2kd k2 tanh k2d) cosh 2kd
6k (tanh2 kd 1) cosh kd2cosh k2(d + z) J'(2ka) n(k)
(4k2 k2) sinh2k2d+ 2k2d HB(k2a)
S ,6k2 (tanh' kd 1) cos )sdcos zi,(d + z)n J (2ka) Kn(i
j 1 (4k' + c-.) sin 2c2d + 2zd K (2a)
"n RV iccosh K(d + z) )J )d
4+ o k cosh Pd(xC tanh cd k2 tanh k2d)
coshk2dcoshk2(d + z) H,,(k2r) /00 ,n,(c)J(ca)d
sinh 2k2d + 2k2d kHa(ka) Jo (<2 kj)
ScosS2idcos j Cj(d+ z) Kn(Oc2jr) f rx2 (iC)J'n(ra)
j=1 sin 2c2id+ 2C2id kK'(rc2ja)Jo (IC2 + Xcj)
ir k2 cosh k2dcosh k2(d + z) b (k)) J(k2a)
2 k sinh 2kzd + 2kid Hn(k2a)
+ c.c (3.80)
In order to discuss the physical features of the solution, it is first convenient to define
several nondimensional coefficients as
Scosh k2d (3.81)
sinh 2k2d + 2k2d
Ci= cos x2id (3.82)
sin 2r2yd + 2C2id
S3 k(tanh kd- 1) (3.83)
1 4 cosh 2kd(2ktanh2kd k2 tanh k2d)
= 6k2(tanh kd- 1) (3.84)
C'1o = 4k2 k2
C 6k(tanh kd 1) (3.85)
4k2 + (.
C ______(3.86)
2 = cosh xd(, tanh id k2tanh k2d)
420 = 00 x!)n.()J'C(ra)) d (3.87)
2j 2b. J'(a) dic (3.88)
Si k2 .(k2) (3.89)
2k
With this notation, the complete second order solution is
e-i2at
i2 = -ikH-- cos n
2 n=0
J,(2ka) r)
C cosh 2k(d + z),Jn(2kr) CoCo cosh k +(d + z)# H(k2a) H(k2r)
C j CiC cos c2(d + z)P# K'(Iia) K( j r
j=1
o II,(k2r)
+ {PV f ccosh (c(d + z)Jn(Kr)dlc Co120 cosh k2(d + ) kH,(k2a)
K,(i2yr)
Cl 2 cos C2(d +z) kK'(2+)
j=1
+ CCocoshk2(d+z) (Jkr)- J(ka) H(kr) +c.c. (3.90)
H,'C(k2a) ( ) i +.(3)
3.6 Discussion of Solution for 42
In the solution given in Equation 3.80 or Equation 3.90, terms enclosed in the first
bracket on the right hand side represent (1) the forced Stokes incident plane wave component
at wavenumber 2k and (2) the free cylindrical wave motion due to the scattering of the
Stokes plane wave. Although notation is different, these terms are identical to the partial
solution given in Equation 2.20 which was derived in a different way by Chen (1979, p.104)
and Chen and Hudspeth (1982) using the method of Green's functions. In deep water, the
terms, Ci, Cio, and Cli, vanish rapidly with increasing kd; this is in agreement with the
50
known behavior of the Stokes second-order plane wave velocity potential in deep water. In
shallow water, the second-order free wavenumber, k2, approaches 2k and resonant quadratic
forcing occurs in the CFSBC. Mathematically, C1 and Clo become singular in shallow water
and the solution "blows up" as expected.
The second bracket includes terms that represent the remaining forced-radiated wave
motion due to nonlinear wave-wave interactions through the forcing terms fi and f",
as well as free scattered waves which are due to the interaction of these forced-radiated
waves with the cylinder. While interpretation is more difficult, the forced wave motions
are composed of local cylindrical standing waves with amplitudes given by the wavenumber
spectrum, based on the integrated effects of the quadratic forcing. The associated scattered
wave motions are free cylindrical waves with an amplitude that is defined in terms of
contributions from all wavenumbers as well. The last components, with the coefficient
Cs in Equation 3.90, represent the "radiated" wave motions that are required to ensure
that the waves generated by fi" and f" satisfy a radiation condition. These motions must
also interact with the cylinder to produce additional scattered wave components.
In deep and shallow water, the behavior of the cylindrical forced-radiated waves, and
the associated scattered waves, is difficult to interpret explicitly. Since all wavenumbers
are present, some long wave motions may be in relatively shallow water even though the
dimensionless depth, kd, may be quite large. Some second-order motions associated with
the f" forcing have very slow depth decay and give the appearance of a "microseism." As
noted earlier, on the upwave side, the f/ forcing has the leading order form
f" a -e-i2 (3.91)
The resulting wave motion, together with other quadratic terms in the equation for the
second-order pressure, is a long standing wave with radial decay (and decay over 0), without
radial oscillation, but with temporal oscillation at frequency 2a. This is the cylindrical
analog to the microseism associated with plane standing waves.
51
As may be verified from the solution procedure, all governing equations and boundary
conditions are satisfied by the solution for 42. The boundary condition of most interest,
the nonhomogeneous CFSBC, is satisfied as follows. Remembering that all terms in the
solution that have cosh k2(d+z) or cos c2i(d+z) behavior satisfy the homogeneous CFSBC,
substitution of 02, from Equation 3.90, into the CFSBC, Equation 3.2 yields the condition
C1(2k tanh 2kd k2 tanh k2d) cosh 2kd#/J,,(2kr)
+ f -I -(, tanh id k2 tanh k2d) cosh odJ,(Kr)dK = + + 7 (3.92)
Jo k 4 +
From Appendix A.4, the forcing term f" is equal to
= 3(tanh2 kd 1)/,,J.(2kr) (3.93)
Based on the definition of C1 in Equation 3.83, the first term on the left hand side of
Equation 3.92 satisfies the first nonhomogeneous forcing term on the right hand side of
Equation 3.92. From the definition for C2 in Equation 3.86, the remaining condition is then
found to be
f ~bD,n,(n)J,,()d = '" + f," (3.94)
where substitution of the expression for D,,(x) yields
fj icJ,,(cr) f r' (f'(r') + '"(r')) J,'(Pcr')dr'dic = Jf(r) + f"(r) (3.95)
From the theorem of Hankel transforms, e.g. Watson (1962) or Sneddon (1951), it is verified
that the integrals on the left hand side represent those on the right hand side; thus the
CFSBC is satisfied.
The solution in Equation 3.90 is found to exhibit the irregularity in the overlap region
of the boundary conditions, at r = a and z = 0, as first observed by Isaacson (1977a). The
irregular nature may be confirmed by considering the equivalence of the following derivatives
in the limit as the overlap is approached from different directions
lim + ol +a g -(2)= (3.96)
Jim- [(2,,t + 02,i),o= + g ('), (3.96)
imr S-0 at2 'zj
52
The solution guarantees that the no-flow requirement is satisfied at r = a; therefore, the
right hand side is identically zero. Substitution of 02 into the CFSBC yields the result
($2:t + '2,),.=o R [e-i2 o nO f(r)] (3.97)
Thus, taking the derivative with respect to r and then taking the limit as r -- a gives
Isaacson's inequality in a different form
lim cosnO n afn(r) 0 (3.98)
T-**64 \ Qr )
n=0
3.7 Numerical Computation of 2
The theoretical second-order velocity potential is derived in closed integral form; how-
ever, numerical integration techniques are required for evaluation. The integrals that must
be evaluated may be separated into two types: (1) the integration from a < r < oo which
yields the wavenumber spectrum, &n(K) and (2) the integration over wavenumber space,
0 < I < oo, of three separate integrands, two of which have singularities at the free wavenum-
ber, k2. Various methods of computing the integrals are available and several methods
are discussed by Monacella (1966), Garrison (1978), Hunt and Baddour (1981), Hunt and
Williams (1982), and Rahman and Heaps (1983). In this study, all integrals are evaluated
with the Simpson's rule.
The r-integrations are difficult since both f,(r) and J,,(r) are oscillatory functions;
thus the integrand is oscillatory and sensitive to the cutoff limit on the upper limit of
integration. In lieu of carrying the integration out to exceedingly large distances in r, the
method used here is to replace the upper limit by an indefinite limit, r. The integration
may then be carried out from r = a to any distance, r yielding a function of r as
b.,(, r) = k' r fi(r'),(,,r')dr' (3.99)
The function, Dn(,nr), is oscillatory, but beyond several pile radii, any additional net
contribution to the integral is found to be very small. By applying a moving filter to
,,(1, r), the asymptotic value of the oscillatory integral can be obtained with a smaller
cutoff limit, Ro, as
D,(c) = lim w(r)D(i, r)dr (3.100)
r-R. Jr-R.
A triangular window
w(r)= 1 (3.101)
where R, equal to 5a has been used. A cutoff, R. = 15a, has been found to be accurate
for cylinder sizes up to ka = 1.5; for larger cylinder sizes the cutoff must be increased.
The evaluation of the i-integrals is surprisingly straightforward. The numerical inte-
gration of the Cauchy Principal Value integrals is performed by removing the singularity as
suggested by Monacella (1966, p. 246). The integrand is first written in terms of a general
function, g(,), as
I= P g() d (3.102)
Sn tanhd k tanh k2d
Then, the value of the integrand at the singularity is removed as
I= g()- dk) d, + g (k) PV 01 di
Jo i tanh Kcd k2 tanh k2d o ctanh d k tanh k2d
(3.103)
The behavior of the first integral is now regular across the singularity and may be integrated
directly. The second integrand is sufficiently linear near the singularity that in the Principal
Value sense, the positive and negative contributions on some small distance, E, either side
of the singularity cancel each other, and a contribution exists only for regions away from
the singularity. The singularity in the integrand, at x = k2, always occurs between k2 = 2k
at the shallow water limit and k2 = 4k at the deep water limit. The contribution to the
integrals is found to be most significant from n = 0 to beyond K = 4k; a cutoff wavenumber
K:e = 12k was found to be required for adequate convergence.
A final numerical feature of the theoretical solution is the appearance of the Gibb's
phenomenon, e.g. Courant and Hilbert (1961), in the numerical transforms involved in
fitting the CFSBC. As noted, substituting the solution for 42 into the CFSBC yields the
Hankel transform of the quadratic free surface forcing as given in Equation 3.95. While
54
this transform identity is generally true, at points of discontinuity in the forcing, f,(r), the
transform can only recover the mean value of the function from either side of the jump. In
this case, the transform method is applied to the quadratic forcing from r = a to r = R,,
with the forcing being zero inside the cylinder; i.e. there is no wave-generating forcing
applied to the free surface inside the cylinder. At r = a, the forcing experiences a jump
discontinuity from zero at r = a- to f,(a) at r = a+, and the Gibb's phenomenon occurs
at the point of most interest in the solution.
An example of this phenomenon is depicted in Figure 3.4 for one order in the solution,
n = 1. The amplitude of the quadratic forcing is shown in the solid curve. By application of
the integral solution, the wavenumber spectrum, b.n(i) is first computed, then the solution
is substituted into the CFSBC. From the Hankel transform identity in Equation 3.95, the
exact form of the quadratic forcing should be recovered. It is found instead that due
to numerically performing a continuous transform with discrete calculations over a finite
interval, there is an error in the reconstructed forcing as
,(r) = f(r) j 00 f .n()J.(cr)d (3.104)
In Figure 3.4, the reconstructed forcing is shown in the dashed line and clearly displays
the Gibb's phenomenon where one-half of the value of the function is recovered at r = a,
while the approximation then overshoots and undershoots the exact value while the error
decreases as r increases.
An intuitive procedure has been used to reduce the effects of the Gibb's phenomena.
The method is based on the repeated application of the transform method to successively
obtain better reconstructions of the quadratic forcing and the velocity potential. From the
initial approximation of the solution, an error is found in the reconstruction of the CFSBC
as in Equation 3.104. This error in the quadratic forcing must be accounted for to obtain an
exact fitting of the CFSBC; therefore, a correction to the wavenumber spectrum, ALin(),
may be obtained by transforming the error as
Ab(5) = k 2 r'n(r')Jn(r')dr' (3.105)
Ja
0
CFSBC EXRCT
Uj0
SP- CFSBC APPROX
- 1 ITERATION
-i
z
--
C. 00 .00 2.00 Y.00 4.00 5.00 6.00 7.00 e.o00 9.00 I'
R/A
Figure 3.4: Amplitude of "' forcing with Gibb's phenomenon after one iteration
o
0
1.0 2_0 3.00 '4.00 5.00 6.00 7.00 8.00 9.00 I_
yR/
Fiue34 mltd fjafrigwt ibspeoeo fe n trto
U,
gf
2.00 3.00 '.oo s.oo
R/R
6.0o 7.00 6.00 9.00
Figure 3.5: Amplitude of f*' forcing with reduced Gibb's phenomenon after 50 iterations
CFSBC EXRCT
CFSBC APPROX
50 ITERATIONS
.o00
1.o00
I .. I
, -
).00
O
O
56
This correction to the wavenumber spectrum may then be added to the initial estimate of
the wavenumber spectrum, to obtain a better approximation as
b() = bn(c) + ADb(sc) (3.106)
By applying the inverse transform using the new wavenumber spectrum, a better fitting of
the CFSBC is achieved with a new smaller error as
.(r) = f.(r) (cD()J,((r)de (3.107)
By successively applying the transform to each error term, additional corrections are
obtained for the wavenumber spectrum. In Figure 3.5, after, 50 iterations, the CFSBC is
satisfied almost exactly and the effects of the Gibb's phenomenon are nearly eliminated.
From repeated numerical testing, the effects of the Gibb's phenomenon are of importance
near the cylinder, as expected; but, differences between the solution after, say 5 iterations
and 50 iterations are typically only a few percent. Other proposed solutions that have been
based on a Fourier-Bessel integral, such as those of Garrison (1979), Hunt and Baddour
(1981), Hunt and Williams (1982), and Rahman and Heaps (1983), certainly contain the
same Gibb's phenomena problem; however, none of the authors mention its existence.
CHAPTER 4
FREE SURFACE ELEVATIONS AND WAVE RUNUP
4.1 Introduction
One of the primary goals of this study is to determine the second-order water surface
elevations surrounding a large circular cylinder. Despite all of the previous studies of the
second-order velocity potential and the resulting wave forces, only Raman and Venkata-
narasaiah (1976b), Raman et al. (1977), and Chakrabarti (1978) have applied second-order
solutions for P2 to obtain the second-order water surface; and, they only considered the
runup distribution around the cylinder for 2 to 4 test cases. No previous solutions have
been used to calculate the three-dimensional wave field around the pile.
The water surface elevation is obtained from the dynamic free surface boundary condi-
tion, Equation 1.10, as
.r = I(V) on z = t (4.1)
pg g 2g g
which is an unsteady form of the Bernoulli equation. Based on the perturbation expansion
technique, 4 and rj are expanded in power series form, and Equation 4.1 is expanded in a
Taylor series about z = 0. The water surface consistent to the second perturbation order
may then be obtained as
1 1 1 1 1
171 + ,72 = -_ u Z2* + 9171Itz I + -a + 7 (4.2)
g g g 2g r i
where the external atmospheric pressure is taken to be zero.
At first order, the water surface is given by one term
171 = --'i on z = 0 (4.3)
9
58
such that the linear free surface depends only on the unsteady dynamic pressure associated
with local accelerations of the linear incident and scattered waves. The second-order free
surface elevation is found from
r72 = 91' -- 1 [(,r) + ( t)2 + (1. 92 on z = 0 (4.4)
g 2g r J
At second order, the free surface elevation is given by two types of components: (1) quadratic
terms that depend on products of derivatives of the first-order potential, associated with
convective accelerations of the first-order fluid motion as well as the free surface displacement
and (2) the local time derivative of the second-order velocity potential associated with the
local accelerations of the second-order motions.
4.2 First-Order Water Surface
Substitution of the solution for $1 in Equation 1.55, into Equation 4.3, yields the first-
order linear wave field as
H L J' (ka) e t
S= -(k) H' (k) H,(kr) cos n + c.c. (4.5)
ZH,'(ka) 2 + 2
n=0 J
The nondimensional water surface may be obtained by normalizing r17 by the incident wave
amplitude as
S 2r7
tl = H (4.6)
to give
71 = p, J.(kr) ) H(kr) cosn0-- + c.c. (4.7)
n=0
The incident and scattered waves are both represented by complex quantities that depend
on the relative cylinder radius ka; therefore, a phase shift is introduced between the incident
and scattered waves that also depends on the size of the cylinder relative to the wavelength.
At r = a, a useful identity may be introduced, e.g. Abramowitz and Stegun (1972)
J' (ka) 2i 1
J(kr)) H(kr) = (4.8)
H (ka) srka H'(ka)
59
The nondimensional free surface elevation around the circumference of the cylinder, qic, is
then
2 01 -47
S(2 Sno)+- cos ne 2 + -c.c (4.9)
(2ka H,(ka) 2
n=0 n
The runup and rundown on the cylinder may be defined as the maximum and minimum
free surface elevations that occur over one wave period at a given angular position, i.e.
the maxima and minima of the wave envelope at r = a. Linear runup profiles around the
circumference of the cylinder for various values of ka are depicted in Figure 4.1. The runup
is greatest at the leading edge of the cylinder and decreases to a minimum between 30 and
60 degrees, or slightly aft of the shoulder; this is shifted farther to the rear as ka is increased.
For small piles, runup amplitudes at the rear of the pile approaches unity, while for larger
piles, the runup amplitude at the rear of the pile decreases as the sheltering effects of the
pile become more important.
The behavior of the maximum runup at the front of the cylinder over the range of
cylinder sizes may be represented by a single curve as a function of ka as shown in Figure 4.2.
For small piles, with ka < 0.2, the incident wave crest is not significantly interrupted by
the pile. As the cylinder radius increases, the cylinder reflects more of the incident wave
energy and an antinode of a partial standing wave system forms at the front of the cylinder
giving larger runup. With increased wave reflection, the maximum wave runup amplitude
approaches twice that of the incident wave.
In Figure 4.3, a vertical cross-section of the free surface region is depicted to show the
linear wave envelope in the upwave and downwave regions, and to show a projection of the
wave envelope around the circumference of the cylinder, for a reference case with ka = 1.0.
The linear wave field is observed to be a partial standing wave system on the upwave side,
with antinodes and nodes that become less pronounced away from the cylinder. On the
downwave side, few diffraction effects are evident. In this region the incident and scattered
wave are propagating in nearly the same direction and produce no modulated wave envelope,
unless ka is very large. Since 41 is composed of terms at frequency a only, the linear water
60
0
ci
-----.... -- KR = 0.25
S ......... Kf 0.50
KR = 1.00
--. KFI 2.00
.. ........
x
\
0
a
o
01o0.00 162.00 1t4.00 126.00 108.00 90.00 .07.0 54.00 36.00 10.00 0.00
ANGLE
Figure 4.1: Runup profiles from linear theory for various values of ka.
0.20 0.o0 0.60 0.80 1.00 1.20 1.40 1.60 1.00
KR
Figure 4.2: Maximum wave runup as a function of diffraction parameter, ka.
-- LINEAR THEORY
Co
UJ 8.00 6.00 k'.00 2.00 0o.00 2.00 4.00 6s.oo e'.00
x R/A R/A
T n
0 0
0 O0
KRA 1.0000 KO 1.5700 KH 0.5000
HAVE DIRECTION
Figure 4.3: Envelope of first-order free surface along axis of wave propagation, ka = 1.0
surface oscillates about z = 0 with equal trough and crest amplitudes.
In Figure 4.4, contours of equal wave amplitude are shown in planform for an area
extending out 7 to 10 times the pile radius for the same reference case. The partial standing
wave system on the upwave side is observed to be fairly uniform along the x axis and decays
gradually as 0 decreases and as r increases. In the downwave region, diffraction effects
are now evident as simple zones of constructive and destructive wave interference. The
most evident diffraction effects are at approximately 30* to 60. In the region behind the
cylinder, the linear theory does not predict any spatially periodic amplitude modulation,
but nondimensional wave amplitudes are reduced at r = a and approach unity as r increases.
Figure 4.4: Contours of first-order wave crest envelope, ka = 1.0
63
Based on the linear solution, some insight may be gained into the nature of the quadratic
free surface forcing which, in turn, generates the second-order wave field. On the upwave
side, the linear scattered waves oppose the incident plane waves and a partial standing
wave field is developed. This region is reflection-dominated, and quadratic interactions of
the incident and scattered linear waves are strong such that this region serves as a generating
area for many second-order wave components. Along the sides of the cylinder, a node forms
in the first-order wave envelope and large quadratic pressures are found near to the cylinder.
In the downwave region, the linear scattered wave propagates in much the same direction
as the incident wave and the linear wave envelope is not strongly modulated. The quadratic
forcing in the CFSBC is weak; and, rather than being a wave generation area, wave motions
in this region originate largely from the front and side regions. The downwave region is,
therefore, diffraction-dominated.
4.3 Quadratic Components of the Second-Order Water Surface
From Equation 4.4, the quadratic terms in the second-order free surface are
1721 = + G ltlt 2 [ lr (1 )2 + (41p)2] (4.10)
In Appendix A.6, these terms are reduced to simpler forms suitable for numerical computa-
tion. From Equation A.88, the reduced form of the nondimensional quadratic free surface
terms, 21j, is found to be
kH H o
S= 8 tanh kd (r) os n
n=0
kH 00 e-i2a
+ (r) cos nO + c.c. (4.11)
+t ianhnkd .,.2
n=O0
The radial functions G,(r) and G((r) are defined as in Equations A.89 and A.90 in Appendix
A.6. Features of i21 are (1) the second-order quadratic components include terms that
oscillate in time at frequency 2
second-order quadratic terms depend on all three nondimensional parameters, ka, kd, and
kH; the dependence on ka and kd is incorporated in the spatial functions G,(r) and GO(r).
4.3.1 Mean Water Levels
The steady components of 121 are
kH
q2H = 8 t k G(r) cos n0 (4.12)
8 tanh kd F c
n=0
where G<(r) is a real quantity. This term includes (1) a uniform setdown due to the self-
interaction terms from the first-order incident plane wave, as well as (2) spatially varying
mean water levels due to the cross and self-interactions of the first-order incident and
scattered waves. In Equation A.85 from Appendix A.6, if no structure is present, then 12
may be shown to reduce to
kH
q21= W (4.13)
8 sinh 2kd13
which is the expected second-order uniform setdown, e.g Dean and Dalrymple (1984, p.
302).
In Figure 4.5, a vertical cross-section of the mean water surface is given to show the
steady water levels on the upwave and downwave side as well as around the circumference
of the cylinder. This figure is based on a set of reference conditions with the dimensionless
radius ka = 1.0, the dimensionless depth kd = 1.57, and the wave steepness kH = 0.5.
In Figure 4.6, the mean water level contours are shown in planform for the same wave
conditions.
General features of the second-order mean water levels are
1. On the upwave side, there is a spatially varying setup and setdown pattern related to
the first-order partial standing wave system. The setup-setdown pattern is completely
analogous to that observed in front of a plane barrier due to standing plane waves,
except that in the present case the amplitudes of the mean water level variations
decrease as r2 and with 0. This "corrugated" mean water level is superimposed on
the usual uniform setdown associated with the incident plane wave.
2. On the downwave side, the mean water level varies monotonically from a small net
setup at the rear of the cylinder to a uniform setdown farther away. The first-order
t
!
./
/
/t \ / \ /
f~: :
'". ." i *
/
00 ,.,v ,,
-
^-----1.4o e Roo-,, 'oo
R/A
N \
:
t~ .j\
~~\
-........ LINEAR THEORY
- SECOND ORDER HERN
i
t
i
i
t
;n
~
'oo/ z2.0oo
~~
I;
j
i
i
i
-f
i
i
4.00
6.00 8.00
KR 1.0000
Figure 4.5: Second-order mean
kd = 1.57, and kH = 0.5
KO 1.5700
WAVE DIRECTION
water levels along z axis for reference case, ka = 1.0,
wave field does not exhibit nodes or antinodes; therefore, the second-order mean water
level does not have spatial periodicity.
3. On the sides of the cylinder, a smooth transition occurs from maximum setup at the
front, to maximum setdown aft of the shoulder, to a secondary setup at the rear. The
large setdown near the shoulder of the cylinder is quite localized and is due to the low
pressures associated with the node of the first-order wave envelope at the shoulder.
KH 0.5000
-
vl
R/A
Figure 4.6: Contours of second-order mean water level for reference case, ka = 1.0,
kd = 1.57, and kH = 0.5
4.3.2 Oscillatory Quadratic Free Surface
The time-dependent terms in 721 are
kH 00 e-i2at
S 8 tanh kd G r) cos n + .c. (4.14)
where G7(r) is a complex quantity. Like the steady water level term, the oscillatory term
consists of components due to the nonlinear self and cross-interactions of the first-order
incident and scattered waves. If no structure is present, then the plane wave components
of qI may be algebraically reduced to the form
2 = kd[3 tanh' kd 1] cos2(k at) (4.15)
8 tanh kd
which is presented in Appendix A.6. In a later section, this expression will be combined
with the plane wave component from the second-order velocity potential to verify that the
series form in Equation 4.14 yields the familiar Stokes second-order plane wave component.
In Figure 4.7, an example of the wave envelope due to the second-order quadratic terms
is given for the same wave conditions used to illustrate the second-order mean water levels
in Figure 4.5. General features of the second-order oscillatory quadratic water levels are
1. On the upwave side, the wave envelope for the oscillatory quadratic components is sim-
ilar in form to the envelope of the first-order terms. One component of the quadratic
wave envelope is the usual second-order plane wave, given in Equation 4.15. The re-
maining nonlinear interactions of the first-order incident and scattered waves produce
a second-order partial standing wave system with antinodes and nodes located at the
same spatial positions as those of the first-order partial standing wave system.
2. On the down-wave side, the second-order quadratic envelope is similar to the first-
order envelope, and represents a near-plane wave at wavenumber 2k with no spatial
modulation. The amplitude in this region is almost identical to that obtained from
two-dimensional Stokes theory; nonlinearity associated with the first-order scattered
wave is usually small in this region.
~
KA a 1.0000 KO = 1.5700 KH = 0.5000
WAVE DIRECTION
Figure 4.7: Envelope of second-order oscillatory quadratic components, ka = 1.0, kd = 1.57,
and kH = 0.5
69
3. On the sides of the cylinder, a smooth transition occurs from maximum amplitude at
the front, to minimum amplitude aft of the shoulder, to a secondary maximum at the
rear. The node of the second-order envelope is in the same position as the node in
the linear envelope.
When the oscillatory quadratic terms are added to the first-order solution, also shown in
Figure 4.7, it is evident they are locked in phase relative to the first-order waves. Since the
second-order terms oscillate at twice the frequency, they contribute to increase the wave crest
amplitudes compared to first-order while they decrease the wave trough amplitudes. These
effects are analogous to the phase-locked effects of the second-order plane wave components
from the usual Stokes plane wave theory. Quadratic free surface terms tend to substantially
increase wave runup and crest elevations compared to first-order.
4.4 Free Surface from Second-Order Velocity Potential
The second-order free surface components from the second-order velocity potential, '2,
are given in Equation 4.4 as
72: = -1 -2 on z = 0 (4.16)
g
Evaluating the derivative and normalizing by the first-order wave amplitude, the nondimen-
sional form of Tr22 is
'722 = 2 = 2i12 on z = 0 (4.17)
H
From Equation 4.17 and from the equation for 02 given as Equation 3.80 in Chapter 3, the
nondimensional free surface from the second-order velocity potential is
-i2ot oo
i22 = kH-- cos n
n=O
K3 k(tanh2 kd 1)
2 (2k tanh2kd k2 tanh kzd)
k (tanh2 kd 1) cosh2 k2d J(2ka)
(4k2 kj) (sinh 2kd + 2k2d) H'(k2a)
S k(tanh2 kd 1) cos2 id nJ,(2ka) Kn( )
j=1 (4k2 + ~ ) (sin 2r,2id + 2K2id) K' (I21a)
j=1 2i
1+ 1 PVr, n(c) J, (r) _
12 Jo k(C tanh kd kz tanh k2d)
cosh2 k2d H(k2r) O rD2n(.()JI(xKa)
(sinh 2k2d + 2kd) kH(k2a) a ( ka)
Scossc21id K,(r-2r) f- 2b,(tc)Ji(#ca)
S(sin 2K2id + 2cz2id) kK(c2ia) Jo (c2 + i,)
j=2
k, COSh2' kd J, (k2) 1
+ ik2 coshk2d (k) J(k2r) J(k 2a)H(kr)
k (sinh2kcd + 2kzd) H( ,(k2a) j
+ c.c. (4.18)
The terms in the first bracket on the right-hand side in Equation 4.18 represent the con-
tribution from the incident Stokes second-order plane wave as well as its associated scattered
wave components. If no structure were present, the only non-zero term in Equation 4.18
would be the leading term in the first bracket
[ 3 k(tanh 2kd- 1) 00
q22 = 3R kH (2k ta 2k -d) ,Jn(2kr) cos nOe-i2' (4.19)
2 (2k tanh 2kd k2 tanh k2) n=O
which may also be written as
3 (tanh2 kd 1)
2 = kH (tanhkd- cos 2(k at) (4.20)
4 (tanh 2kd 2 tanh kd)
Combining Equation 4.20, the plane-wave term from the second-order velocity potential,
with Equation 4.15, the plane-wave quadratic component fij, yields
kH cosh kd(2 + cosh 2kd) c -
S= cos 2(kx at) (4.21)
8 sinh3 kd
which is verified to be the usual Stokes second-order plane wave component at wavenumber
2k, e.g. Dean and Dalrymple (1984, p. 303). This plane wave term is always phase-locked
to the first-order incident wave, while its associated scattered components, especially the
outwardly scattered free wave at wavenumber k2, are shifted in phase with a phase shift
that depends on the relative cylinder radius.
The terms in the second bracket represent the remaining forced-radiated wave motions
and their associated scattered waves. These terms are all related to the presence of the
first-order scattered wave and may be traced to both f"' and f" forcing in the second-
order CFSBC. It should be remembered that all of these terms yield local standing wave
71
motion in the vicinity of the free surface quadratic forcing, but far away, give an outwardly
propagating wave motion with wavenumber k2. All terms, the forced-radiated motions and
the scattered motions, become more important as the cylinder size increases.
In Figure 4.8, the wave envelope from the second-order velocity potential is depicted
for the reference wave conditions ka = 1.0, kd = 1.57 and ikH = 0.5. General features of
the free surface components from the second-order velocity potential are as follows
1. Along 8 = r, the free surface contribution from the second-order velocity potential
appears as a complicated partial standing wave system with generally decreasing am-
plitude as r increases. The asymptotic behavior of these terms is a combination of
the uniform amplitude incident plane wave component, a standing long wave, and
outwardly propagating free waves.
2. Along 8 = 0, the direct quadratic forcing applied to the free surface is small; however,
a substantial outwardly propagating wave field is predicted. This is a manifestation
of diffraction effects in which waves are generated by the strong forcing along 0 = W
but then propagate around the cylinder into the shadow region. Since the waves that
propagate outward from the region of large quadratic forcing are second-order free
waves, the dominant wavenumber in the diffraction-dominated region is k2.
3. Around the cylinder, the wave amplitudes from the second-order velocity potential
are more complicated. One feature of the second-order velocity potential is that large
wave amplitudes occur along the shoulders of the cylinder, at the location of the node
in the first-order envelope. The quadratic forcing on the free surface is large in this
area due to the convective accelerations from the first-order flow field, and this acts
as a generating mechanism for second-order waves.
The free surface terms from the second-order potential are added to the first-order
terms in Figure 4.8 to investigate the phasing relationship. Except for the incident plane
wave term, the remaining components of the second-order velocity potential are mostly
KA = 1.0000 Ko = 1.5700 KH = 0.5000
HWVE DIRECTION
Figure 4.8: Envelope from second-order velocity potential and effect of second-order poten-
tial on free surface, ka = 1.0, kd = 1.57, and kH = 0.5
73
out-of-phase with the first-order solution. Along 0 = x, the second-order terms are strongly
out-of-phase near the cylinder such that the trough amplitudes are accentuated while the
crest amplitudes are reduced. Comparing Figures 4.7 and 4.8, it is found that the oscillatory
quadratic terms and the components from the second-order potential are generally out of
phase near the cylinder and will partially cancel each other. As a general conclusion, the
second-order velocity potential is not exactly out-of-phase with the quadratic second-order
terms or the linear solution; but it seems to always act to reduce the overall second-order
effects.
On the down-wave side, the combination of free surface terms from the second-order
potential and the terms from the first-order potential leads to a substantial change in the
form of the free surface envelope, unlike the effects of the oscillatory quadratic terms. The
wave envelope in this region is characteristic of a wave system in which a free second
harmonic, with wavenumber k2 and frequency 2a, is added to a first-order wave, with
wavenumber k and frequency o, and its bound second harmonic at wavenumber 2k and
frequency 2a, e.g. Buhr Hansen and Svendsen (1974).
4.5 Free Surface Complete to Second-Order
The complete theoretical solution for the free surface, consistent to the second pertur-
bation order, is obtained by superimposing the following components
1. The first-order solution, i1, given in Equation 4.7.
2. The second-order quadratic mean water levels, i1, given in Equation 4.12.
3. The second-order oscillatory quadratic terms, 11, given in Equation 4.14.
4. The terms from the second-order velocity potential, i22, given in Equation 4.18.
For the reference conditions, ka = 1.0, kd = 1.57, and kH = 0.5, the cross-section
of the total wave envelope to second-order is given in Figure 4.9. General features of the
complete solution are as follows
KA = 1.0000 KO = 1.5700 KH = 0.5000
WAVE DIRECTION
Figure 4.9: Total wave envelope to second-order for reference case ka = 1.0, kd = 1.57, and
kH = 0.5
75
1. All second-order oscillatory terms are complex quantities such that phase shifts exist
between the various components. The plane wave terms are phase-locked to the first-
order wave, while the quadratic terms also generally in phase, especially near the
cylinder. The forced-radiated waves and their scattered components, in the second-
order potential, are generally out-of-phase with the first-order solution. As a result,
partial cancellation occurs to reduce the net second-order contribution to the total
wave envelope.
2. Along 0 = 7r, the partial standing wave system at first-order is largely preserved; how-
ever, wave crest amplitudes are increased and trough amplitudes are decreased. For
smaller values of ka, additional nodes and antinodes associated with the second-order
velocity potential components may be introduced (see Appendix E). The maximum
wave runup at the cylinder is increased substantially, and in this example the max-
imum crest amplitude also exceeds twice that of the linear crest amplitude; this is
partly due to the increased incident crest heights at second-order and partly due to
the nonlinear reflection process. Maximum wave rundown on the cylinder is reduced
relative to the linear theory and, in general, trough amplitudes become more uniform
everywhere.
3. Along 0 = 0, the second-order terms alter the diffraction pattern significantly to
produce a modulated wave envelope, unlike that obtained from linear theory. Again,
crest amplitudes are generally increased while trough amplitudes are decreased. For
larger values of ka, like the example shown, the envelope will be modulated due to
significant diffracted and scattered second-order free waves. For smaller cylinder sizes,
the scattered and diffracted terms are small. Plane wave and oscillatory quadratic
terms dominate at second-order such that the resulting envelope will not be strongly
modulated, although crest elevations will still be increased. The maximum runup at
the rear of the cylinder is increased due to second-order diffraction. Maximum wave
rundown may be greater or less than that at first-order.
76
4. Around the circumference of the cylinder, the wave runup profile is generally increased
relative to the linear solution, except that for small cylinders, runup is often less than
that predicted by linear theory in the region between 30 and 60 degrees. Runup
maxima at both the front and back of the cylinder are much greater than the linear
maxima, a feature that is in qualitative agreement with laboratory data. Wave run-
down is more complicated and rundown amplitudes are generally smaller than those
predicted by linear theory, except at the position of the first-order node. Much of
the departure from linear theory at this point seems to be due to the second-order
mean water levels which, according to potential flow theory, experience their largest
setdown in this region.
In Figure 4.10, the total wave crest envelope to second-order is shown in planform for
the example case; and, by comparing Figure 4.10 to Figure 4.4, differences in the first and
second-order theories may be identified. As second-order waves radiate and scatter away
from the cylinder, the interaction of second-order crests with the first-order crests occurs at
different distances from the cylinder at each angular position. This leads to more localized
and isolated maxima and mimima; this is more dramatic for somewhat smaller cylinder
sizes as observed in Appendix E. The most striking differences between linear and nonlinear
theories is in the diffraction pattern at the rear of the cylinder. While linear theory contained
only a broad region with nearly uniform amplitude in the downwave region, second-order
theory predicts distinct and localized maxima and minima due to nonlinear diffraction, with
locations where the crest amplitudes are nearly 50 percent smaller and 50 percent larger
than the incident wave amplitude.
As a more detailed illustration of the theoretical second-order effects, Figures 4.11
through 4.12 present an oblique view of the wave field for two wave phases before and at
the phase of maximum runup. In Figure 4.11, the cylinder is in the wave trough and a wave
crest is approaching from the left while a previous wave crest moves away from the cylinder
to the right. The incident crest is distorted by scattered waves such that its height is not
Figure 4.10: Contours of total wave crest amplitude to second-order for reference case
ka = 1.0, kd = 1.57, and kH = 0.5
Figure 4.11: Oblique view of wave field near cylinder to second- order, phase of maximum
runup-xr/2, for reference case ka = 1.0, kd = 1.57, and kH = 0.57
uniform along the crest but is lower along the 0 = 7 axis as it passes into a node of the
wave envelope. The wave form is asymmetrical as the face of this wave in the direction of
propagation is steepened while the back side of the wave is elongated by the superposition of
the scattered waves. The previous crest is also highly distorted, with significant amplitude
variation along the crest due to the second-order scattered and diffracted waves. This crest
is no longer straight but is bowed behind the cylinder due to diffraction of the wave crests
from each side.
In Figure 4.12, the wave crest is at the phase of maximum wave runup. The crest runup
is highly localized on the front of the cylinder. Water surface slopes along the shoulders
of the cylinder are near their maximum and, the solution shows that water particles near
the cylinder moves faster than those in the wave crest farther away. For very steep incident
waves, localized breaking is often observed at or near this phase position. Second-order
free waves are evident in both figures moving away from the cylinder. These are not too
pronounced on the upwave side as second-order motion is dominated by long standing waves
Figure 4.12: Oblique view of wave field near cylinder to second- order, phase of maximum
runup, for reference case ka = 1.0, kd = 1.57, and kH = 0.5
for large cylinders. In the downwave direction, the radiated-scattered free waves are quite
significant; it is also evident that their phase speeds are much slower than linear wave. For
smaller relative depths, the phase speed of these waves approaches the linear phase speed as
the free wavenumber, k2, approaches 2k.
4.6 Summary of Nonlinear free surface
The reference case cited above serves as a useful example of the principal features of the
second-order solution; however, the second-order free surface is dependent on the relative
magnitudes and phases of several independent components, which vary with the cylinder
size, ka, and the relative depth, kd. In Figures 4.13 through 4.16, examples are given for
the free surface envelopes for the following additional cases (1) large cylinder, intermediate
depth, in Figure 4.13, (2) small cylinder, intermediate depth, in Figure 4.14, (3) deep water,
moderate cylinder size, in Figure 4.15, arid (4) near-shallow water, moderate cylinder size,
in Figure 4.16.
80
Varying the pile radius, ka, for a given water depth and wave steepness has a great
effect on the second-order solution. In Figure 4.13, large cylinders are associated with
more pronounced scattered wave components at first-order and; therefore, larger nonlinear
quadratic pressure terms in the CFSBC and the Bernoulli equation on the upwave side.
This is manifested in (1) larger spatially varying mean water levels (2) larger quadratic
corrections to the free surface due to increased convective fluid accelerations, and (3) larger
contributions from the second-order velocity potential. Due to the sheltering effects of large
cylinders, the wave amplitudes on the rear of the cylinder also vary greatly with ka; for ka
greater than about 0.8, wave amplitudes decrease in the downwave region.
In Figure 4.14, small cylinders do not disturb the linear wave field significantly and the
resulting wave envelope has little modulation anywhere. The wave crest elevations are nearly
uniformly increased in the front and rear; however, the wave crests are locally disturbed
at the cylinder, leading to second-order runup elevations that are larger than would be
expected simply from the increased nonlinear incident crest height. Plane wave terms in
the solution are not effected by the cylinder size. However, as these plane waves reflect from
the cylinder, their resulting scattered components introduce additional modulations in the
wave envelope. These scattered effects are most readily observed for small cylinders where
the other forced-radiated motions are not as large.
The behavior of the nonlinear theory from deep to shallow water is depicted in Fig-
ures 4.15 and 4.16. Plane wave terms, found in Equation 4.15 and in the first terms on
the right-hand-side of the second-order velocity potential in Equation 4.18, increase rapidly
from deep to shallow water. The remaining second-order terms tend to be of comparable
magnitude in deep or shallow water; the cylinder size has more of an effect on these than
does the depth. It is noted that since the plane wave terms become singular due to resonant
free surface forcing, the entire second-order theory is limited in shallow water by exactly
the same conditions that limit the usual two-dimensional Stokes theory at second-order.
KA a 1.5700 KD 1.5700 KH 0.5000
WAVE DIRECTION
Figure 4.13: Example of second-order wave envelope for large cylinder, ka = 1.57, kd = 1.57,
kH = 0.5
RSS
HIDE
