Irregular wave kinematics from a kinematic boundary
condition fit (KBCF)
Deift University of Technology
Ship Hydromechanics Laboratory
Library
Mekelweg 2, 2628 CD Deift
Shell Development Co., Box 481, Houston, Texas 77001, USAThe Netherlands
Phone: +31 15 2786873 - Fax: +31 15 2781836 Calcuiation of the kinematics of random waves above themean water line presents great difficulties.
The kinematic boundary condition fit (KBCF) method approximates the solution through the
numerical calculation of a potential function which fits the kinematic boundary condition on a
specified surface. Comparisons with a high order regular wave show that the method converges to the true solution when the surface is accurately specified. Tests of the method for irregularwaves were made with measurements from a laser-Doppler current meter in the Delft wave tank. These tests showed good agreement between theory and measurement when the surface evolutionwas calculated
correct to second order. Stretched linear theory was also compared to the measurements. The stretched velocities were reasonably good when the phases of the component wavelets were measured but somewhat low when the phases were selected from a uniform distribution.
GEORGE Z. FORRISTALL
INTRODUCTION
Predictions of water particle velocities are needed for calcu-lations of the forces on structures in irregular waves. Monte
Carlo techniques can efficiently simulate realizations of
wave kinematics while remaining faithful to the concept of the sea as a random process. Forristall' has shown that the statistics of such simulations based on linear wave theory agree well with measurements made below mean water level in storms at sea.
It is much more difficult to accurately calculate particle velocities above, mean water level in irregular waves. Tung has shown how to account for the intermittent immersion of a location in the splash zone when calculating the prob-ability distribution of particle velocities, but his velocities are given by linear wave theory. Unfortunately, the linear
suprposition principle breaks down above mean water level. Long waves can carry short wave components to
elevations that are a substantial fraction of their wavelengths above mean water level, so that the exponential factor in the formula for their particle velocities grows unreasonably large.
In acteal
calculations, the unreasonable exponentialfactor is usually avoided thEough use of empirical modifica-'
tions such as the coordinate stretching introduced by
Wheeler.3 Although this technique is widely used and com-pares reasonably well with measurements, it has no basis in hydrodynamic theory.
With the conventional assumption that the fluid flow is irrotational, wave kinematics can be found from a potential function' which satisfies Laplace's equation. This function must satisfy two boundary conditions on the free surface: a kinematic condition which requires that the fluid on the surface follows the motion of the surface, and a dynamic condition which requires that the pressure computed using Bernoulli's equation is constant on the surface.
Received March 1985. Discussion closes December 1985.
'202 Applied Ocean Research, 1985, VoL 7, No. 4
In this paper, we formulate a method for the accurate
calculation of kinematics in irregular waves both above and below mean water level. The baic -concept of this- lcine-. matic-b.oundary condition fit (KBCF) rnethodis th if the - shape of the surface-and its time derivative are known, then the=kineiriatIc 1öündaiy condition-deflnes a NeIII?IäniI problem-for- theT velocity potential. The potential can thus be found through standard numerical relaxation techniques'. Specifying the evolution of a random wave surface is equivalent to kiiowing the dispersion relation. Simply using the linear dispersion relation is surprisingly effective, since KBCF fits the kinematic boundary condition on the actual surface in addition to having the first order correct disper-sion relation. Greater accuracy can be obtained by using
the second order
relations which were calculated by Longuet-Higgins4 for deepwaterand extended tointer-mediate water depth bySharrna and Dean.5 These disper-sion relations satisfy the kinematic and dynamic boundary conditions correct to second order in wave steepness. The
KBCF calculations then find a potential function which matches the kinematic boundary condition much better than that given directly by the second order formulas. Although the dynamic boundary condition is not
expli-citly included in the numerical calculations, we will see that the results also fit the dynamic boundary condition better than does the unmodified second order solution.
First we check the results 'of the numerical calculations by comparing them to a high order regular wave theory. For a regular wave that propagates without change of form, the numerical theory developed by Chappelear6 gives both the surface and the wave kinematics very accurately. Tests of the KBCF scheme with the Chappelear surface given as
input gave particle velocities essentially identical to those
from Chappelear theory.
-We then compare the KBCF calculations to laboratory data on velocities in wave crests. The measurementswere
made in the Deift Hydraulics Laboratory by Bosrna and
Vugts,7 using a laser Doppler current meter. Direct applica-0141-1187/85/040202-11 $2.00
tion of linear or second order theory gives particle velocities much higher than those that were measured, while KBCF applied using the linear dispersion relation underpredicted by about 10%. When the second order dispersion relation was used as input, the KBCF calculations were an unbiased predictor of the measured velocities.
Stretched linear theory based on the measured surface height also matched the wave tank data reasonably well. Since the waves were two-dimensional,, matching the
surface at the one measurement point established the cor-rect phases of all the wavelets, even though linear theory could not give the wavelets the correct phase velocity. The
stretched linear calculations thus had the advantage of
starting with a surface that had the correct skewness. When Monte Carlo simulation is used in design work, the phases are picked at random from a uniform distribution. Random simulations of the tank data made using this method gave a distribution of velocities about 15% below the distribution
of measured velocities. The KBCF theory with random
phase input but with second order terms included gave a velocity distribution only slightly lower than the measure-ments.
WAVE KINEMATICS AS TEE SOLUTION OF A NEUMANN PROBLEM
For two-dimensional, irrotational, incompressible flow, a potential function 0 can be defined such that the velocity
components u in the x direction and w in the z direction
are given by
U = - çb (1)
w = -
(2)where subsripts denote partial differentiation, and the z
direction is upward. The potential function satisfies Laplace's equation in the interior of the fluid.
(3) At the free surface given by (x, t), the potential function must satisfy both kinematic and dynamic boundary condi-tions. The kinematic condition requires that a point on the
urface remain on the surface during the motion so that
+0-0v=O atz=(x,t)
(4)The dynamic boundary condition uses Bernoulli's equation to specify that the pressure on the surface remains constant.
n - 0rI
-I- (Ø + Ø)/2g = const.at z = (x, t)
(5) At the bottom, the vertical velocity must be zero so that0=O atz=d
(6)Kinsman8 gives a well written derivation 'of 'equations
(l)-.(6).
Linear or Airy wave theory is constructed by simplifying
the boundary conditions under the assumption of small amplitudes. The products and squares of derivatives are
neglected in equations (4) and (5), and these equations are
applied not at i but at the constant level z = 0. The
solu-tion for a sinusoidal wave is then ag cosh IC (z + d)
sin(icxat+e)
(7)a
cosh,d
where a is the amplitude of the wave, a is its radian fre-quency, ic is the wavenumber, e is the phase angle, and c is
Irregular wave kinematics from a kinematic boundary condition fit (KBCF): C. Z. Forristall
w
the phase velocity. The frequency and wavenumber are
related by the dispersion relation
&2=glctanhKd (8) and
c = a/ic (9)
Since the equations have been linearized, sums of terms
of the form given by equation (7) also satisfy them. The
sinusoids form a complete basis, and thus any wave staff record could be synthesized from equation (7). This fact is the basis of the powerful methods of spectral syave analysis.
Although sums of linear waves satisfy the linearized
equations, they do not necessarily satisfy the assumption of small amplitude that allowed the linearization. To illus-trate this point, it is useful to consider the errors in a linear
wave or a sum of linear waves. Figure 1 shows the
kine-matics of a single wave with period 10 s and height 20 ft in infinitely deep water. The abscissa for each of the stacked plots is the phase of the wave in degrees, which might be
converted to a length scale running from 0 to approxi-mately 512 ft. The solid line in the bottom panel is the
sinusoidal Airy wave profile, and the middle graph is the horizontal particle velocity at the instantaneous free surface in ft/s. The top panel gives the' error in equation (4), also in ft/s. The dashed lines show comparable results for a high order Chappelear wave with the same height and period.
The profile of the Chappelear wave has a slightly higher
crest and a shallower trough. The velocity at that free
sur-0
0-LJ
0 90 270 360
0 90 iso 270 360
Phase
Figure 1. Kinematics of a regular wave. Solid lines are for a linear wave and dashed lines are for a high order Quip-pelear wave. The bottom panel shows the urface elevation,
the middle panel shows the horizontal velocity at the
surface, and the top panel shows the error in the kinematic boundaiy condition
Irregular wave kinematics from a kinematic boundmy condition fit (KBCF): G. Z. Forrisrall face is almost identical to the velocity in the linear wave at
its free surface. The error in the kinematic boundary condi-tion for the Chappeleai wave cannot be distinguished from zero at the scale of the top panel. The wave in Fig.l is only moderately steep, with H/gT2 = 0.0062. Greater differences between Airy and Chappelear theory would be expected for
steeper waves or waves in h1low water. However, the
results shown indicate that linear theory can give a reason-able solution of the equations for waves typical of oceanic conditions.
Linear theory does not perform nearly as well for short
waves suósëd on long waves. Figure 2 shows a 3 s, 2 ft
wave superposed on the wave of the previous example. The wave profile now has a small ripple running along the long wave, but the character of the horizontal velocity at z = is dramatically changed with very large oscifiations at the crest of the long wave. The unrealistic velocities produced
by linear theory are reflected in the kinematic boundary
condition error, which is an order of magnitude larger than it was for the long wave alone.
The long wave and the short wave combine to give large errors since the long wave carries the short wave away from z = 0. In terms of its wavelength, the short wave is then far
away from where the boundary conditions were applied in the Airy solution. This argument is the motivation for
the coordinate stretching transformation
z=(z')d/(d+)
(10)where z' is the location at which the kinematics are desired
and z is the value to be used in equation (7). Forristail'
LJ 0 -20 0 ie 0 -12 0 90 180 Phase
Figure 2. Kinematics of a ripple on a long wave.The
bottom panel shows the surface elevation, the middle panel
shows the horizontal velocity at the surface, and the top
panel shows the error in the kinematic boundary condition 204 Applied Ocean Research, 1985, Vol. 7, No. 4
160
compared simulations using equation (10) with field
measurements and showed that they generally
under-predicted velocities measured in the crests of waves.
KBCF theory is an attempt to eliminate the worst
effects of the inappropriate coordinate system in the solu-tion for superposed linear waves on a more rasolu-tional basis. The key to the development is the division of the problem
i
into two steps: specification of the surface motion and
calculation of a potential functiOn that matches the sur-fa If (x, t) were known, equation (4) would give the normal derivative of at the surface. Equation (6) gives the normal derivative at the bottom. Suitable conditions can be imposed on lateral boundaries placed far enough away to
have little effect on the solution under the crest of the wave. The problem is then completely specified as the calculation of a potential function with a given normal
derivative, a Neumann problem. Standard numerical tech-niques are available for the solution of such problems.
Knowledge of (x, t) could be gained from a line oFi wave staffs in a tank experiment or perhaps by a
stereo-scopic motion picture of the sea. The mOtion of the water would then be completely determined as described above. However, we do not usually have such complete informa-tion. The usual specification of the surface comes from a single wave staff or simply a power spectrum, perhaps a spectrum hindcast from historical winds. Some method for estimating (x, t) from this temporal or spectral
informa-tion is needed. The simplest soluinforma-tion is to use the linear
dispersion relation given by equation (8). The linear relation is not precisely correct for waves of finite steepness, and the solution which results from this assumption will there-fore not precisely satisfy the dynamic boundary condition, equation (5). A better-solution-will be obtained usingsecond -otdr--correctionS1ioTthe dispersion-
relationwhich--lock-somethehigherjfrequency omponents to the
funda.-meiflalcornpfiti
In the past few years, Ramamonjiarosa and his col-leagues have made measurements of the phase- speed of waves which seem to contradict the assumption that the
linear dispersion relation is nearly correct. Ramamonjiarosa9 made his first measurements in a wind wave tunnel. Rama-rnonjiarosa and Giovanangeli'° then made field measure-ments in the Mediterranean. These results and others have been summarized and interpreted by Ramamonjiarosa and
Mollo-Chuistensen." In both the laboratory and field
experiments, the phase velocity was found by cross spectral analysis of the data from two wave probes separated by a
short distance in the direction of wave propagation. For frequencies above that of the spectral peak, the phase
velocity was found to be nearly constant and equal to the
velocity of the components at the spectral peak. These results imply that waves are dominated by modulated trains of finite amplitude waves rather than the freely
propagating cOmponents assumed in linear simulations.
The work of Ramamonjiarosa et al. has caused
con-siderable controversy because it was carefully done, but the results contradict conventional wisdom. Phillips'2 pointed out that in a laboratory wind-wave tank, the ratio of wind
speed to wave celerity is much greater than it is in the
ocean. In tank experiments, the- ratio of the celerity to the fricticn velocity is of order one for waves near the spectral peak. Breaking processes associated with the surface drift current effectively limit the growth of waves with phase speeds less than the friction velocity. Thus the only high frequency components which exist in laboratory wind-wave experiments are harmonics of the fundamental components.
270 360
Fhi]lips later elaborated on these ideas in a more rigorous fashion.
Interpretation of cross spectra from measurements in the
ocean is complicated by the directional spreading of the
waves. An increase in the width of directional spreading will result iii an increase in the apparent phase velocity, as
dis-cussed by Komen.14 Huang'5 calculated the dispersion
relation that would result from a spreading function of the
form cos
0, and pointed out that the cross spectral
analysis only gives good results when the probe separation is a small fraction of the wavelength. Dudis'6 carried this
argument further and showed that the analysis picks the
phase function of the component having the largest spectral
density and essentially ignores contributions from other
modes.
Both free and, locked components exist in any wave.
system. The real question is their relative importance. In laboratory wind-wave experiments, the dominant contri-butions at frequencies above the spectral peak seem to be locked. In the open ocean, individual wave crests persist for
only a few wave lengths, suggesting that the system is
animated by dispersive components. Calculations of wave teractions to second order confirm this suggestion.
How-ever, we will see-that. the results obtained fromKBCF theory are -definitely -improved when the sednd order
locked components are included. FINITE DiFFERENCE SOLUTION
First we will describe the solution to the Neumann bound-ary value problem and check it for a regular wave where the boundary condition is known almost exactly. The partial differential equation (3) can be solved using finite difference
methods. Figure 3 shows the computational star in the
interior of the grid. The horizontal grid spacing is Lx. The
horizontal index I increases to the right and the vertical
index / increases downwards. A centered difference approxi-mation to the derivatives gives
{(I+ 1,/) +0(1 l,J)20(I,J)}/x2
(11) and,{(I,
J 1) + 0(I,J + 1) 20(1, J)}/z2
(12) en the finite difference analog of equation (3) is written for each grid point, the resulting set of linear equations is coupled and must be solved simultaneously. One simple and economical method of solution is successive over-relaxation. Define the residual at a given node and step of therelaxa-
2-8O$-Figure 3. Computational star for interior points
Irregular wave Icinematics from a kinematic boundary condition fit (KBCF): G. Z. Forristall
tion by
r = +
Then at the next step, is replaced by
= 0 + c)r/a,
where
akk = 2(l/x2 + 1/Az2)
and
1 <w <2 is chosen to maximize convergence. A
thorough discussion of this standard method is given byForsyth and Wasow.'7
Thedifferenceschemeirnist
be:modifleheböund-caries. Figure 4 shows an infinitesirnai section of the wave surface. The normal derivative of 0 at the surface is given byFigure 4. Geometry of the wave swface
A section of the finite difference grid with the wave
surface running through it is shown in Fig. 5. To form the finite difference analog of equation (19) at grid point (I,.T), we must estimate Ø, there; First drop a perpendicular from the wave surface as shown. Then
= Ø. cos.0 + sin9 (16) LI
But from the geometry in the figure,
cos 0 = - n(l +
rj)1"2 (17) sinO = (1 + 2)1/2 (18)0
VP
0
(i+1,J)
and thus the ldnernatic boundary condition (4) can be
0/
¼!written---= - '?t(l + 2y11z
(19) 0(1-i + I) 0(E) 0(1,J + 1) 82-309-SS
I.
V
Irregular wave kinematics from a kinematic boundaiy condition fit (KBCF):G. Z. ForrLstall206 Applied Ocean Research, 1985, VoL 7, No. 4
35 30 25 20 15 10 -15 Legend Chap Linear KBCF 0 5 10 15 20 25 30 Velocity (ft/sec)
Figure 6. Horizontal velocity profile under the crest of a regular wave
dition. The simplest method of extending the solution to random waves is to find the surface using linear theory.
Standard linear simulation methods' calculate the potential function as a sum of terms of the form given by equation (7). The amplitudes are chosen to give the specifiedwave spectrum, and the phase angles are taken from a uniform random distribution. The wave surface is then given by
N
acos(Kxat+e)
(27)n=i
and the derivatives of the surface are
N
nx=_
ajK sin(x - a,t + c,)
(28)n=1
N
=
a1cr, sin(,cx - at + e)
(29) n=iEquations (28) and (29) provide the information needed in equation (19) which gives the surface boundary condi-tion for the numerical calculacondi-tion of the potential funccondi-tion. The resulting solution will satisfy the kinematic boundary condition to within the accuracy of the numerical approxi-mations However, since it is basedöh the linear disprsion
relation,- itwifl Thöt also satisfy the dinannc boundaiy ciditioir (5). The seriousness of this failure can best be
assessed through comparison with a carefully controlled experiment.
A long series of pertinent measurements was made at the 50 m long, 1 m wide wave flume at the Delft Hydraulics Laboratory as a Marien Technologisch Speurwerk (MaTS)
project. The still water depth during the experiment was
85 cm. Bosnia and Vugts7 give a good description of the experiment. Unidirectional irregular waves were produced using a cradle type wave generator programmed to reproduce Pierson-Moskowitz spectra. The simulations were obtained
by recording filtered white noise onto an analog tape
= flxI.Z (20).
A linear estimate of the potential at E is
(E)=cb(I,J+ 1)c/Ax {(I-1,J+ l)Ø(I,J+ 1)}
for e <0
q(E)=b(I,J+ 1)+e/Lix{(J+ 1,J+ l)Ø(I,J+ 1)}
fore0 (21)
The normal derivative at (I,J) is then
= {Ø(í, J) - Ø(E)}JAz(l + (22) The residual at the surface node is
r = , + {(I,J) Ø(E)}/Az
(23)and the potential is replaced at the next step by
'(I,J) = (I,J) wrAz
(24)At the bottom where Ø, = 0,
41(1,1) = (I,f)+ wrAz
(25) wherer = (,.J - 1) - ?p(I, J)}/Az (26) The numerical solution was checked by comparing it to
the results from a high order regular wave theory which
satisfies both boundary conditions (4) and (5) nearly
exactly. Chappelear6 solved for the kinematics of regular
waves by employing the type of series used in Stokes
expansions, but then calculating the coefficients through a
least squares minimization of the errors in the boundary
conditions. Since the series is fit rather than truncated, the errors are somewhat less than those for Stokes waves even if the series is of the same length. In the implementation we used, there are 15 terms in the series for the surface profile and 9 terms in the velocity series. The errors in the solution are quite small even for very steep waves.
The KBCF solution for a Chappelear wave uses the surface profile and phase speed calculated by Chappelear theory. The x derivative of the surface is found by numeri-cal differentiation. Since the wave propagates without change of form, the time derivative of the surface is simply the phase speed times the x-derivative. Because thewave is regular, the end boundary conditions can be specified by taking the solution over one wavelength with a reflection condition at the lateral ends of the grid.
Several tests showed that the KBCF solution converged well to the velocities given by Chappelear theory. Figure 6 shows velocity profiles under the crest of a 47 ftwave with a 10 s period in 80 ft of water. This wave is relatively steep, with H/gT2 = 0.0 146, and its nonlinearity is enhanced by the relatively shallow water depth. The KBCF solution used a grid with 360 points horizontally along the wave so that Ax = 1.34 ft, and Az = 1.00 ft. With this fine grid,
converg-ence was rather slow, and the solution shown is for 600
iterations with o = 1.95. The solid line in Fig. 6 shows the horizontal velocity from Chappelear theory, and the dashed line shows the KBCF solution. The chaindashed line shows the velocity from a linear wave with the same height and wavelength as the Chappelear wave, which was used as the
starting point of the iteration. The convergence to the
Chappelear wave velocity was excellent. FIRST ORDER RANDOM WAVES
The tests with Chappelear waves established the accuracy of the potential function solution for a given boundary
con-Iby
measurements at the heights above mean water level listedlinear wave theory. Here we will only consider the in the table above.Since the waves are two-dimensional and a wave staff
record of the surface height is available, it is possible to
perform deteirninistic simulations in which the simulated
and measured velocities under particular crests are
com-pared. A Fourier transform of the wave staff record gives the amplitude and phase of each component wavelet. We
then use the linear dispersion relation to calculate n(x), (x), and flt(X). This procedure differs somewhat from
that which would be used in design, since the phase angles
are not uniformly distributed. The consequences of this
difference are discussed later.
The MaTS data was divided into records of 2048 samples.
For the first 10 records in each test, we selected the four
waves with the highest crests for analysis. The maximum measured horizontal velocity under each of these crests was compared with the maximum calculated velocity. Figure 7 shows the result of simply summing all the linear
compon-ents up to 2.5 Hz to get the particle velocity at 4.4 cm
elevation.
higher than the measurements because of the large expo-As expected, the calculated velocities are muchnential factors in the high frequency components.
The grid for the KBCF solution was set up with 1x = 4.0 cm and z = 1.1 cm. The lateral boundaries were
placed at the points where the linear potential function was zero. These points correspond roughly to the troughs that precede or follow the crest under investigation. The poten-tial at these lateral boundaries was kept constant during the iterative solution for the KBCF potential function. Numeri-cal experiments determined that moving the lateral bound-aries farther out had negligible effect on the solution under the crest. There were generally about 100 grid points in the
x-direction and 90 grid points in the z.direction. With
= 1.95, satisfactory convergence was generally obtained in 300 iterations.
The maximum horizontal velocity measured under each crest is compared with the maximum velocity at the same
elevation in the KBCF waves in Fig. 8. The theoretical velocities are almost always lower than the measured
velocities, and the error increases in the waves with large
velocities. Figure 9 shows that stretched linear theory
actually performs better, although there is still some
under-Irregular wave kinematics from a kinematic boundary condition fit (KBCF): G. Z. Forristall
Figure 7. Comparison
of peak
horizontal velocitiesmeasured under wave crests and calculated using linear
theory 80- 70GO - 50- 40- 30-20-j 10-0 Legend o 533 523 0 10 20 30 40 50 60 70 80 Me osured
Figure & Comparison
of peak
horizontal velocitiesmeasured under wave crests and calculated using KBCF
based on the linear dispersion relation
prediction of the largest velocities. The cumulative distri-butions of measured, stretched, and linear KBCF velocities are compared in Fig. 10. The curves show the fraction of the 40 waves in test 533 for which the maximum horizontal velocity exceeded a given value. Linear KBCF theory under-predicts the measurements at almost all probability levels,
while stretched linear theory underpredicts at low
prob-ability.
-SECOND ORDER INTE ACTIONS IN A WAVE SPECTRUM
Since linear wave theory is reasonably successful, it is natural that more accurate solutions would be based on
which was used as input to the wave maker. All of the tests used a cutoff frequency of 3.0 Hz. The pertinent parameters for the tests we studied in detail are given below:
Period at Significant Measurement Test spectral peak wave height level
number (s) (cm) - (cm) lao- 140- 120- 100-ID 80- 4°- 20-D0 0 00 523 2.05 11.96 2.9 533 1.86 17.97 4.4
The free surface elevation was measured using a resist-ance wave staff about 20 m from the wave generator. The vertical and horizontal velocities were measured in the same plane nortnal to the direction of wave propagation using a
laser-Doppler current meter. Simultaneous measurements of surface elevation and water particle velocity were made in each test for a duration of about 35 mm. All data were passed through a 10 Hz low pass analog filter before digiti-zation at 25 Hz.
Measurements were made at several elevations for each input wave condition. Bosma and Vugts7 showed that the measurements below mean water level were well matched
o 20 4.0 Measured 60 80 0 000 0
Jar wave kinematics from a kinematic bounthzry condition fit (KBCF): G. Z. Forristall 0 Li Cc / 0 o0 c A
p
Legend
C 533 523 Legend Measured Stretched Linear KBCFFi&ure 9. Comparison
of
peak horizontal velocities measured under wave crests and calculated using stretched linear wave theory10 20 30 40 50 60 70 80
Velocity (cm/sec)
Fig. 1 LX Distribution
of
peak horizontal velocities in test 533perturbation expansions. Aördorder Stkespasion
itht'iè the
fdhavenüñib of the fundamental
ThjscoredóëjiQtbey fhflmëä dispersion
relbut-insteadir lock4toTrhve th phase with the
:fiirnèntal Similar expansions can be carried out for
continuous or discrete wave spectra. Each pair of wavelets
then contributes a second order interaction term which
has the sum of frequencies and the sum of wavenumbers of
the fundamental wavelets. The Stokes expansion can be
thought of as the diagonal term of this more general inter-208 Applied Ocean-Research, 1985, VoL 7, No. 4
action matrix. These. second order interactions which
affect the local kinematics of the wave field should not be
confused with the weak higher order interactions which contribute to the slow modification of the thape of the
spectrum.
-Tick'8 characterized the first order solution as a Fourier-Stieltjes integral and calculated the secoild order interaction kernel for unidirectional waves. Weber and Barrick'9
per-formed the expansion in such a way that the third order
correction to the dispersion relation was also obtained, and Barrick and Weber2° calculated explicit formulas for the second order spectrum. Their work was motivated by the fact that the second order wave spectrum is directly
observ-able in high frequency radar echos from the sea surface.
The fact that measurements of the second order portion of
the spectrum can be used to calculate the whole wave
spectrum. is a striking verification of the accuracy of the interaction theory.
For our purposes, the expansion of Longuet-Higgins4 is more useful, since he begins with a representation in the
form of equation (27). Longuet-Higgins performed his calculations for infinite water depth, but they have been
extended to finite depths by Sharma and Dean.5 Since this latter result is not widely available, it will be included in this report.
If the fIrst order wave height is given in the form of equation (27), then the second order correction for the
water surface elevation is, according to Sharrna and Dean5,
iN N
ajaj{Kcos(,Iict'1)
4+ K cos(' + 'j)}
(30) whereK = [D - (,c, + R,R1)] (R1R1)' + (R + R1)
K = [D - (c1ic1 R1R1)] (R1R1y' + (R1R1) Dii(Rj R,"2) {R2(ic Ri)
(xj RJ}
RI2 R)'22 -
tanh Kd2 RI2 R)2K!c1 + RR1)
+(RI2 RJ2 ic
tanhK1jd2(R12 +R2)2(1, R1R1)
(Rr+Rr)2tanhKd
(R'2 + R1") RI2(,c7 R7) + R)12(K R)}
+ -(Rr+RJ')2-4itanK&d
= K +
c3 Rt=Kz tanhic1d= a/g andi/i=icxat+e
Equation (30) was derived for a directional wave spectrum, so that the wavenumbers can be thought of as vectors and
products of wavenumbers interpreted as dot products.
However, we will only consider the case of plane waves. Inspection of equation (30) reveals that the second order terms are quadratic in the first order amplitudes. There are two sets of terms whose phase functions are sums and dif-ferences of the first order phase functions, each multiplied
by a complicated interaction kernel. It turns out that the 0
0 0 20 30 40 50 60 70 80
V
.emeIs peak at i=j, so that the sum terms will beconcen-trated near twice the frequency of the peak of the first order spectrum, and the difference terms have low fre-quencies. None of the terms-obey the linear dispersion
relation since their frequencies and -wavenumbersare sums
and differences of- the first order frequencies and
wave-numbers. -
-The second order potential function associated with
equation (30) is
---
LU c-.J-(2)
f
a1a1 cosh ,c(z + d) D4,11a1og2
cosh.gd
(aa1)
cosh 4. (z + d)
D.
XSm(/i,-11I1)+ /
'
r cosh K1d (aj + o) Tcr
Direct. use -of equation :(31-) to calculate particle velocities above -mean-water level leads to.the saine pioblèrii
èperi-enced with the -use of thiinaitCñtial given by: equation
(7). That is, the exponential increase of the velocity from the high wavenurnber terms gives unrealistic total velocities. will therefore not use ejuation (31). directly but
rar
use--the- second- ordi corrècti to the wave surface givenby equation (30) as the basis for a second order KBCF solution. However, it is instructive to study the foi:m of
equation (30) for some example spectra first.
Equation (30) includes the phase of the second order waves. This form is most useful for simulation, but for
comparison with previous work, it is useful to convert the amplitudes in that representation to spectra. Consider the first order wavenumber spectrum given by
S(,c)=0.00S/2K3
for ac'0.0436
(32) where ic is in m' and S(ic) is spectral density with units of in3. The waves are unidirectional and the water is infInitelyS
0 0 0 1-0 2.00/
0 0000 -O 0.16 0.24 WA V EN U MBERFigure .11. Second order spectrum as a function of wave number and frequ ency
Irregular wave kinematics from a kinematic boundary condition fit (KBCF): G. Z. Forristall
0-32
60
40
0.5 1 1.5
Frequency (Hz)
Figure 12. Measured (i5lid line) and calculated second order (dashed line) spectra for test 533. Forfrequencies
above LO Hz, both curves have been amplified by a factor of 10 to make them more visible.
deep. Figure 11 shows the second order spectrum given by
applying equation (30) to the spectrum in equation (32).
Sigma is the radian frequency in s, and the density for the wavenumber-frequency spectrum is in in3 s. The figure is a representation of the three-dimensional surface of second order spectral density above the wavenuxnber-frequency
plane. The rmooth curve lying in the base plane is the
linear dispersion relation. Most of the energy in the second order spectrum is at frequencies higher than the linear waves,
and that energy peaks sharply on the curie a= (2gic)"2. However, the energy is spread over a range of wavenurnbers and Trequencies as opposed to the first order energy which
lies precisely above the dispersion curve. The spectrum
given by the difference terms lies at lower frequencies a.nd
is considerably less energetic. Integrating the surface in
Fig. 11 over frequency gives a second order wavenumber spectrum whiäh can be compared with Fig. 2 in Barrick and Weber.18 The results are the- same once allowance is made for the fact that Barrick arid Weber plotted the two-sided spectrum. Our spectra are defined so that the integral from zero to infinity gives the total variance of the wave height, and thus their curve must be multiplied by 2 to match our results.
Figure 12 shows the measured first order and calculated second order frequency spectra for the first 2048 points in
MaTS test 533. Above 1.0Hz, the scales for both curves
have been thvided by 10 to make the spectra more visible. The second order spectrum is negligible compared to the
main peak of the measured spectrum, and it becomes
important only above 1.0 Hz. Above 1.5 Hz the calculated
second order spectrum is about equal to the measured
spectrum, which indicates that there was no freely
propa-gating energy at these frequencies in the wave tank test.
Although the second order energy is small compared to the total spectrum, we will see that it ir dynamically significant.
Irregular wave kinematics from a kinematic bounda'y condition fit (KBCF): G. Z. Forristall
A second order KBCF simulation of the MaTS data
begins as before with a Fourier transform of the wave staff measurements. The second order surface is then calculated from equation (30). This procedure is not precisely cor-rect, since the measurements include all the higher order waves as well as the fundamental wavelets. Since the second order contribution is small, this error should be of a higher order; However, in order to recover the measured height at the wave staff, we modified the first order amplitudes. If the measured complex amplitude of a wavelet is defined
as
a' = an exp(ien) (33)
and the complex sum of the second order contribution at frequency a, is a2, then we make the amplitude of the first order wavelet
(34)
The sum of the first order wavelets defined by equation (34) and the second order wavelets defined by a will match the measured profile at x = 0, defined as the wave
staff location. Equation (34) might be thought of as the
first step in an iterative process which would yield consistent first and second order components, but it is well to remem-ber that the calculations are only correct to second order.
Both the first and second order spectra were determined
up to f
2.5 Hz. Including higher frequencies tended to make the calculations unstable, probably because of poorresolution of the wavelength on the grid used, while
decreasing the cutoff frequency below 2.0 Hz producednoticeable reductions in the calculated velocities.
In order to avoid storing the N2 second amplitudes implied by equation (30), the amplitudes were summed
in wavenumber bins of width x = 0.001 cnf'. The wave
surface and surface derivatives, were then calculated by
E 150-.2. I-P -50 '20 10 300-P 250- 200- 50-0 Legend Un.r Scond lcecr -100 -50 0 50 100 x
Figure 13. Dynamic boundary condition error from linear, second order, and KBCF calculations. The bottom panel shows the wave profile
210 Applied Ocean Research, 1985, VoL 7, No. 4
40-
20-I I I
0 10 20 30 40 50 60 70 80
Measured
Figure 14. Comparison
of peak horizontal
velocities measured under wave crests and calculated using KBCF based on second order theory0
0 10 20 30 40 50 60 70 80
Velocity (cm/sec)
Figure 15. Distribution of peak velocities in test 533
using KBCF based on second order theory'
summation over wavenumber. The KBCF iterative solution method for the potential function was applied to this surface. Figure 13 shows that the KBCF solution fits the dynamic
boundary condition much better than the potential
func-tions calculated directly from either the linear or second
order theories. The bottom panel of the figure gives the
shape of the wave crest for one of the large waves in the
MaTS data set. The top panel is the Bernoulli 'constant'
calculated from the left hand side of equation (5). Although the second order potential function matches the boundary
condition better than the linear potential does, both of
them are seriously in error when the wave elevation exceeds
0.a-
0.2-Legend
Measured KBCF 0 0' 0' 0 0 a Legend o 533 523 30-p
10 0 80- 70- 60-50-ocm. The KBCF solution matches the dynamic boundary
condition much better than either the first or second order potential functions since it eliminates unrealistic high velocities at heights far above mean water level.
The results of the second order KBCF calculations are presented in Figs. 14 and 15. Figure 14 shows thatKBCF
is an unbiased predictor of the measured velocities,and
Fig. 15 shows that cumulative distributions of the measured and predicted velocities are nearly identical. Comparison with Figs. 8-10 shows that second order KBCF performs notably better than either stretchingor linear KBCF. RANDOM PHASE SIMULATIONS
Since the waves in the MaTS experimentwere two-dimen-sional and a wave staff record was available, the phase of each wavelet in equation (27) could be established. Thus
the linear simulations described above used the phases which result from the natural higher order interactions,
even though they did not use the correct phase speed for all the wavelets. When the..simulation methodis used in design., the power spectrum is specified arid the phases are typically ot known. Monte Carlo simulations are then made, with the first-order phase angles chosen from a uniform distri-bution. If a second order simulation is made, the phases of
the wavelets in the correction term are determined from
equation (30).
The random and deterministic simulations are thus
somewhat different and it is important to check the behavior of the Monte Carlo simulation. Comparisons with data and other simulations must be through probability distribution functions, since the waves no longer correspondon a one to one basis. Figure 16 shows that a random simulation based on stretched linear theory considerably underpredicts the measurements at all probability levels. Comparison with Fig. 10 indicates that specifying the correct phase angles
for the stretched linear simulation greatly improved the
0.8- 0.2-0 0
Legend
Me as u red Stretched KBCF Velocity (cm/sec)Figure 16. Distribution
of
peak velocities in test 533using random input phases
Irregular wave kinematics from a kinematic bounda,y condition fit (KBCF): G. Z. Forristall
0.4 0.3 - 0.2 a) 0.0 -0.1 a) -0.1 0.0 0.1 0.2. Measured
Figure 17. Comparison of measured and simulated skew-ness for data segments in test 533
0.3
a)
simulation. At the 50% probability level, the measured
velocity is 54.76 cm/s, the deterministic stretched velocity is 54.10 cm/s, and the random stretched velocity is 45.80 cm/s.
The second-order KBCF simulations should recover the proper relation between the phase angles, at least to second
order, through use of equation (30). One check on the
calculated phases can be made by comparing theskewness of the measured and calculated profiles. Hereskewness is defined by
= m3/m/2 (35)
where m3 and m2 are the third and second moments of the
wave profile. The expected value of the skewness for a
simulation with uniformly distributed phase angles iszero. Second order interactions tend to sharpen thecrests of the waves and flatten the troughs, leading to positive values of skewness. Figure 17 ves a comparison of the skewness
from measured and computed profiles. Eaèh data point
shows the measured arid simulated skewness for one 2048 sample record. The simulations had a high frequency
cut-off of 2.5 Hz for both the first and second order terms. Second order difThrence terms with wavelengths longerthan the tank were not included in the simulation. The second
order terms calculated from equation (30) were simply added to the first order profile, with no adjustment to the first order terms.
The skewness is seen to be an unstable statistic; its value varies greatly from one record to the next. The skewness of the simulated waves is correlated with themeasurements
but slightly smaller. Figure 14 shows that the simulated velocities in the crests using KBCF theory are also a bit on
the low side. The simulated velocity at the 50% level is 52.50 cm/s versus 54.76 cm/s for themeasurements.
Figure 16 shows that second order KBCFcan be used to accurately determine the statistics ofwave crest kinematics ven only the wave spectrum as input. Monte Carlo simula-tions using KBCF may thus be an effective way to model the kinematics of naturally occurring irregular waves for structural design.
0.4
Irregular wave kinematics from a kinematic boundary condition fit (KBCF): G. Z. Forristall CONCLUSIONS
The kinematic boundary condition fit (KBCF) theory
described in this report is based on a numerical calculation of a potential function which fits the kinematic boundary condition for water waves. The theory gives an excellent fit to measurements of particle velocity made in a wave tank if the phase relationships between component wavelets are calculated correct to second order The probability distribu-tion of the measured velocities can aiso be fit rather well when only the spectrum of the measured waves is used as input. Stretched linear theory only fits the data when the correct wavelet phases at the wave staff are specified, infor-mation which is not available for design applications.
Naturally occurring waves are usually directionally spread as well as dispersive, and KBCF must be extended to three dimensions to include this complexity and verified with field data in high and short crested waves.
ACKNOWLEDGEMENTS
I thank J. E. Chappelear and G. Rodenbusch for many
helpful discussions as well as a review of an earlier version of the manuscript. The excellent MaTS data set was provided by J. H. Vugts with permission of the Steering Group for Fluid Mechanics Research of the Netherlands Maxine Tech-nological Research Program. The results of the MaTS project, from which the experimental data used in this report were
taken, have been fully documented in the following two
reports: 'MaTS Report VM-I-4A: Wave Ki±iernatics in
Irregular Waves, Parts I, II, lilA and IIIB', and 'MaTS
Report VM-I-4B: Three Crest Kinematics Models Compared to MaTS Data and OTS Data'.
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