A NEW M E T H O D FOR T H E GENERATION
OF SECOND-ORDER R A N D O M WAVES
by Peter Jan van Leeuwen and
Gert Klopman
Report No. 11-91
A NEW METHOD FOR THE GENERATION
SECOND-ORDER RANDOM WAVES
by
Peter Jan van Leeuwen^^ and
Gert Klopman^^
December 1991
Delft University of Technology
Department of Civil Engineering
P.O. Box 5048
2600 G A delft
Delft Hydraulics
P.O. Box 152
8300 A D Emmeloord
CONTENTS
Abstract
1
1.
INTRODUCTION
2
2.
PROBLEM F O R M U L A T I O N A N D SOLUTION
4
2.1
Problem formulation
4
2.2
Perturbation expansions
5
2.3
First-order solution
7
2.4
Second-order solution
8
2.5
Implementation
12
3.
EXPERIMENTS
4.
5.
3.1
Experimental arrangement
14
3.2
Data processing
15
3.3
Noise level and error analysis
16
TEST OF T H E NEW CONTROL SIGNAL
18
4.1
Bichromatic spectra
18
4.2
Continuous spectra
18
Table 1
18a
CONCLUSIONS
21
References
23
Appendix
25
Figures
28-34
Abstract
For the generation of second-order random waves in a flume the control signal for the
wave board has to be correct up to second order. An expression for this control signal
is derived with the perturbation method of multiple scales. It is much less complex and
requires less computation time than the expressions obtained from the full second-order
theory. A verification of the new method for second-order subharmonics is provided for
bichromatic and continuous first-order spectra. The data are analysed with the complexharmonic principal-component analysis to reduce the influence of noise. The validity of
the new method is confirmed.
1
1.
INTRODUCTION
The generation of second-order random waves is important for laboratory experiments
in which the problem under investigation is sensitive to second-order effects in the wave
field. For instance, second-order subharmonics are important for studies of the surf-beat
mechanism, the generation and evolution of sand bars and the slow-drift motion of
moored vessels. The second-order superharmonics sharpen the wave crests and flatten
the wave trougths and are important for sand transport, among others.
Sand (1982) and Bathel et al. (1983) calculated the second-order wave-board motion for
the correct generation of second-order subharmonics, the bound long waves. They base
their work on the transfer function for these low-frequency waves in absence of a wave
board, as first given by Ottesen-Hansen (1978). Sand & Mansard (1986) did the same
thing for the generation of superharmonics. The transfer function for the superharmonics
in absence of a wave board is given by Dean & Sharma in 1981.
The expressions obtained for the control signal of the wave board are exact to second
order. To obtain this signal a convolution-type integral has to be performed. The
integration is in the frequency domain and the integrand is a combination of products
of the Fourier components of the first-order surface elevation at two different
frequencies and the transfer function. In this way the nonlinear interactions of all
first-order spectral components are taken into account.
A first look at the resulting equations for the wave-board movement reveals the
disadvantage of their use: they are very complex, and it requires considerable computing
time to obtain a second-order signal for the wave board, mainly due to the convolution.
When the first-order spectrum is narrow, this procedure seems unduly expensive. This
is because only the frequency components near the peak frequency will give rise to
substantial sub- and superharmonics.
If the first-order spectrum is narrow the first-order waves can be described by an
oscillation with a slowly-modulated frequency and amplitude. This was the motivation to
2
use the method of muUiple scales to describe the water motion. The same method is
used by Mei (1983) to calculate the second-order waves in absence of a wave board. The
modulation acts on a longer time and length scale than the periods and wave lengths of
the first order waves. To incorporate these slow modulations new time and length scales
are introduced to describe these phenomena. So a cascade of new variables is introduced,
hence the name of the method.
In this method the calculation of the second-order surface elevations is reduced to a few
multiplications in the time domain. In principle, the theory is valid for narrow first-order
spectra, but it can even be applied to a Pierson-Moskowitz spectrum. For a detailed
discussion of the applicability of this method the reader is referred to Klopman & Van
Leeuwen (1990).
In this paper we will give simplified expressions for the control signal for the wave board
for the generation of second-order waves in a flume based on the method of multiple
scales. This control signal will be such that, in theory, it wiU produce the second-order
surface elevations away from the wave board as found by Mei (1983). The use of the
multiple-scales method to find the wave-board motion to second order resembles the use
of the same method by Agnon and Mei (1985), who determine the slow-drift motion of
two-dimensional bodies in beam seas to second order.
The structure of this report is as follows. In the next section the method of multiple
scales is briefly dicussed. The boundary-value problem for the water movement is
formulated and the control signal for the wave board is presented. Then the experimental
setup is described together with a short explanation of the method of data analysis. This
is followed by an experimental test of the new control signal for bichromatic and
continuous first-order spectra. The report is closed with some conlusions.
3
2.
PROBLEM F O R M U L A T I O N A N D SOLUTION
2,1
Problem formulation
In this section the boundary-value problem for the water movement is given followed by
a short outline of the method of multiple scales and the resulting expressions for the
wave-board movement. Details on the method of multiple scales can be found in i.e. Mei
(1983), see also Klopman and Van Leeuwen (1990).
In figure 1 a sketch of the situation is given. The flume is equipped with a translating
and rotating wave board. The effective centre of rotation is at a distance
/
from the
flume bottom. The water depth in absence of waves is h.
[figure 1]
The basic equations for the velocity potential 0 are the following. With the assumption
of incompressibility and irrotationallity the continuity equation reads
A(t)=0
(1)
in which A is the Laplace operator. The kinematic free-suface boundary condition reads
C,-cDX=(j)^
o n z = C (2)
in which C is the surface elevation and the lower index indicates partial differentiation
to the index variable. The equation states that the particles at the surface have to follow
the surface displacements. The dynamical boundary condition at the free surface is given
by
5C+4>,4(*'^*?)=0
onz=C (3)
It states that the pressure at the surface should equal the atmospheric pressure, which
4
is taken zero here. The boundary condition on the horizontal rigid bottom reads
<l>rO
o n z = -h (4)
and states that the vertical velocity of the particles is zero on the bottom. The boundary
condition at the wave board reads
(|),cos6 -(j)^sine =Az)—^l +tan20
o n x = f(z)X (5)
in which X is the wave-board position at the water surface and 0 is the angle between
the wave board and the vertical (see figure 1). The factor f(z) is given by
Condition (5) states that the particles on the wave board should follow the wave-board
motions. Finally, far from the wave board the solution must describe the first-order waves
with the bound second-order waves as given by Mei (1983). Free second-order waves
should not occur.
2.2
Perturbation expansions
Because the free-boundary conditions are nonlinear, perturbation techniques are used
to reduce the nonUnear boundary-value problem to a set of linear boundary-value
problems. To this end the surface elevation C, the velocity potential </> and the waveboard position X are expanded in a non-linearity parameter e, which is equal to the
wave steepness. Because the amplitudes of these variables are finite but small, Taylor
series expansions are carried out at the free boundaries.
In the conventional method the variables are decomposed in their Fourier components.
The first-order problem is solved for each frequency component separately. To solve the
second-order problem all non-linear interactions of the first-order waves have to be
taken into account. As argued before, this is an unduly complex method for a narrow
5
first-order energy-density spectrum.
In tlie method of multiple scales the first-order surface elevation is assumed to oscillate
with the peak frequency of the first-order waves. This is a reasonable assumption as long
as the energy-density spectrum of the first-order waves is narrow.
Second-order waves are described in the following way. The superharmonics follow from
nonlinear interactions of the first-order components and have a frequency of twice that
of the first-order waves. The subharmonics arise due to first-order wave amplitude
modulations. These modulations act on a longer time and length scale than the firstorder wave oscillations. This is the motivation to introduce multiple independent
variables, thus introducing a new ordening parameter
\i . The new variables are given
by
x„=\i"x
and
A differentiation with respect to time or space can now be split up in a differentiation
to the fast variable (
or
) plus a differentiation to a slow variable (n>0). The
latter is of lower order than the former. We have chosen the modulation parameter
equal to the nonlinearity parameter, which gives the most general expansions. This
assumption is justified for for instance a standard Jonswap spectrum.
We now introduce the following expansion for the surface elevation
C=Ee"E
ti'l
^
ni'-n
6
(7)
in which w is the angular frequency of the first-order waves and C „.-m = Cn.m* to keep
Cn real. The Cnm
dependent on the slow variables. The same notation is used for 0
and X.
2.3
First-order solution
The perturbation expansions for the variables are substituted into the boundary-value
problem and the equations are solved. The solutions to first order read:
2(0 chq
loifnt
'cosp.
^ '
^••=ifi^
in which
c/2=cosh
A=A{t^
q=kh
(10)
, the slowly-varying complex first-order wave amplitude,
Q=k(z+h)
p.^l^h
P.=lj{z+h)
k is the positive root of the dispersion relation
(ji^=gk \2sMh
1 j is the positive and real root of
-oi^=gl. tanhi^z
with 0-1/2) 7r < Ij h < j TT for j = 1,2,3,...
The constants B and Cj are given in the appendix. The first terms in the expressions for
7
d l and
are the same as those obtained by Mei (1983). The second term describe the
so-called evanescent modes. They are vertically standing waves, with horizontally
decaying amplitudes. They arise because the wave board does not produce the correct
velocity profile over the water depth, but only an approximation. So the progressive
waves do not fulfill the boundary condition on the wave board and evanescent modes are
generated so that the sum of progressive waves and evanescent modes does fulfill this
condition.
2.4
Second-order solution
We now consider the second-order solution. The solution to the second-order boundary
value problem for the velocity potentials is given in the appendix. Here we only deal with
the second-order control signal for the wave board.
First, we deal with the second-order first-harmonic case. In the multiple-scales method
the first-order waves oscillate with the peak frequency of the first-order wave spectrum
while more frequencies are present near the peak of the first-order wave spectrum. This
gives rise to the second-order first-harmonic term, which descibes the modulation of the
frequency of the first-order waves. In full nonlinear theories this term is present implicitly
in the first-order solution.
The control signal for the second-order first-harmonic waves is given by
0) " ' I h+2lo>h
f K / z ^ ! ^ 2 i f i - ^ X , , ^
-h
s
-h
iz=0)
(11)
The first and second terms arise from the fact that the first-order wave-board motion is
not correct on the longer time scale slow motion of the wave board. The third term
produces the correct bound second-order
first-harmonic waves. The fourth term
compensates for the finite stroke length of the wave board. Note that the slowly varying
wave-board position, which will produce the second-order low-frequency waves, is
responsible for this term.
8
For a piston-type wave board (only translation) equation (11) reduces to
8
-21
dA
I
g
2co^ö ^t^ 2(^^Cg
in which
tanpj\
1+-
J ! , C,
1-1 Pi+Pj
dA
dt^
is the group velocity of the first-order waves and
The first two terms in equation
proportional to
^10'
Ih
^(12)
is given by
12 describe the frequency modulation; they are
— . The third term still contains the slow wave-board motion
X.q .
An expression for this quantity will be given below.
The expression for
is modified for a rotating wave board by multiplying all but the
first term on the right-hand side with the factor
h+l
h+ll
(13)
For the second-order super harmonics the control signal reads
h+2l /iw
(15)
-h
The first term on the right-hand side produces the second-order super harmonics and the
second term compensates
for the finite wave-board stroke. This equation
can be
evaluated for a piston-type motion as
3
gk
g
^
r
[4 b'
(16)
'
9
The factors Gj and Hy are given in the appendix. In the case of a rotating wave board
all terms in the right-hand-side have to be multiplied with the same constant as in the
first-harmonic case.
Finally, the sub-harmonic control signal can be calculated from
0
(17)
The first two terms produce the low-frequency water motion close to the wave board and
the last term describes the influence of the finite wave-board stroke.
(J),^
describes
the bound low-frequency waves which arise from the slow modulation of the first-order
wave amplitude, ^^o
describes the low-frequency motions with an x-dependence, which
arise from the first-order evanescent modes.
and (p^Q are the integration constants of the first-order solution. Their magnitude is
first-order, but they appear in the problem only at second order, so their physical
influence is of second order. This is due to the fact that they vanish when being
differentiated with respect to the fast variables; they only appear at differentiation with
respect to the slow variables.
From equation (17) we get the impression that X^g is influenced by the evanescent
modes. However, because the bound low-frequency waves far from the wave board are
not z-dependent, continuity tells us that the wave-board motion must not depend on zdependent quantities. Indeed it can be shown that the z-dependent terms in equation
(17) cancel and that the wave-board motion can be written as
(18)
10
in which the overbar indicates time averaging over the slow time variable. This equation
shows that the volume flux due to the wave-board motion produces the surface elevation
of the low-frequency waves, just as one would expect. The control signal is then given as
ƒ»+/
^io=2f:^^
h+2l h
)
!(^20-C^
0
(19)
dh
which for a piston-type wave-board motion can be evaluated as
^ 1 0 = — ^ x — / ( i ^ i ' - i ^ ^ i
2hiCg^-gh)\
(20)
2/J
in which n is a depth dependent quantity given by
n = U - ^
2 smh2g
(21)
For a rotating wave board the right-hand-side of equation 20 has to be multiplied by the
factor given in equation 13.
From equation 20 one can get the impression that
because it is proportional to
X,o is of second-order magnitude
|/lp . However, due to the integration with respect to
the magnitude of this term is first-order. (Its magnitude is of order
e^/n .)
The total control signal for random wave generation up to second order now becomes
X(f,fi)=X,i(fi)e-'"'+Xio(fi)+X2i(fi)e-'"'+X22(r,)e-^"'
(22)
The quantities on the right-hand side are found in equations 10, 12, 16 and 20 for a
piston-type wave-board motion. For a rotating wave board the expressions for X21, X22
and Xjo have to be corrected with the factor given in 13 in the indicated way. Note that
the control signal consists of four terms only!
11
2.5
Implementation
Now that we have obtained the control signal for the wave board up to second order we
will give the recipe for the generation of the complete control signal.
- First, the peak frequency of the given first-order energy-density spectrum has to be
determined. From this frequency the wavenumber
and the quantities
q
k , the group velocity
and Pj (given after equation 10) have to be calculated.
- Secondly, a time series for the required first-order surface elevation
Cn
has to be
generated from the given first-order energy-density spectrum. In this way A(tj)
is determined. We used the random-amplitude/random-phase method described
by Tucker et al.(1984).
- Third, the control signal for the wave board has to be calculated in the time domain
from equation 22. We perform the time integration in equation 20 with the
modified midpoint rule and the time differentiations with central differences. The
accuracy of both operations was second order.
These three steps are sufficient to obtain the control signal for the wave board correct
up to second order. To obtain this signal an FFT and a few extra multiplications have
to be performed. In the conventional method an FFT, a few extra multiplications and a
convolution are needed. An FFT needs 4pN multiply-add operations in which N the
number of time steps and
p=HogN
(see for instance Bendat and Piersol, 1986). For
the convolution in the conventional method N^ operations are needed, as can be
observed in Barthel et al.(1983) and Sand and Mansard (1986). I f we neglect the extra
multiplications (of which more are needed in the conventional approach) the gain in
computational speed of the new method compared to the conventional method is
4pN+N^
4pN
N
4p
In a typical experiment
N will be of the order
12
i C , so that the gain in speed will be
a factor 250, which is a big factor indeed.
Note that the equations can also be used to generate second-order monochromatic and
bichromatic waves.
13
3.
3.1
EXPERIMENTS
Experimental arrangement
To verify the theory, experiments were conducted in a wave flume of 40 m length and
.8 m width. The water depth ranged from .42 to .50 m. Wave-height meters of the
conductance type were placed in the horizontal part of the flume. The water-surface was
measured by the meters simultaneously at 20 to 50 Hz. At 19 m from the wave board a
1 in 25 concrete slope began.(See figure 2.) In one experiment a concrete bar of 10 cm
height was placed on the sloping bottom. The shape of this bar like that of a Gaussian
distribution.
The experiments were performed with bichromatic and continuous first-order spectra. In
the case of the bichromatic energy-density spectrum of the first-order waves four waveheight meters were placed at distances of respectively 10,14, 16 and 18 m from the wave
board. The reflection of the beach was reduced by absorbing material on the water Hne
to be able to fully concentrate on the generation and leave the reflection on the wave
board out. The still-water depth was .50 m and the sample frequency was 50 Hz.
In the case of the continuous first-order spectra six wave-height meters were used at
respectively 6, 10, 12, 14, 15 and 18 m from the wave board. The absorbing material on
the waterline was not present. The still-water depth was .42 m and the sample
frequencies were 10 and 20 Hz.
The flume is equipped with a hydraulically-driven wave board with an active waveabsorption system. To this end wave-height meters are fwed to the wave board. They
measure the instantaneous water-surface elevation on the board. This signal is integrated
in time to obtain a wave-board position. This position is then compared with the
previously calculated position and the difference is compensated for by an extra
movement of the wave board. In this way waves which are reflected from the beach are
absorbed. This absorption system has been used succesfuUy by Kostense (1984).
14
We performed a test of this absorption system. To tliis end tlie slope was replaced by a
vertical wall 38 m from the wave board. The reflection coefficient of the wave board at
a range of frequencies was determined. This coefficient was obtained in the following
way. The wave board produced waves of a certain frequency until a steady situation
occurred with standing waves. A wave-height meter was placed at a surface-elevation
maximum. Then the absorption system was switched on. The reflection coefficient is
given by the ratio of the amplitude of the standing waves after the absorption system was
turned on to that of the standing waves before the absorption system was turned on.
In figure 3 the variation of the reflection coefficient as function of frequency is given. For
frequencies higher than .1 Hz the reflection coefficient is well below 10%, which is
acceptable for our purpose. High-frequency waves will break on the beach and their
reflection back to the wave board will be very small. Low-frequency waves will reflect
nearly 100% on the beach, but are of second order. The reflection at the wave board will
reduce them to third order and we want to test the second-order theory.
A problem is formed by the waves with frequencies lower than about .05 Hz. The
reflection coefficient of the wave board is too high in this case. It means that these waves
are rereflected by the wave board and will travel to the beach again. Of course, this does
not happen on a natural beach. The reason for this high reflection is probably leakage
of water below and along the sides of the wave board. In our case it means that we can
only test the model down to .05 Hz. However, for instance surf-beat phenomena are
usually above this frequency (after scaling up, of course).
3.2
Data processing
The data are Fourier transformed and the influence of noise which is uncorrelated with
the wave signals is reduced by means of Complex Harmonic Principal Component
Analysis (CHPCA). This last method was developed by Wallace and Dickinson (1972)
and has been widely used in meteorology and oceanography. Recently the method found
its way in coastal engineering. An example of this is given by Tatavarti e.a.(1988), who
showed with CHPCA that the reflection coefficient of low-frequency waves can easily be
15
Sand, S.E. (1982) Long wave problems in laboratory models, J. Waterway, Port, Coastal,
Ocean Engtig. 108,(WW4) 492-503.
Sand, S.E. and Mansard, E.P.D. (1986) The reproduction of higher harmonics in
irregular waves, Ocean Engng. 13(1), 57-83.
Tucker, M.J., P.G. Challenor and D.J.T. Carter (1984) Numerical simulation of a random
sea: a common error and its effect upon wave-group statistics, AppL Ocean Res.
6(2), pp. 118-122
24
Appendix
The constants B and Cj in equations
(8), (9) and (10) are found by substituting the
expression for 0|, in the first-order version of the boundary condition at the wave board
(equation (5)). Multiplication of this equation by chQ or by cosPj and integration over
depth gives B respectively Cj as
_2o>
2cli'q
B=sh2qkish2q+2q)
kQi+T)
C,=
2(0
lj{sin2p.^2p)
sin2pj-
(Al)
2cos'pj
(A2)
These expressions reduce in the case of as piston-like motion of the wave board to
P_ (0
k
C,= -
2sli2q
(A3)
sh2q+lq
2sin2p.
(0
(A4)
/. sin2p.+2pj
The solution to the second-order
first-harmonic equations is given by
(}),, =Dc/iQe'^+y; EcosP^-'/^-"
^
Itjij:^
^
J
J
C
PsinP,
I Xj-^
Bcospj
2(0^
chq dx.
U{Xj-l)xcosPj
,-'/(dA_
(A5)
dx,
in which D and Ej are constants which still have to be determined. The term with the
factor D describes free waves which will travel away from the wave board. Because we
don't want these second-order free waves we put D equal to zero. Ej is found by applying
Greens' theorem on 0,, and cpji-The area of integration is
25
[ix,z) | O^x^L , -h^z^O)
,
in wliicli L is a point far from the wave board where the evanescent modes have died
out. We then find after straightforward but tedious algebra
dA
^ 2o>cosp.[U
Bil^^l)
(A6)
231. dx,
The constant Xj is found from the condition that the first-order wave amplitude is
independent of x as
Xj=iCg
(A7)
For the superharmonics the solution for
is given by
<^,,=Fch2Qe'""^Yl GjCosiPj-iQ)e'"-''^^Y,
j-\
ij-o
H.cosiP.^Ppe'"'''''
(A8)
in which
F=-
(A9)
16 sh^q
1 Cj
4(0 B
i6u>'-48^kl/-g'k'^8Hj)
4<n2cos(pi-iq)+g(l.-ik)sm(pi-iq)
(AlO)
and
1 C f j
4(0 B2
Ooy'^lg^lj^gHj)
-iA' .
4(^^cos(p^+Pj)+g(l^+ipsm(p.+pp'
(All)
In the subharmonic case we will give the surface elevation of the bound waves. I t
contains all information of the solution because the waves are not z-dependent. It is
identical to the expression obtained by Longuet-Higginns and Steward in 1962. The
26
surface elevation is given by
gh-Cl
in which
S
the radiation stress of the first-order waves. It can be evaluated as
27
28
as
0.28
026
Reflection
022
coeflcient
02
0.18
OJ
Frequency (Hz)
Figure 3: Reflection coefficient of the wave board with active wave absorption
function of frequency.
29
ENERGY
DENSITY
(cm^/Hz)
6 -
3i
0
0.5
1
1.5
2
FREQUENCY (Hz)
Figure 4: Measured one-sided energy-density spectrum in the Jonswap case. e,=,06 (see
text).
30
ENERGY
DENSITY
2
(mm /Hz)
FREQUENCY (Hz)
Figure 5: One-sided low-frequency wave energy-density spectra. The solid line is the
theoretical bound low-frequency wave spectrum (Laing 1986). The other lines are
the spectra of the decomposed waves: the line with the + for the outgoing bound
waves, the line with the x for the outgoing free waves and the line with the boxes
for the incoming free waves. The spectral densities of the incoming free waves are
divided by 10 to fit in the figure.
31
1
Coherence
0.5
1
0.1
0.2
0.3
frequency (Hz)
Figure 6: Colierence between tiie liigii-frequency envelope and the outgoing free lowfrequency waves for the Jonswap case. The boxes are the estimates for the
coherences and the drawn line indicates the 95% confidence interval on zero
coherence.
ENERGY
DENSITY
(cm^/Hz)
0.5
1
1.5
FREQUENCY (Hz)
Figure 7: Measured one-sided energy-density spectrum for the broad spectrum case.
e,=0.08 (see text).
32
FREQUENCY (Hz)
Figure 8: One-sided low-frequency wave energy-density spectra. Tlie solid line is the
theoretical bound low-frequency wave spectrum (Laing 1986). The other lines are
the spectra of the decomposed wave components: the line with the + for the
outgoing bound waves, the Hne with the x for the outgoing free waves and the line
with the boxes for the incoming free waves. The spectral densities of the incoming
free waves are divided by 10 to fit in the figure.
33
1
Coherence
0.5
1
O -\
;
^
^
0.1
0.2
0.3
1
Frequency (Hz)
Figure 9: Colierence between the high-frequency envelope and the outgoing free lowfrequency waves for the broad spectrum case. The boxes are the estimates for the
coherences and the drawn line indicates the 95% confidence interval on zero
coherence.
34