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 **

CONTENTS

**Abstract 1 **
1. INTRODUCTION 2

2. PROBLEM FORMULATION 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
**3.1 Experimental arrangement 14 **
**3.2 Data processing 15 **
**3.3 Noise level and error analysis 16 **

**4. TEST OF T H E NEW CONTROL SIGNAL 18 **

**4.1 Bichromatic spectra 18 **
**4.2 Continuous spectra 18 **
**Table 1 18a **
5. 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 complex-harmonic principal-component analysis to reduce the influence of noise. The validity of the new method is confirmed.

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

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.

2. PROBLEM FORMULATION AND 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)

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 wave-board 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

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
first-order 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 = E e " E ^ (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. ^ ' **

**^ • • = i f i ^ (10) **

*in which A=A{t^ , the slowly-varying complex first-order wave amplitude, *

*c/2=cosh q=kh 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,...

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

(11)

**0) " ' I h+2lo>h ****f K / z ^ ! ^ 2 i f i - ^ X , , ^ iz=0) **_{-h -h}_{ s }

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.

For a piston-type wave board (only translation) equation (11) reduces to
*8 dA g *
*-21 *
*2co^ö ^t^ 2(^^Cg *
**I tanpj\ J ! , C, ***1+-1-1 Pi+Pj *
*dA* ^ 1 0 '
*dt^ Ih * ^(12)

in which is the group velocity of the first-order waves and is given by

The first two terms in equation 12 describe the frequency modulation; they are
* proportional to — . 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 *

*-h *

(15)

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 *

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

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 dependent, continuity tells us that the wave-board motion must not depend on z-dependent quantities. Indeed it can be shown that the z-z-dependent terms in equation (17) cancel and that the wave-board motion can be written as

**0 **

(17)

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»+/ ) **

*h+2l h *

^ i o = 2 f : ^ ^* !(^20-C^ dh* (19)

**0 **

which for a piston-type wave-board motion can be evaluated as

**^ 1 0 = — ^ x — / ( i ^ i ' - i ^ ^ i (20) **
**2hiCg^-gh)\ 2 / J **

in which n is a depth dependent quantity given by

*n = U - ^ (21) *

2 smh2g

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 X,o is of second-order magnitude because it is proportional to |/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!

**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 k , the group velocity *
*and the quantities q 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). If 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^ N *
*4pN 4p *

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.

3. EXPERIMENTS

3.1 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 wave-height 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 wave-absorption 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).

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

*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

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

*B=-* _2o>
*kish2q+2q) *
*sh2q-2cli'q *
*kQi+T) *
**C,= ** **2(0 **
**lj{sin2p.^2p) ***sin2pj-2cos'pj *
(Al)
(A2)

These expressions reduce in the case of as piston-like motion of the wave board to

**P_ (0 2sli2q **

*k sh2q+lq * (A3)

**C,= -** **(0 ** *2sin2p. *

*/. sin2p.+2pj * (A4)

The solution to the second-order first-harmonic equations is given by

(}),, =Dc/iQe'^+y;
**EcosP^-'/^--" ^ ^ J J**** 2(0^*** chq dx. *
*C I *
*Itjij:^ Bcospj *
*PsinP, *
*Xj-^ U{Xj-l)xcosPj ,-'/(dA_ *
*dx, *
(A5)

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 [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

**^ 2o>cosp.[U Bil^^l) 231. ***dA *

*dx, * (A6)

The constant Xj is found from the condition that the first-order wave amplitude is independent of x as

*Xj=iCg *

For the superharmonics the solution for is given by

(A7)

*<^,,=Fch2Qe'""^Yl GjCosiPj-iQ)e'"-''^^Y,*

*H.cosiP.^Ppe'"'''''*

*j-\ ij-o * (A8)
in which
*F=-16 sh^q * (A9)
**1 Cj i6u>'-48^kl/-g'k'^8Hj) *** 4(0 B 4<n2cos(pi-iq)+g(l.-ik)sm(pi-iq) * (AlO)
and

**1 C f j Ooy'^lg^lj^gHj)***( A l l ) In the subharmonic case we will give the surface elevation of the bound waves. It 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*

**4(0 B2 4(^^cos(p^+Pj)+g(l^+ipsm(p.+pp'**-iA' .surface elevation is given by

*gh-Cl *

**Reflection **
**coeflcient **
**as **
0.28
**026 ****022 *** 02 *
0.18
OJ

**Frequency (Hz)**

Figure 3: Reflection coefficient of the wave board with active wave absorption function of frequency.

**ENERGY **
**DENSITY **
**(cm^/Hz) **
**6 **
**-3 i **
**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).

**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.

**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
low-frequency 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).

**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.

**1 **
**Coherence **
**0.5 1 **
**O -\ ; ^ ^** 1
**0.1 0.2 0.3 **
**Frequency (Hz) **

Figure 9: Colierence between the high-frequency envelope and the outgoing free low-frequency 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.