Second order non-linear effects in random ocean surface waves

SECOND-ORDER NON LINEAR EFFECTS

IN RANDOM OCEAN SURFACE

_WAVES

by

A.R. OSBORNE (*)

M. DOGLIANI (**)

Technical Bulletin N. 102 Genova, August 1988

PREAMBOLO

Gil effetti idrodinamici di tipo quadratic° sono spesso importanti nelle strut ture marine, ne sono esempio ii momento flettente sulle navi

o le oscillazioni lente di uno scafo ormeggiato.

L'ingegneria navale si occupa di questi effetti non lineari da

molto tempo; tuttavia mentre molte risorse sono state spese nell'analisi degli scostamenti della risposta da un comportamento lineare non ci si 6

soffermati a lungo sulle non linearita delle onde.

Per questo motivo ii RINA, in collaborazione con Snamprogetti

(Fano) e Tel (Milano), ha lanciato un programa di ricerca congiunto per

Jo studio delle non linearitd delle onde di on mare tempestoso.

In tale ambito al professor Osborne 6 stato assegnato l'incarico di studiare un modello di secondo ordine delle onde di super/ice.

Ii rapporto finale di tale studio, pubblicato nel seguito, puO

risultare utile al let tore per una migliore comprensione della fisica delle onde.

L'impostazione generale del problema sard illustrata in on

successivo Bollettino di prossima pubblicazione.

PREAMBLE

Hydrodynamic quadratic effects are often very important in marine structures, examples are wave bending moment on ships or slow drift oscillations of a moored vessel.

Engineers and marine architects have been studying non-linearities for many years and much effort has been given over to the analysis of

the displacements of the response from linear behavior but often disregarding the inner non-linearities of sea waves.

For this reason BINA, in collaboration with Snamprogetti (Fano) and Tei (Milano), has started a _{joint industrial research program the goal}

of which is to take into account also non-linearities of waves in a

confused sea.

In this context professor Osborne has been given the task of

studying a second order model of sea surface waves.

The final report of this study is published here in the hope that

it may be useful to the reader for a better understanding of wave physics.

The general formulation of the _{problem will be discussed in a}

SUMMARY

New deterministic procedures are developed for estimating second-order nonlinear

effects in random ocean surface waves. The methods are equivalent to the familiar stochastic, hi-spectral approach for computing second order nonlinearities. The new formulation developed herein, however, has the advantage that no bi-spectral calculation need be made. This is because all second order corrections may be made in terms of first order quantities. In order to simulate random sea state which includes second order nonlinearites it is sufficient to first assume some power spectrum such as Pierson-Moskowitz or JONSWAP. One then uses any of several

standard approaches (which assume uniformly distributed random phases) to compute a

realization of a linear, random wave train. The second order correction is then made by a simple nonlinear correction to the first order theory. The correction requires knowledge of the envelope function of the linear wave train: we show that the envelope is easily computable using the

Hilbert transform We give several examples of second-order: nonlinear random wave trains using the approach outlined here.

CONTENTS

I.

Introduction

5

IL Theoretical Aspects of Second Order Nonlinear Effects in Ocean

Surface

Wave

Motion

6

Estimating the Degree of Nonlinearity in Wave Data

16

Ma. Estimating the Level of Nonlinearity of a Measured Time Series

17

Mb. Estimating the Contribution of Radiation Stress of a Measured Time

Series

20

Determining the Envelope and Phase of a Random Wave Train with the

Hilbert

Transform

Estimating Second Order Nonlinearities of the Measured Wave Data of

Snamprogetti

and

TEI

Analysis

of

Snamprogetti and TEI

data

30

Computation of Second Order Nonlinearities in Random Waves

Simulated by the Pierson-Moskowitz Spectrum

31

Nonlinear Fourier Analysis of Simulated, Nonlinear Random Waves 46

Bispectn-1 Analysis of Simulated, Nonlinear Random Waves

51

Conclusions

and

Recommendations

52

References

₅₄

Table

Captions

57

Figure

Captions

58

3

..

.

Flow Charts

Flow Chart I

A Second Order Integrable

Theory for Narrow-Banded,

Nonlinear Surface Wave Motion in Shallow Water

9

Flow Chart II

A Generalized, Second Order Integrable Theory for Narrow

Banded, Nonlinear Surface Wave Motion in Arbitrary Water

Depth

12

Flow Chart III

Procedure For Graphing the General Nonlinearity Parameter

for Ocean Waves as Derived from the Theory of Hasimoto and

Ono

19

Flow Chart IV

Procedure

for

Determining

the

Contribution

of

Radiation Stress of a Measured Time Series

21

Flow Chart V

Procedure for Determining the Hilbert Transform of a Random

Wave

Train

- 27

Flow Chart VI

Radiation Stress from the Pierson-Moskowitz Spectrum in

Shallow

Water

35

Flow Chart VII

Radiation

Stress from the

Pierson-Moskowitz Spectrum

in

Arbitrary

Water

Depth

38

...

... . ...

...

...

.

I. Introduction

Here we study the general aspects of second order nonlinear effects in ocean surface waves. Others have studied this problem from the point of view _{of random ocean surface waves}

(Tick, 1963) (Hasselmann, 1967). Generally speaking these formulations consist

_{of a}

stochastic linear first order motion which is assumed to be gaussian and a second order

nonlinear correction which is responsible for the nongaussian component of the sea surface elevation. In formulations of this type one generally speaks of a power spectrum at first order

and a bi-spectrum at second order. Thus the power spectrum describes the major linear

component of the sea surface and the bi-spectrum contains contributions at second ordur in nonlinearity. In an engineering sense one is comfortable with dealing with such familiar forms for the power spectrum as the JONSWAP and Pierson-Moskowitz spectra. At second order

there is as yet no well formulated consensus as to an appropriate physical form for the

bi-spectrum. Thus stochastic formulations, while theoretically very rigorous and satisfying have yet to yield a formalism directly applicable to the engineering study of nonlinear wave motion.

One of the major aims of this study has been to develop an entirely _{new formalism}

which allows rigorous determination of second order nonlinear effects in random _{ocean waves;} the second order contribution to the motion_{is determined directly from the first order formalism} normally used in engineering applications. Almost everyone is familiar with linear _stochastic simulations of the sea surface elevation used, say, in compliant structure design, drilling riser design, or floating or moored structure design. This procedure assumes a Pierson-Moskowitz

or JONSWAP power spectrum and generates, in the_{random phase approximation, a linear}

gaussian realization of the sea surface(see Osborne (1982) and references cited therein). With the new formulation presented herein, _{one can modify in a simple way such simulations to} include the effects of second order nonlinearities. We caution that while the mathematics of the formulation presented herein is not simple, the final results are simple and easily applicable to engineering problems normally treated_{stochastically only at first order.}

II. Theoretical Aspects of Second Order Nonlinear Effects in Ocean

Surface

Wave Motion

In order to develop the new concepts needed herein we consider radiation (pressure) stress in ocean surface waves in two dimensions (1 space + 1 time) and derive the concept from the Korteweg-deVries (KdV) equation using the multiscale expansion procedure developed by Whitham [1973] and subsequently used by Zalcharov and Kuznetsov [1986] to study the theory of systems integrable by the inverse scattering transform. Here we summarize results presented elsewhere (Osborne and Boffetta, 1988a) in which we derive the nonlinear Schroedinger (NLS) equation directly from KdV and in which we show that the amplitude of a shallow water wave

train or wave field may be described as a modulated second order Stokes wave whose

(complex) modulational envelope is a solution of the NLS equation. The second order Stokes wave is characterized by a (nearly constant) slowly varying mean sea level which results as a consequence of radiation stress. We discuss how radiation stress has nonlinear Fourier spectral representations (in terms of hyperelliptic functions) of both the KdV and NLS equations (Tracy, et. al, 1987a, 1987b). A natural consequence of the formulation is that phase locking occurs

among the linear Fourier components of a wave train evolving according to KdV/NLS

dynamics; the phase speed of both the fundamental carrier and the first harmonic are therefore nearly the same. We further discuss how radiation stress (proportional to the squared envelope of the wave field, is a solution to the ordinary KdV equation and how the NLS phase function is a solution to the potential KdV equation. A generalization of these methods to arbitrary water depths as found by Osborne and Boffetta (1988b) is also given.

Principle Physical Results

The NLS equation may be derived from the KdV equation (using the multiscale expansion approach); NLS has _{as its solution a complex function A(x,t)ei4)(x,t) which serves to} modulate an associated second order Stokes wave. This modulated Stokes field describes the second order nonlinear time evolution of the amplitude of the sea surface and consists of a linear superposition of a cos() term, a cos20 term and a slowly varying term -yA2(x,t) (see Figure

II. 1). The linear Fourier spectrum thus has three peaks centred on these frequencies; the

dominant first harmonic peak occurs at the frequency (f1 of Figure 11.1) of the cos0 term, the

second order peak at cos20

_{occurs at twice this frequency (2f1 in Figure 11.1) (second} harmonic) and the -yA2 term appears at low frequency. The cos° term describes the carrier or fundamental together with associated (nearly continuous) side bands.

The cos2O term in the second order Stokes wave is such that phase locking occurs between the fundamental (carrier) cos0 and the first harmonic cos20, i.e. the phase velocities of these two terms are nearly equal. This effect has been also observed in the Zakharov equation (Yuen and Lake, 1982). The cos20 term describes what is normally referred to as the second harmonic.

The -yA2(x,t) term in the Stokes wave expansion may be identified with radiation

stress, a concept first considered in the context of water waves by Longuet-Higgens and

Stewart (1964) (see also LeBlond and Mysak, _{1978). Radiation stress physically is a slow}

space-time variation in the mean level caused by the presence of an overlying rapidly oscillating wave train, which depresses the mean sea level proportional to the squared amplitude of the modulation envelope.

Radiation stress may be shown to have a nonlinear spectral decomposition in terms of the basis functions (nonlinear normal modes) of both the KdV and NLS equations (Osborne and Bergamasco, 1985, 1986) (Tracy, et al , 1988a,b). Thus nonlinear Fourier analysis _may be used to described the dynamics of radiation _stress.

Given that A(x,t)e4(x,t) is a solution to NLS it may be shown by a multiscale expansion of the NLS equation that radiation stress -yA2(x,t) is a solution of the KdV equation and that 04,t) is a solution of the potential KdV equation. (4

The intrinsic nonlinearity of a wave train may be estimated from the coefficient of the cos20 term. This general rwnlinearity parameter, which we refer to as Ug, reduces to the Ursell number in shallow water and the sea slope in deep water. Graphing Ug for data in terms of significant wave height and dominant or zero crossing period provides estimates of the relative nonlinearity of different sets of data.

On page 7 Flow Chart I presents the principle results of our study for the case of

shallow water. This case is simpler than the deep water case and hence serves to introduce the

reader to the concepts before attempting to understand the more complicated deep water

problem. Flow Chart H summarizes our results for the general water-depth case. Recall that the physics of the theoretical formulation given herein is illustrated in Figure 11.1 which describes,

roughtly speaking, several important features of unidirectional random, nonlinear wave propagation. Figure 11.1 shows a typical wave train (la) with its modulation envelope and

radiation stress components. A typical Fourier spectrum is given in (lb). Note the dominant spectral peak (the first harmonic centred on frequency f0) is complimented by a low frequency,

second order (radiation stress) contribution and a second order contribution (the second

harmonic centred on frequency 2f0). The effect of phase locking between the dominant peak and the second harmonic results in the phase speedsbeing nearly equal (1c).

9

Begin with the ordinary Korteweg-deVries equation in shallow water which has the surface wave elevation as its solution: ri = rgx,t)

rlx + coilx + 00111x + 1311xxx = 0 with constant coefficients

co = (gh)1/2 (linear phase speed),

h (depth), g = 9.8m/sec2 (acceleration

_{of gravity)}

3c0

term)

a =

_-)11

(coefficient of nonlinear

h2 co6 dispersive 13 =

(coefficient of

_term)

with the linear dispersion relation

(Do = cak0(1 h21c.02/6)

Flow Chart I

A Second Order Integrable Theory for Narrow-Banded, Nonlinear Surface

Wave Motion in Shallow Water

A

Do a multiscale expansion

_{on KdV in the case of a narrow banded wave train or wave}

field (Figure 1) to get the nonlinear

_{Schroedinger equation}

iwt + icgvx

Pvxx + viv2w = o

with solution w = iv(x,t) = A(x,t) e-idt + io(x,t) where A(x,t) is the real modulation

envelope, sl)(x,t) is the real modulation phase and o.)' is the nonlinear frequency

correction to the dispersion relation codko) _{coo +}

Here A(x,t) and 0(x,t) have _{a nonlinear spectral decomposition from the}

periodic NLS equation, i.e. A(x,t) and 45 (x,t) _{may be represented as the linear superposition of}

nonlinearly interacting nonlinear

_{waves or basis states (hyperelliptic functions).}

+

I

The constant coefficients of NLS are given by

acoo

Cg = 1-;TzT = c0(1 h2k02/6) (Group Velocity)

1.1 = ₂ h2kor₀

(Coefficient of Dispersive Term)

9c

V

16h4k0

(Coefficient of Nonlinear Term)

Note that phase locking arises naturally between the carrier term cos() and the second

harmonic term cos20 in the Stokes field since the phase velocity may be computed from

_co_

a/at

azeiat

f

' Associated with the NLS equation is a Stokes field for the surface wave elevation as an

approximate solution to the KdV equation (in the narrow-banded a. proximation)._

33

A2(x,t) A(x,t) A2(x,t)

i(x,t)

+ cos20(x,t)

4k02h3 cos8(x,t) + 4k02h3

1

-where the phase 0(x,t) is given by 0(x,t) = kox (coo + co1)t + 0(x,t)

The NLS spectrum is said to be "empty" (i.e. there is no modulation) when A =ao and (4 = 00, where ao and 00 are constants. In this case the Stokes field reduces to a

second order Stokes wave

3a02 3a02

li(x,t) ,

4k02h3 + ao cose +4k02h3

cosn

where the phase 0(x,t) is given by 0(x,t) = kox (c)o + o')t + 400,

and the nonlinear frequency correction is

9coao

'

co =

Note that radiation stress results by taking

field <1-1(x,0> =

a simple space or time average of the Stokes

3

A2(x,t)

41(02h3

IG

Do a multiscale expansion on NLS to show that given the modulation A(x,t) from NLS,then

there is an associated KdV equation for radiation stress

ii(x,t) 3 A2(x,t)

41c02h3

I

1 H

Radiation stress has a nonlinear Fourier decomposition for the KdV equation:

N

<11(x,0> =

i=1

1 I

In the above derivation of a KdV equation for radiation stress we also find from the multiscale expansion that the modulation phase (1)(x,t) of NLS is a solution to the

potential KdV equation (pKdV)

Ot + COx + p(.1),()2 + 130,,,x + const

= 0

3 _1,

P = Tcoropiis02

1

The modulation phase has a nonlinear Fourier decomposition for _pKdV:

N

e(x,t) = / Vj

F'

-Flow Chart II

A Generalized, Second Order Integrable Theory for Narrow Banded, Nonlinear

Surface Wave Motion in Arbitrary Water Depth

Begin with the Hasimoto and Ono (1972) form of the nonlinear Schroedinger equation

(NLS) for arbitrary water depth h, as reformulated by Osborne and IBoffetta, 1988b. This nonlinear second order equation was derived by them from the Euler equations in 1 space

and 1 time dimensions; the dimensional form of NLS is: iNft + iC2V + v1W12tv = 0

The NLS equation has the complex modulation solution

N.r(x,t) = A(x,t)e-ico't + ickx,t)

Here A(x,t) and 0(x,t) have a nonlinear spectral decomposition for the periodic NLS

equation. The NLS spectrum is said to be "empty" (i.e. there is no modulation) when A =

The constant coefficients of NLS are given by

Cg =

au). 1(1 (52)k0h 1

-3r-co- = -5c [1 + )j

(Group velocity)

G

with dispersion relation

2

coo = gkoa, a = tanh koh and

1 a2(0 ,___...9_ - ___S____ { [a_koho_cy2)]2 +

42h22(1-2))4

= z ak02 skocywo V = ' _t_ _{[4c2 4(1-62)cCg} -, + +gh(1-(32)2] + (9-10(52+90-4) 2co, ,2c5

- .gh

20.2 coo gc5\1/2

c =

=

(Phase speed)

ko

AO

A

13

-The associated Stokes Field approximation to the free water surface elevation is given by:

yA2

r,

CgA0 i

_CA

.

ri(x,t) = + ALI + - xj cos() +

--

sinar, + cos20

4k0a _wo coo 4k0cr

where A = A(x,t) is the real modulation envelope and

0 = 0(x,t) = lcox - (coo + d)t + (1)(x,t) is the total phase. Here

00(x,t) = kox - wot

is the carrier phase and (1)(x,t) is the modulation phase. The constants y, 6 are given by

1 Y -

_{gh _ Cg z}

f2cookoCg + (1 - a2) ghk02] (3 G2)ko2 5

-1

(7-and the nonlinear frequency correction is

a2i6T2.

c)' = v

4(1)02

When the spectrum of NLS is empty (A=ao=const, 0=00) then the Stokes field reducesto a

second order Stokes wave

yao26a02

ri(x,t) =

_{+ ao cos0 + cos20}

4Ic0a 4k,oa

where the phase 0(x,t) is given by 0(x,t) = kox

_{- (wo + (-0t + $o}

and 2,a02

44)2

-Note that phase locking arises naturally between thecarrier term cos() and the second harmonic term cos20 in the Stokes field since the phase velocity may be computed from

co aeiat a2elat

c

-7

ao/a,

a20/ax

and is hence the same for the carrier and second harmonic.

Recognize radiation stress as being a simple space or time average arising from the constant term in the second order Stokes wave (proportional to A2):

yA2(x,t) <ii(x,t)> =

4koa

Do a multiscale expansion on NLS to show that given its modulation envelope A(x,t), then A2(x,t) is a solution to the KdV equation. Thus we obtain a KdV equation for radiation

stress in terms of

yA2

ri(x,t) =

4koa The KdV equation is given by

lit + 0Th + &Mix +

13*Tlxxx = 0

where the coefficients are

2p.vn*

c* = Cg

Cg c

21..tvY

a*

; Yo = Yo(Cg c) 4k0cr i 112

15

-The modulation phase has a nonlinear Fourier decomposition for pKdV:

0(x,t) = v j

j=I

Radiation stress has a nonlinear Fourier decomposition from the KdV equation:

N

<11(x,t)>. = 1 p.i

j=1

J

The modulation phase 0(x,t) is a solution to a potential KdV equation pKdV Ot + COx + 1 p()2 + 13Oxxx + const

= 0

3v

P =

2no' (Cg c)2 no 2ttv -

-HI. Estimating the Degree of Nonlinearity in Wave Data

Consider the constant coefficient in front of the cos 20 term in the Stokes field (in arbitrary water depth, box D of Flow Chart II) which we here give the label I:

= =

A28 A2(3 0-2)ko

I

4alco

4a-In the shallow water limit koh << 1, a a koh and we find

3A2

I =

- AU

4k02h3

Thus in shallow water I is the amplitude "A" times the Ursell number U = 3A/41(02h3. In the deep water limit a a- 1 and we find

1

I = -2k°A2 = AUd

In deep water I is the amplitude "A" times a deep water Ursell number Ud. It is worth noting that Ud = k0A/2 (= the sea slope) is the nonlinear parameter that the NLS spectral problem sees in the deep wate- limit of the theory.

We are thus motivated to introduce a generalized Ursell number Ug, based upon

(3.1), valid for all water depths. Hence I = AUg where:

U g

-(3.3) (3.4) AS

A(3 - 02)k°

40k° 403 (3.1) (3.2)

-In deep water Ug is the deep water Ursell number (sea slope).

ug =

koA = Ud _(3.6)

The point is that if we want to make a shallow water Stokes expansion of the Euler equations

we use (3.5) (Ursell, 1953) and if we want a _{deep water Stokes expansion we use (3.6)}

(Kinsman, 1965). This suggests that instead of plotting the Ursell number we should plot the more general depth dependent parameter Ug, (3.4).

Ma. Estimating the Level of Nonlinearity Of a Measured Time Series

We may graph (3.5) in a rather special way; rewriting (3.5) for the shallow water case

ug = 1(0((k÷.1)2)

Then we rewrite the second bracket using_

coo _2rc

koh h

2itJ

_{where coo =}

and c =,coo/lco is the phase. speed. Then we get for Ug

3

(a) (cry

ug

4(2702 LIT) L h )

Take the log of this to get the form,

log Ug = log (4(237)2) + lo",

+ 210g cT

h (3.10) (3.7) (3.8) (3.9) as = = 17

-Thus for a fixed value of the wave slope a/h the above is a straight line with positive slope of 2.

Now lets go to the deep water case which we rewrite

=

The log of this is then

a CT1

log Ug = log it + log

(17)

- log

h

_

IT) (koh) = (t) (C-f- 1

)

Thus for fixed a/h this is a straight line of negative slope as a function of log (CT/h).

In general we want to graph equation (3.4) (Ug) as a way to characterize some particular time series. We have developed the procedure given in Flow Chart HI. Here we have used a = H5/2, where Hs is the significant wave height. Also we have set T = Tz, where Tz is the zero crossing wave height.

Note that Ug physically represents the fraction of the wave amplitude represented by the second harmonic (cos 20) contribution to the sea surface elevation.

(3.11)

(3.12)

Ug,

Flow Chart III

Procedure For Graphing the General Nonlinearity Parameter for Ocean Waves

as Derived from the Theor of Hasimoto and Ono

Measure a time series of surface waves in water depth h

Compute Hs and Tz from the time series using standard procedures

Iteratively solve co02 = gko tanh(koh) to get ko

27r I co

-Tz Hs(3 - (3.2)k0 U 8(33 wo

Compute the phase speed c =

Graph Ug as a function of cl'hz = k021

-IIIb. Estimating the Contribution of Radiation Stress of a Measured Time

Series

We now consider how to analyze measured time series for their radiation stress

contribution. Basically the idea is this: Given the time series, determine its Hilbert transform to get A(0,t) (see Section 4 below). Then the radiation stress contribution is

1 gA2

f

C

L4 -

1]

+ constant

<11> =

4 gh - Cg2 (3.13)

In the following Flow Chart (IV) we give a sequence of steps for constructing the radiation stress contribution to the mean sea level variation as determined from the second order general water-depth formulation of the theory. Note that in the Flow Chart we have selected the

arbitrary constant (3.13) to give a zero mean for the time series. This sequence of steps may be summarized in the following way. Given a measured time series 11(0,t), use the second order

nonlinear theory given herein to estimate the radiation stress component to the slowly

varying mean sea level.

The time series of the low frequency radiation stress component <T1(0,t)> can now be analyzed by spectral transform, Fourier transform or some other procedure.

Flow Chart IV

Procedure for Determining the Contribution of

Radiation Stress of a Measured Time Series

Measure a time series of surface

wave elevation in water depth h

Compute Tz from the time series using standard procedures

IIteratively solve wo2

= gko tanh(koh) to get lc,

co

G =

_gko

wo

Phase speed of the carrier C

=

a [Group

speed of the carrier Cg =

wo C +

(1 G2)koh

Determine the envelope of the time series using the Hilbert transform: A(0,t)

<710,0>_

1

grA2(0,0_,,72]

14

c

4

gh c2

L -21 -1 IMeasure (0.02 -IDetermine

IV. Determining the Envelope and Phase of a Random Wave Train with the

Hilbert Transform

As seen in the last Section, one of the major new results of the present formulation is the introduction of envelope and phase functions in the representation of the sea surface elevation. It is convenient at this juncture to introduce the Hilbert transform as a tool for determining the envelope and phase time series from time series of wave elevation. This section is adapted from Osborne, Bergamasco and Cavaleri (1987). Hence we discuss the discrete Hilbert transform as a tool for the time series analysis of data. The specific application is to determine the complex envelope function of the time series and then to compute the associated envelope and phase time series. This type of analysis is important for the study of wave groups and their properties and for computing second order nonlinear effects in the wave field and may be extended to include nonlinear Fourier analysis (Osborne and Boffetta, 1988a,b).

A fast and efficient method is implemented for taking the Hilbert transform of discrete time series of measured or computed wave motions. The method is based upon the discrete

Fourier transform used in conjunction with a simple 90-degree phase shifting filter; the

approach is equivalent to directly evaluating the Hilbert transform integral, but is considerably more efficient. It is assumed that the time series (1) is recorded at equal time intervals At for a period T and (2) is periodic over the interval T. These assumptions are compatible with the fast Fourier transform; it is this algorithm which forms the basis of the methods described herein

(Oppenheim and Schafer, 1975). For numerical check-out of the algorithm we then apply this

method to determine the Hilbert transform of a sine wave. Examples of the use of this

procedure in the analysis of ocean wave data are given in later Sections.

23

-It is often convenient to regard_{some wave motion (possibly random) as the real part of (4.1):}

X(t) = A(t) cos 0(t) = Re [A(t) ele(01 _(4.2)

The imaginary part of the complex time series C(t) is given by

Y(t) = A(t) sin 0(t) = Im [A(t) eie(t)] _(4.3)

The amplitude and phase may be determined from X(t) and Y(t):

A(t) = [X2(t) + y2(t)i1/2 _(4.4)

0(t) = arc tan [Y(t)/X(t)] _(4.5)

In the study of wave groups one normally measures X(t) and then seeks its envelope _function A(t); the behavior of the envelope gives information about the behavior of wave groups and second order modulation effects. To find A(t) one also needs the time series Y(t); to determine Y(t) one requires that C(t) be an analytic function. Thus one seeks to determine the analytic

function C(t) = X(t) + iY(t) given the real _{function X(t) along the real axis. This is a}

well-known problem in the theory of functions_{of a complex variable; the function Y(t) may be}

determined by the Hilbert transform:

Y(t)

L

x(T) dt

_(4.6)

It t

where the dashed symbol overlying the _{integral denotes the Cauchy principle value. For a fuller} discussion of the above topics_{see Beckmann (1967).}

Practically speaking we are concerned with a time series which is assumed to be periodic (since the 1-1-1 is periodic). This leads to problems of causality which, when dealt with rigorously, allows the discrete Fourier transform to be used, in combination with a 90° phase shifting filter, to compute the Hilbert _{transform. Specifically we write F(o) for the discrete}

Fourier transform of X(t) and F(w) for the discrete Fourier transform of Y(t).. Then the Fourier transform of Y(t) may be found by

F(o) = H(0)) F(0))

(4.7)

Where H(co) is a response function (900 phase shifting filter) given by

!()_ co<it

7c5co<0

(4.8)

Thus given either of the time series X(t) or Y(t) the Other can be determined. The procedure

which we follow is this. X(t) is assumed given and its discrete Fourier transform,Fx(c)), is

computed by the Ft- I. Then the Fourier transform of Y(t), Fy(co), is computed by (4.7).:Finally the inverse FFT of F(c)) is computed to give the time series Y(t). Finally the complex envelope function is given by C(t) X(t) + iY(t); the envelope A(t) and phase OW are computed from (4.4) and (4.5). These results, are summarized in Flow Chart V.

In reality the ideal Hilbert transform or 900 phase shifter is similar to theideal lOwpass filter in that it provides a valuable theoretical 'concept corresponding to a noncausal system. In applications the shape of H(o) should be modified by windowing and frequency sampling to improve the characteristics of the filter. These considerations are not discussed here; the reader

11 h

II

111

1111

Figure IV.1 shows the shape of this filter. It also follows that

F(w) = [1/H(co)] F(co) --H(co) F(c)) (4.9)

11 1 11 411 i = =

We note one further point with regard to the expression (4.5) for the phase 0(t). The complex envelope formulation discussed herein is most physically applicable to systems with

narrow banded spectra, i.e. for time series centred on some carrier frequency o)0, plus the

frequency shift correction given in Section II. Then one can write the time series as being the real part of the complex envelope:

u(t) = Re (C(t) e lak)t)

-icoot 0,

I Re { A(t) e-4% + d)t + 4(0)

u(t) = Re {A(t) e + i0(

This follows from the definition for the relative phase OW:

0(t) = -iw't +

It is the relative phase which is used throughout the rest of this report. With regard to spectral analysis for the nonlinear Schroedinger equation, it is the relative phase OW, together with the real envelope function A(t), which is described physically by this equation. We thus seek to obtain the spectrum of the relative phase OW and not the phase itself 0(t).

In order to verify the computer program we used a simple sine wave as the time series input (Figure rv.2). This waveform is given specifically by

X(t) = X0 sin(27if + (0) = X0 cos(27cf + 4 + it/2) _(4.10)

where the parameters have been defined _{as X0 = 1, f = 0.05 Hz and = 0. A discretization}

interval At = 1 sec and 512 points _{were generated. The resultant time series for Y(t) is shown in} Figure IV.3. Note that it is not a perfect sine wave; at the beginning and end of the record there is some discrepancy. Our interpretation of this effect is that we are using the filter (4.8) rather

-than the more sophisticated approach used in Rabiner and Schafer (1974). The envelope

function A(t) is given in Figure IV.4; again some extraneous effects are seen at the beginning and end of the time series. The phase OW is shown in Figure N.5. Note that we find a constant value of ic/2 except for end effects. In Figure IV.6 we present X(t) and A(t) graphed above the input signal. Figure IV.7 gives Y(t) together with A(t). Note that in both cases the envelope function behaves as required: it is always above the signal and occasionally touches it.

Practically speaking we resolve the end effects shown here by extending the signal about 10% on each end before Hilbert transforming [Osborne, Bergamasco and Cavaleri,

27 -Flow

Procedure for Determining the Hilbert Transform of a Random Wave Train

Measure or computer generate a time series of surface waves in water depth h, call it X(t).

of X(t) and call it F(w) Compute the I-I-1

Compute the 900 phase shifting filter H(co) =

{

-i

(;1 .co<rc

i _--7t5.co<0

Compute the FFT F(o) of the Hilbert transformed signal _Y(t)

F(o) = H(c)) F (co)

Compute the Hilbert transformed signal Y(t) with the inverse I-1-l

Compute the envelope A(t) of the original signal X(t) A(t) = [x2(t) + y2(011/2

Compute the phase OW of the original signal X(t) 0(t) = arc tan fY(t)/X(t)]

V. Estimating Second Order Nonlinearities of the Measured Wave Data of

Snamprogetti and TEI

Here we estimate the average degree of nonlinearity of the Snamprogetti and TEI data by graphing the generalized Ursell number Ug given by equation (3.4). Details of the graphing procedure are given in Section III. We note that Ug may be viewed as the average ratio of the amplitude of the second order term cos 20 divided by the amplitude of the first harmonic cos 0. Hence if Hg = 0.1, say, then the second harmonic nonlinear correction to the sea surface is 10% of the amplitude of the first harmonic. Generally speaking Hg must remain small for the second order approximation considered herein to be valid. Hence a crucial result of the present analysis must be that Hg remain small for the data set in order for the theories advocated herein to be valid.

In Figure V.1 we have graphed Ug for the Snamprogetti data as a function of cTSh on a log-log scale. Note that cTz may be viewed as the average zero crossing wave length L, hence cTSh is Li/h, the wavelength-to-depth ratio. Thus in Figure V.1 shallow water is to the right

while deep water is to the left. These two regions are separated by the vertical line which

corresponds to the location of the Benjamin-Feir instability; waves to the right are stable to infinitesimal perturbations in the direction of wave motion while those to the left are unstable. The curved parallel dotted lines correspond to lines of constant Hs/h ratio. To the right of the

graph these tend to be straight lines with slope +2; in this region the shallow water Ursell

number is a good approximation for Ug. To the left of the graph the constant Hs/h curves tend

to be straight lines of slope -1; in this region the sea slope is a good approximation for Ug.

Moving upward on the graph corresponds to large waves while small waves are toward the

bottom. Each point plotted in Figure V.1 corresponds to one of the time series of the

the theoretical formulation presented herein may be used to estimate the effects of second order nonlinearities.

In Figure V.2 we present the results of an analysis of the TEI data. Again each point corresponds to a single time series. The total amount of data here is approximately _{equal to that} considered above from Snamprogetti. However the TEI data were divided by them into a larger number of time series of shorter length. Note that all of the data fall into the deep water part of the graph. The range of generalized Ursell numbers in this case is 0.02<Ug<0.15. Hence the

TEI data fall into the range of small nonlinearities and therefore lie within the range of applicability of the theory given in Section II. Both data sets are plotted together in Figure V.3.

-VI. Analysis of the Snamprogetti and TEL data

One original goal of this study was to analyze the data of Snamprogetti and TEI to

determine if radiation stress is present in ocean waves. Unfortunately, for technical reasons to be discussed below, this goal has not been met. Sample time series data from Snamprogati are given in Figures VI. 1-5 and from TEl in Figures VI.6-11. The Snamprogetti data were obtained

with Datawell Waverider buoys, while the TEI data were obtained with Endeco Type 956

Wave-Track buoys. Generally ,speaking the data are quite typical of ocean wave records taken in

a variety of locations around the world. One observes some up-down asymmetry which is

normally attributed to nonlinear effects in the wave motion. Hence wave crests tend tobe higher and more peaked than adjacent troughs,

In order to properly assess the capability of the two, types of buoys used in the data analyzed. here we give the amplitude response functions of the instruments as computed. by their

manufacturers (Figure VI.14). Note that the, frequency response characteristics of the

instruments are not good. at low frequency. This is due to the rack of responseof accelerOffieters to low frequency, long wave motions.. Evidently all instruments of this type suffer from the. same lack of sensitivity at low frequency. The inference is that in order to experimentally study radiation stress more precise measurements of the wave motion must bemade at low frequency. This is &relatively easy task in shallow water because fixed platforms can be used to mount instrumentation which is sensitive in this frequency range. Greater difficulties occurin deep Water where. fixed platforms may not exisit. A major conclusion of this 'study is that wave. measurements with buoys cannot give precise measurementsof radiation stress.

111

1

101

T

VII. Computation of Second Order Nonlinearities in Random Waves Simulated

by the Pierson-Moskowitz Spectrum

We consider a procedure for estimating radiation stress from random ocean waves

described by the Pierson-Moskowitz spectrum. The theoretical formulation is based _{upon the} derivation of the nonlinear Schroedinger equation from Euler's equations by Hasimoto and _Ono

[1972] as extended by Osborne and Boffetta, 1988b (See Section II). While the procedure

described herein is specific for the Pierson-Moskowitz_{spectrum, it may be easily generalized to} other types of ocean wave spectra. Perhaps the most important point to realize in the procedure

described herein is that: These are general procedures for computing a second order,

unidirectional, nonlinear, random sea state. Applications consist of the simulation ofnonlinear, random wave forces on platforms, complaint structures, drilling risers, etc. A major advantage of these procedures is that they are identical_{to those normally used for the simulation of random}

seas (see Osborne, 1982)plus nonlinear corrections (given in Flow Charts VI and VII

below) which are simple algebraic formulas.

The procedure is briefly summarized in what follows for the

_{shallow water case}

(Flow Chart VI):

(I) Given some value of significant wave height and zero crossing period Hs and To at some particular spatial location x=x0, compute the Pierson-Moskowitz spectrum, P(f) as a function of frequency f. (See first box A in Flow Chart VI).

(2) Using this form of the P-M spectrum, compute a random sea state at x = xo assuming a

linear Fourier representation of the sea surface and random phases on (0,2n): ri(x0,t) (see

Osborne (1982) and references cited therein). (See box B in Flow Chart VI).

-Compute the envelope A(x0,t) and phase 0(x0,t) of the linear sea surface representation n(x0,t) using the Hilbert transform. (See box C in Flow Chart VI).

Verify algorithmically

Ti(x0,t) = A(xo,t) cos 0(x0,0

where

e(x0,t) koxo

(wo + d)t + ((x0,t)

and ko and coo are to be computed from To using the dispersion relation (002 = gko tanh koh, for coo = 27c/To. (See box D in Flow Chart VI).

Now compute the time series of radiation stress by

3

ir(x0,t) _4k02h3A2(x0,t)

where h is the water depth. (See box E in Flow Chart VI).

Compute the second order nonlinear sea state rIn(x0,0 using the Stokes field representation of the sea surface. (See box F in Flow Chart VI).

Note that the Stokes field representation of the sea surface reduces to a second order Stokes wave when the modulation envelope and phase are constants. (See box G, Flow Chart VI).

We now summarize the procedure for the case of arbitrary water depth (Flow Chart

V11):

compute the Pierson-Moskowitz

spectrum, P(f) as a function of_{frequency f. (See box A, Flow} Chart VII).

Using this form of _{the P-M spectrum, compute a random sea state at x = xo assuming}

a

linear Fourier _{representation of the}

sea surface and random phases _{on (0,270: 11(x0,t) (see}

Osborne (1982) and _{references cited therein).}

(See box B, Flow Chart VII). This part is

identical to that for_{shallow water.}

Compute the envelope A(x0,t) and phase 0(x0,t) _{of the linear}

sea surface representation

n(xo,t) using the _{Hilbert transform. (See}

box C, Flow Chart VII). This is also identical to

shallow water.

Compute the spatial derivatives of the envelope _{A(x,t) and phase 0(x,t) (See box D, Flow} Chart WI).

Verify algorithmically

11(x0,t) = A(xo,t)cos 0(x0,t)

where

0(x0,t) = koxo _{(c)o + (oe)t + (1)(x0,t)}

and ko and coo_{are to be computed from}

To using the dispersion relation too2 = gko tanh koh, for

coo = 27c/T0. (See box E,

Flow Chart VII). Again this duplicates the shallow watercase. Now compute the time_{series of radiation stress by}

yA2(x0,t)

"Tlr(xo,t)

4kocs

where h is the _{water depth and y (See boxes}

F and G, Flow_{Chart VII).}

33

-Compute the second order nonlinear sea state in(xo,t) for the Pierson-Moskowitz spectrum using the Stokes field representation of the sea surface. (See boxG, Flow Chart VU).

Note that in the absence of modulation the Stokes field reduces to a second order Stokes wave (See box H, Flow Chart VI).

Compute a realization of the random sea state at some particular spatial location_{x = xo} for the above Pierson-Moskowitz spectrum by the following algorithm:

11(x0,t) = E Cn cos(koxo

_{- coot + On)}

n=1

where the coefficients are given by = [2P(fn) Af] 1/2

and uniformly distributed random phases On selected on (0,21r).

35

-Suppose it is desired to simulate a certain sea state based upon the Pierson-Moskowitz spectrum. We assume that what follows will be computed for some fixed spatial location

x=x0

Given some particular value of significant wave height Hs, compute

the dominant period Td by the formula:

r

5 11/4

in

Hs ' -

.0022 Hs1/2 = 1/fd

Td =

_{It L - i}

g2 a The Pierson-Moskowitz ag-1

spectrum is given by

rfd] 4 1 a

r

b 1 fi

- T

5iT,

_{= -75-t exp L- f7-4 j} P(f)

-(27t)4f5 exp -) g-`--.1-] 4

5-b = p

[

= fri4 27tW5

4

-a -

a,

(2704 g = 9.8 m/s2,

a = 0.0081,

13 = 0.74

Flow Chart VI

Radiation Stress from the Pierson-Moskowitz Spectrum in Shallow Water

A

-Compute the zero crossing period To of the resultant time series using the standard

definition in terms of zero up- or down-crossings.The zero crossing frequency is then

given by coo = 27t/T0 and the zero crossing wavenumber 1(0 is given by the shallow water dispersion relation coo = cok - h2k02cW6 where the linear phase speed is co = (gh)1/2.

Using the Hilbert transform compute from the simulated sea state Ti(x0,t) the spatio-temporally varying wave envelope A(xo,t) and phase 0(x0,t). First compute

N

c(xo,t) = / Cn sin(knxo

coat + On)

n=1

Then the envelope of the simulated time series is given by A(xo,t) = [112(x0,t) + C2(xo,t)1112

and the associated spatio-temporal phase is then

0(x0,t) = tan -I [C(xo,t)/Ti(xo,t)]

Verify algorithmically that given A(x0,t) and 0(x0,1) one can recover ri(xo,t) by the formula: ti(x0,t) = A(x0,t) cos e(xo,t)

and

c(xo,t) = A(xo,t) sin 0(x0,t) where

e(xo,t) = koxo (coo + co')t + 0(x,,t)

Thus this stochastic formulation in terms of the envelope A(xo,t) and phase 0(x0,0 is equivalent to the one given above in terms of the spectrum P(f) and random phases On.

The goal now is to introduce second order nonlinearities into the formulation. These include two types:

3

(1) Radiation stress,

4uc02h3A2(x0,0

3

(2) Second

harmonic'

4k02h3A2(xo,t) cos20(x0,t)

37

-Hence we compute a Stokes field for the surface wave elevation

3 3

Ti(x0,t) A2(x0,t) + A(x0,t) cose(xo,t) +

4k02h3 A2(x0,t) cos20(x0,t) 4k02h3

where the phase 0(x0,t) is given by

0(x ,t) = k x (coo + co')t + 4:1(x0,t)

and the nonlinear frequency correction is

, 9c,A-2

co =

koho4

When there is no modulation A(x0,t)_{= ao and cb(xo,t) = In this case the Stokes field reduces}

to a second order Stokes wave

32

32_2

ri(x,t) = --",

4k02h3 + ao cos0 +4k0- h3 cos20

where the phase 0(x,t) is given by 0(x,t) = kox _{(coo + d)t + (1)o}

and

, 9c0a02

co

lcoh04

-ii

Suppose it is desired to simulate a certain sea state based upon the Pierson-Moskowitz spectrum. We assume that what follows will be computed for some fixed spatial location

x=x0 Given some particular value of

the dominant

r 5 ii/4Hsin

significant wave height Hs, compute

period Td by the formula:

'

5.0022 Hs1/2 = l/fd 1-g2a-1 The Pierson-Moskowitz

aol

spectrum

{ T5 [-fd 4 1

is given by

=

a

exp

r Li

f5 f4 P(f) '''' exp (2n)4f5 g2 _{b = 13 L}

r_fr2_14

_5 4 a

a,

(2704 27[Wsj d g = 9.8 m/s2,

a = 0.0081,

13 = 0.74

Flow Chart VII

Radiation Stress from the Pierson-Moskowitz Spectrum

in Arbitrary Water Depth

A

Compute a realization of a random sea state at some particular spatial location x = xo for the above Pierson-Moskowitz spectrum by the following algorithm:

li(xo,t) = Cn COS(knx0

Wnt + On)

n=1

where the coefficients are given by Cn = {2P(fn) Afi 1/2

and uniformly distributed random phases On selected on (0,27t)

-Compute the zero crossing period To of the resultant time series using the standard

definition in terms of zero up- or down-crossings.The zero crossing frequency is then

given by coo = 27t/1-0 and the zero crossing wavenumber ko is given by the dispersion relation coo2 = gko tanh(lcoh).

Using the Hilbert transform compute from the simulated sea state ii(x0,t) the spatio-temporally varying wave envelope A(x0,t) and phase 0(x0,t). First compute

N

(x0,t) = / Cn sin(kox,

coot + On)

n=1

Then the envelope of the simulated time series is given by A(xo,t) = fri2(x0,t) + c2(x0,t)11/2

and the associated spatio-temporal phase is given by e(xo,t) = tan-1K(x0,t)hi(x0,0]

Derivatives of envelope A(x,t) and phase 0(x,t) with respect to space x.

This is done by computing A(x0 + Ax,t) and 41.(x0 + Ax,t) for an arbitrarily small Ax: A x(x0,t) A(x0 + Ax,t) A(xo,t)

Ax

Ox(xo,t) (I)(xo + Ax,t) 0(x0,t)

Ax

39

-Verify algorithmically that given A(x04) and 00(04 one oanTecover rgx0,t) by the formula: ri(x0,t) = A(x,o,t) cos 9(x.0,t)

and

(x0,t) A(x0,t) sin 0(x0,0 where

8(x0,t) koxo. (coo + o.)1)t + 0(x0,0

Thus this stochastic formulation in terms of he envelope A(x0,t) and phase 0(x0,t) is equivalent to the one given above in terms of the spectrum P(f) and random phases On.

The goal now is to introduce second order nonlinearities into the formulation. These

include three t Apes:

(1) Radiation stress,, yA2(x0,t)

4koa

First harmonic,

C°a

A(x0,00x(x0,0

+ CzAx(x0'0

(2) cos() sine

coo Wo

-(2) Second harmonic,

5A2(x0,t) cos20

4koa . 11 , , 11

Hence we compute a Stokes field for the surface wave elevation

--yA2 r CaOxi C A 5A2

AO + °--- + 1-1-C +

ri(x0,t) = + ] cose sino cos2e

4k0a _coo coo 4k0cr

. where A = A(xo,t) is the real modulation envelope and

0 =10(x0,t) = koxo (0)0 + 01))t + <gx0,t) If if 11.1 1 1 =

41

-Definitions for the Modulated Stokes Field

Here 00(x0,t) = koxo wot

is the carrier phase and 0(x,t) is the modulation _{phase. The constants y, 5 are given by}

7 1

gh Cg 2 [20uolcoCg + (1 cr2) gh1c02]

(3 _ 02A02

S

-(72

and the nonlinear frequency correction

_is

g2A-2

'---U) =V

4(002

When there is no modulation A(x0,t) = ao and (1)(x0,0 = 00. In this case the Stokes field reduces

to a second order Stokes wave

ya02 6a02

ii(x,t) = - + do COS() + -COS20

4kocr 4k0cr

where the phase e(x,t) is given by 0(x,t) = kox

_{(co0 + d)t + 00}

g2a 2

a

= Vv' °

4(1)02

-The group velocity above is given by

ao) 1 r

(1 - a2)koh ii

Cg 1-c L i + ) J

6

with the dispersion relation

coo = gicoa, a = tanh koh and

ko4 ( c 2 { 1 r _{1 1}

L4C2 + + gh(1a2)2 j +

(9-10a2+9cr4)/

2coo 2.a) Cg2 gh 40-62)CCg 2(52

with phase velocity

coo

(gap2

c = 1c, = Oc.0)

4

-An Example of a Nonlinear Random Sea

We now discuss a particular example of the above procedures for computing a nonlinear, random sea state. We follow Flow Chart VI and begin with box A for computing the Pierson-Moskowitz spectrum. We select a significant wave height of Hs = 8.0 m which has the dominant period Td = 14.1 sec. We then generate a realization of a linear random _{sea using the} approach given in box B (see Osborne (1982) and references cited therein). The Fourier phases

were selected to be uniform on the interval (0,2n) and the amplitudes were taken to be

proportional to the square-root of the power spectrum (box B). The physical spatial location for computing the wave motion was chosen to be x = x,= 0 in the present example; the value of xo can however be chosen to be any desired value, say at a fixed location within some space-frame structure. The resultant time series ti(x0,t) is shown in Figure VII.1. In box C we compute T, from the series in Figure VII.1 using the zero up-crossing approach and find 'I',= 10.6 sec. In box D we compute the Hilbert transformed signal c(xo,t). In order to compute this signalone should compute c(x0,0 using the Hilbert transform as discussed in Section IV. The formula

given for C(x0,t) in box D is symbolic in that the cosine of box B is replaced with a sine

function; no difference in phase information has been taken into account. The actual Hilbert transform operation should be done directly as in Section IV; the signal for C(x0,t) determined in the present case is given in Figure VII.2. The next step is to determine the envelope A(x0,0 by the formula in box D; this is shown in Figure VII.3 together with the original signal 1(x0,t). The phase 4)(x0,t) is then computed from the formula in box D. The raw phase is given in Figure VII.4 while the phase corrected for the effect of cut (as discussed in Section IV) isgiven in Figure VII.5. Note that the latter varies more slowly in time that the former. In box E we

show how to recover the signals n(x0,0 and c(xo,t) from the envelope A(xo,t) and phase

0(x0,t). _{This corresponds to the "inverse problem" (recover the original time series from the}

envelope and phase) and serves as a check that the Hilbert transform algorithm works well; Figures VII.8 and VII.9 demonstrate that this is indeed the _{case (compare to Figures VII.1, 2).}

-We now compute radiation stress as -0.1A2(x0,t) and graph it in Figure VII.6. The

constant 0.1 has been chosen arbitrarily for this example. As seen in box F this corresponds to 0.1 = 314k02h3. Since ko is fixed to Tz = To by the dispersion relation (box A, Flow Chart I) then this relation fixes the water depth, which in this case is about 15 m. Note that the radiation stress signal is characterized by long, low frequency variations in the mean sea level. The

largest excursions (actually plunges in the mean level) occur when a large wave packetappears in the original signal i(x0,t). The second harmonic contribution is given by formula (2) in box

F; in the present case we have 0.1A2(x0,t) cos 20(x0,t). A graph of this signal is given in

Figure VII.7. Note that it oscillates about twice as rapidly as the carrier A(xo,t) cos 0(x0,0; the envelope of the second harmonic is the signal for radiation stress (in the present shallow water example).

The total nonlinear signal 0.1A2(x0,t) + A(x0,t)cos 9(x0,t) + 0.1A2(x0,t)cos 20(x0,t) is shown in Figure VII. 10 for this shallow water case. Note the presence of the peaked crests

and the shallow troughs. One feature not observable in the present simulation is up-down asymmetry; this occurs because the radiation stress and second harmonic are of equal magnitude and hence cancel any tendency to render the nonlinear signal asymmetric. This can be seen by removing the radiation stress and graphing only A(x0,t)cos 0(x0,0 + 0.1A2(x0,t)cos 20(x0,t) (Figure VI1.11). Here the asymmetry is very clear. Ironically, the nonlinear appearance of most ocean wave data may be due to the filtering of low frequencies by buoy accelerometers!

To consider a more realistic case we go to deeper water (about 100 m) and repeat the

above simulation with a reduced radiation stress term: 0.05A2(x0,t) + A(x0,t)cos 8(x0,t) +

0.1A2(x0,t)cos 20(x0,t). This smaller value of radiation stress is consistent with the deeper water case considered. We have neglected other small contributions as given in box G of Flow

To show that the last statement is true we now Fourier analyze the signal of Figure VII.14. We first show that the signal A(xo,t) cos 0(x0,0 is truly described by the

Pierson-Moskowitz spectrum; this is clear from the FFT of the original linear signal (Figure VII.1) as

shown in Figure VII.15. The characteristic shape of the Pierson-Moskowitz spectrum is

unmistakable. We contrast this result to the 1-F1 of the nonlinear signal of Figure VII.14. This is shown in Figure VII.16. Note that the spectrum is much more irregular and contains low frequency and high frequency contributions not present in the linear Pierson-Moskowitz part. In Figure VII.7 we give the FFT of only the radiation stress contribution; it is seen to clearly be dominated by low frequency motions. Figure VII.18 gives the FFT of the second harmonic

contribution, which occurs only at hid) frequency. As additional information we show in

Figure VII.19 the FFT of the linear Pierson-Moskowitz series in log-log coordinates; Figure VII.20 shows the FFT of the nonlinear signal in log-log coordinates.

Finally we show the behavior of the I-1-1 phases of the original linear signal (Figure VII.21), radiation stress (Figure VII.22), the second harmonic (Figure VII.23) and the entire signal (Figure VII.24). All of these results are indistinguishable from uniformly distributed random phases. These results are somewhat surprising in view of the fact that nonlinearities are

present.

-VIII. Nonlinear Fourier Analysis of Simulated, Nonlinear Random Waves

Here we consider a new approach to the analysis of time series which we have referred to as nonlinear Fourier Analysis (Osborne and Bergamasco, 1985, 1986). The idea is based

upon a new method of mathematical physics which is referred to as the inverse

scattering/spectral transform (1ST, see for example Ablowitz and Segur, 1981). We discuss here some of the approach for the shallow water case, although the method can be extended also to deep water. Basically the idea is that the KdV equation with periodic boundary conditions can be integrated and its exact solution can be determined using the periodic 1ST. The reason for assuming periodic boundary conditions is that they have historically been useful for the study of wave trains or wave fields in a variety of contexts. Periodic 1ST has a theoretical structure

which is analogous to, but very much more complex than, the mathematical structure of the

ordinary linear Fourier transform. 1ST, although a linear theory, solves a nonlinear wave

equation (KdV); this spectral theory reduces to the Fourier transform in the small amplitude, linear limit. This limit, while theoretically satisfying, is although satisfying from a practical standpoint: The discrete analogue of the theory reduces to the discrete Fourier transform in the

small amplitude limit. Since the discrete Fourier transform is the basis for the fast Fourier

transform (FYI) algorithm used in computers, one finds that the natural small-amplitude, linear limit is also met in the computational algorithm. Since the I-1-1 is a periodic algorithm, this constitutes anothei justification for the use of periodic 1ST algorithm.

Many of the details of the implementation of 1ST are discussed elsewhere (Osborne and Bergamasco, 1985, 1986) and will not be reiterated on here. Basically the idea is simple. While linear Fourier analysis can be described as the linear superposition of linear waves (sinusoids), nonlinear Fourier analysis can be described as the nonlinear superposition of nonlinear waves

mathematical formulation of the problem. The direct transform _{is described by F1c>quet theory} for the linear Schroedinger equation and the inverse transform is given by the Jacobian inverse problem of algebraic geometry. Here we briefly discuss the direct transform, as it will be used below in the analysis of nonlinear, simulated random waves.

The direct spectral transform (DST) yields a nonlinear spectrum for the time series under investigation. Physically the depth is assumed to be small and the waves to be long. The degree of nonlinearity is described by the Ursell number, U; when U is small the DST reduces to the linear Fourier transform, while when U is large the theoretical structure is very far from the linear theory. The spectral components may be described in order of increasing Ursell number:

sine waves, Stokes waves, cnoidal

_{waves and solitons (or solitary waves). The linear}

superposition of these nonlinearly interacting spectral components gives the solutions to the KdV equation. The spectral problem for DST is the Schroedinger eigenvalue problem in_which

one introduces the usual Bloch eigenfunctions. The monodromy matrix, defined to

operationally shift the two basis functions of the theory by one period, contains the spectral elements of the theory. For present_{purposes the most important information in this matrix is the} trace, which when it coincides with +1 or -1 (as a function of E = f2), produces a periodic or antiperiodic Bloch solution to the Schroedinger equation. Figure VIII.1 shows typical examples of the trace of the monodromy matrix. Essentially it is an oscillating function T(E) which _selects eigenvalues when it crosses +1 or -1. When T(E) is above +1 or below -1 a forbidden band _is defined by the band edges, which are the adjacent eigenvalues. In Figure VIII.1 the eigenvalues are labeled by an integer of increasing value. Inside a forbidden band (see that defined by Et and E5 for example) there is a spectral component which we here label p2. While the Ej are time invariant, the pi vary in time, and are interpretable as the nonlinear Fourier components of the theory. The pi in fact oscillate in time between the band edges and are described by nonlinearly coupled ordinary differential equations which define them as hyperelliptic functions. Sincethe pi oscillate between the band edges as fundamental limits, it is simple to show that the width of the band edges is the wave height of the nonlinear wavelet pi. Practically speaking this is a great advantage since one need non compute the _{(which can be a very complicated, difficult task) if}

-one wants to graph the nonlinear DST. One needs only to compute the widths of all the

forbidden bands and to graph these as a function of frequency, much in the same manner as the linear Fourier transform. Thus one has the great fortune to have a nonlinear spectral theory in which many aspects are directly analogous to linear Fourier theory. This simple structure is not preserved in other nonlinear wave equations such as the nonlinear Schroedinger equation or the sine-Gordon equation.

We now turn to the nonlinear Fourier analysis of a random shallow-water wave field.

The approach is identical to that given in the last section, with but one exception. Here we

generalize the wave field solution to the KdV equation to infinite order. This is a relatively recent development (Osborne and Boffetta, 1988) and allows higher order nonlinear effects to be considered. In effect the approach extends the Stokes solution to KdV beyond second order

to infinite order. The relevant expression is given by

00

ri(x,t) = U(x,t) A(x,t) + A(x,t)

1,

nUn-1(x,t) cos nO (7.1)

n=1

where the Ursell number U(x,t) is the same as before

3A(x,t)

U(x,t) =

41(02h3

The advantage of this extension of the solution to KdV to infinite order is that, in the absence of modulation, the above expression reduces to the Stokes series solution to KdV. This solution is identically a spectral component of the DST. The results given here form a natural generalization of those given in previous sections.

given in Flow Chart VI, with the exception that the expression for ii(xo,t) given in box G has

been generalized to that given above (to infinite order) in (7.1). The envelope A(x0,t) is given in Figure VIII.3 and the radiation stress component is given in Figure VIII.4. The modulated

Stokes field to infinite order (in the absence of radiation stress) is given in Figure VIII.5 The complete, simulated nonlinear signal is given in Figure VIII.6; it is this signal which we analyze in the remainder. We take the linear Fourier transform of the signal in Figures_{VIII.7, 8. Figure} 7 and 8 differ only by the upper limit in the frequency given; this was done to investigate higher frequency content of the record. One notes the presence of radiation stress and the higher

harmonics. Finally in Figures VIII.9-12 we give the nonlinear Fourier analysis. Figure VIII.9 shows the Floquet diagram (trace of the monodromy matrix as a function of E=f2). Note that

the oscillations are quite wild here and span thirty orders of magnitude! Each tick mark

constitutes a power of ten on the vertical scale. This wild behavior indicates _substantial

nonlinearity in the original time series. To the left of E=0 is the soliton spectrum, while to the right is the radiation spectrum.. In the soliton spectrum the oscillations are much more violent than those in the radiation spectrum, a result typical of nonlinear Fourier_{analysis. An exploded} view of the soliton part of the spectrum is given in Figure VIII.10; note that the vertical scale

has been changed and now spans 60 orders of magnitude and still does not contain the

oscillations. The exponential nature of the theory is clearly evident in this Figure! Please note that the amplitude of the oscillations in no way effects the actual nonlinear spectrum. Only the values of the discrete eigenvalues give information about the spectrum.

Finally in Figure VII1.11 we show the nonlinear Fourier spectrum as a function of

frequency. We have graphed the nonlinear Fourier amplitudes _{(.1.i) as the continuous curve. The} radiation components are on the right side of the graph. This part of the spectrum is analogous to the linear Fourier spectrum (see Figure VIII.12 on the _{same scale). It is useful to compare the} radiation spectrum to the Fourier spectrum of Figure VIII. 12. Note that there is a net depletion of the radiation spectrum compared to the linear Fourier spectrum. This is because some of the radiation energy is stolen by the DST to supplement the soliton part of the spectrum. This is a

surprising result and is evidently due to the large measure of nonlinearity in the present - 49 -_-

-example. The soliton part of the spectrum is shown on the left side of the graph in the region of negative E and are represented by the discrete vertical lines. The solitons reside in the radiation stress part of the original signal. Note that the maximum soliton amplitude is about 0.5 m. Since much of the soliton spectrum arises from energy stolen from the original radiation spectrum, one must infer that, for the nonlinear case considered here, the radiation stress signal is indeed even larger than that computed in the simulation of the original signal. This point bears further consideration and should be a target of any possible future studies, especially in shallow _water where nonlinear effects grow with decreasing water depth.

IX. Bispectral Analysis of Simulated, Nonlinear

_{Random Waves}

We now briefly review a bispectral analysis of _{some of the simulations considered}

herein. The bispectral analysis approach is _{a standard one and will not be reviewed. One should}

recall that it provides information about _{second-order nonlinear interactions in a random}

process. We consider two cases in the present analysis. The second case considered in Section VII and the more nonlinear_{case considered in Section VIII. The case from Section VII is given} in Figures IX.1, 2. We given in Figure IX.1 the relief map, something which is normally not

done, because we feel that it enhances the

_{understanding of the location of nonlinear} interactions. Figure IX.2 gives a contour map of the bispectrum of Figure IX.1. The isolated

enhanced "islands" corresponds to the harmonics and are spaced by the differences in the

harmonic frequency, about 0.13 Hz of the input nonlinearly simulated time series (maximum

frequency is 0.5 Hz). The low frequency (at the_{left) peak (island) corresponds to radiation}

stress. Subsequent peaks correspond to the interactions between _{harmonics and radiation stress.}

Figures DC.1, 2 give the bispectral analysis for the nonlinear, random wave simulation of Section VIII. Here the spacings are due to the harmonics at 0.12 Hz of the input nonlinear signal. The maximum frequency shown is again 0.5 Hz. The enhanced activity to the left is of course radiation stress and its interactions with the higher harmonics. The bispectra presented here illustrate the spiked nature of the harmonic interactions in nonlinear waves.

-51-X. Conclusions and Recommendations

We have given a simple formulation for the effect of radiation stress on ocean surface

waves (low frequency variations in the mean sea level) up to second order. In the spirit of

engineering applications we have given the major results in Flow Chart form. The principle theoretical results in shallow water are given in Flow Chart I, while those for arbitrary water

depth are given in Flow Chart II. A procedure for estimating the level of nonlinearity in a

particular sea state is given in Flow Chart III. In order to estimate the contribution of radiation stress (in time series form) to the wave motion we give Flow Chart N. Most of the calculations

presented herein require the use of the Hilbert transform; a procedure for obtaining the associated envelope and phase of a time series is given in Flow Chart V. Finally, procedures for simulating time series of nonlinear waves associated with a chosen spectrum is given for the shallow water case in Flow Chart VI and for deep water in Flow Chart VII.

Highlights of the present study are the generation of nonlinear random ocean waves

(Sections VII and VIII) and their analysis by nonlinear Fourier analysis (Section VIII) and

bispectral analysis (Section IX).

Procedure For Computing Particle Velocities

We recommend a rather standard procedure for estimating particle velocities. The usual procedures for computing particle velocities in time series simulations should be applied. These will not be discussed in detail here, but they consist in applying a decay in particle velocities (actually the particle velocity spectrum) as a function of depth relative to the surface-wave

power spectrum. This is done of course as a function of frequency. In simulations we