/JP7S3 «^ NAVAL POSTGRADUATE SCHOOL

INFORMAL REPORT

NCSL 144-72 Formerly NSRDL/PC 3472

DECEMBER 1972

THE DIRECTIONAL ANALYSIS OF OCEAN WAVES:

AN INTRODUCTORY DISCUSSION (SECOND EDITION)

CARL M. BENNETT

Approved for Public Release;

Distribution Unlimited

NAVAL COASTAL SYSTEMS LABORATORY /j PANAMA CITY, FLORIDA 32401

copy- 52

ABSTRACT

An introductory discussion of the mathematics behind the direc- tional analysis of ocean waves is presented. There is sufficient detail for a reader interested in applying the methods; further, the report can serve as an entry into the theory. The presentation is basically tutorial but does require a reasonably advanced mathe- matical background. Results of a program for the measurement of directional ocean wave bottom pressure spectra are included as an appendix. This second edition makes corrections to the first and adds some details of an iterative directional analysis method.

ADMINISTRATIVE INFORMATION

This report was prepared to document the mathematical methods used in connection with work done in support of Task SWOC SR 004 03 01, Task 0582, and applied on Task ZR 000 01 01, Work Unit 0401-40.

This report was originally issued in September 1971 as NSRDL/PC Report 3472.

TABLE OF CONTENTS

Page No.

1. INTRODUCTION. . 1

2. WAVE MODELS 1

3. A DIRECTIONAL WAVE SPECTRUM 7

4. CROSS SPECTRAL MATRIX OF AN ARRAY 12

5. SPECIAL CROSS SPECTRAL MATRICES 18

6. A MEASURE OF ARRAY DIRECTIONAL RESOLVING POWER 23

7. DIRECTIONAL ANALYSIS FROM THE, CROSS SPECTRAL MATRIX ... 26

8. SUMMARY 39

REFERENCES 40

BIBLIOGRAPHY. 41

APPENDIX A - A COLLECTION OF DIRECTIONAL OCEAN WAVE BOTTOM PRESSURE POWER SPECTRA A-l

APPENDIX B - A FORTRAN II PROGRAM FOR SINGLE-WAVE TRAIN ANALYSIS B-l

APPENDIX C - A FORTRAN PROGRAM FOR ITERATIVE WAVE TRAIN ANALYSIS C-l

LIST OF ILLUSTRATIONS

Figure No. Page No.

1 Simple Ocean Wave 2

2 Real Wave in Wave Number Space 7

3 Directional Wave Spectrum at a Fixed Frequency,

*o ■ • 8

4 Wave Direction Analysis < 27

5 Direction Analysis from a Pair of Array Elements 28

6 Directional Estimates for a Pair of Array Elements 30

7 Least Square Single Wave Fit Directional Spectra A0(f,9o) 33

8 Detector Geometry 34

ii

1. INTRODUCTION

The report presents an introductory discussion of the mathematics pertaining to the directional analysis of ocean waves. The presenta- tion is tutorial in form but does require a reasonably complete mathe- matical background; a background equivalent to that required in reading Kinsman's textbook Wind Waves (1965).

The level of the presentation is moderate at the beginning. The level picks up rapidly toward the middle but there should be sufficient detail and redundancy in the mathematics to allow the reader to follow the development without having to rediscover too many omitted steps. It is in this sense that the report is tutorial. In some places the mathematical development is intuitive rather than rigorous. This is deliberate in order to provide insight and understanding. In most such cases, references' to rigorous reports are given.

The development is reasonably detailed so that the interested reader may apply the methods presented and use the report as an entry point into the rigorous theory of the directional analysis of ocean waves. In this respect, if the report serves as a bridge across the gap between a handbook and a rigorous and sparse theory on the subject then the objective of the report will have been fulfilled.

The report first presents an intuitive development of a sea sur- face model that assumes the sea surface to be a two-dimensional random process definable in terms of a directional power spectrum. A discus- sion of the space and time covariance function and its relationship to the directional power spectrum follows. Both one- and two-sided power spectra are discussed; however, the main development is in terms of the two-sided spectrum. Next, the relationship between the power and cross power spectrum for two fixed locations and the sea surface directional spectrum is developed. Explicit relationships for the special cases of an isotropic sea and a single wave of a given direc- tion and frequency are then obtained. The related topic of the direc- tional resolving power of an array of wave transducers is then presented.

Using the preliminary developments as a basis, several methods for the directional analysis of ocean waves based on the information obtainable1from an array of wave transducers are presented. The methods

are basically a direction finder technique, a least square single-wave train fit, and a Fourier-Bessel expansion fit. In conclusion, a generalized Fourier expansion method is suggested. Extensive results of the application of the least square single-wave train fit are pre- sented in Appendix A. Appendix B is a FORTRAN II listing of a program for this analysis.

WAVE MODELS

In its simplest form an ocean wave can be thought of as a single frequency, sinusoidal, infinitely long crested wave of length \, moving in time over the ocean surface from a given direction 6. Such a wave is illustrated in Figure 1.

Spatial Frequency Along v axis is zero

Spatial Frequency Along y axis is m = K sin 6

Direction of 9 Wave Travel in Time t

u Spatial Frequency along u axis K = l/X

Water Depth

Spatial Frequency along x axis is i = K cos e

FIGURE 1. SIMPLE OCEAN WAVE

Assume that the wave is frozen in time over the surface (the x,y spacial plane). The coordinates (u,v) are a 6 degree rotation of the (x,y) coordinates. The positive u axis lies along the direction from which the wave is traveling. The wave surface n(u,v), shown frozen in time in Figure 1 can be described mathematically by

n(u,v) = cos(27rKu + 27TCp) (2.1)

where K = 1/X is the wave number of spacial frequency in cycles per unit length along the u axis, and 2TT<J> is a spacial phase shift.

To make the wave move in time across the spacial plane with a time frequency f = 1/p, where p is the wave period, it is necessary to add a time part to the argument of the cosine function in the model above. The time part is a phase shift dependent only upon time. As time passes, the time part changes causing the cosine wave to move across the (u,v) plane, in this case the ocean surface. Adding the time part we get (where 2TT^ is a fixed time phase shift)

n(u,v,t) = cos(2irKu + 27T<J> + 2TT ft + 2TT^) .

If we combine the effect of the <J> and \p phase shifts as u a f + f, we get

n(u,v,t) = COS(2TT(KU + ft + a)) (2.2)

as a simple model of a sinusoidal wave moving in time over the ocean surface.

Since the coordinates (u,v) are a rotation of the coordinates (x,y) through an angle of 6 degrees, we know

u = x cos 6 + y sin 6

v = -x sin 6 + y cos 6.

Using the above relations, and letting & = k cos 6 and m = K sin 6 be the spacial frequencies along the x and y axes, respectively, we have

n(x,y,t) = A COS(2TT(£X + my + ft + a)) (2.3)

as a model for a wave of height 2A moving from a direction

6 = arctan (m/H)

with a phase shift of 2irot. A wave crest of such a wave system is infi- nite in length. A crest occurs at a set of points (x,y,t) which satisfy the relation

Äx + my + ft = a constant = (n - a)

where n = 0, -1, +1, -2, +2, ... . Each value of the index n relates to a particular crest. The intersections of the crests with the x and y axes move along the respective axes with time velocities Vx = -f/£ and Vy = -f/m. This follows from the differential expressions

(2.4)

(2.5)

obtained from the wave crest relationship given above.

From Euler's equation we know that cos y = (Exp(iy) + Exp(-iy))/2. If we consider y as 2TT(£X + my + ft + a) we can write

n(x,y,t) = 1/2A Exp(i2ir(J!,x + my + ft + a) +

1/2A Exp (i2TT(-£x - my - ft - a)) (2.6)

where -» < I 1 +_ °° > -°° < m < + °° and -°° < f < + °°...

In the above we have introduced the notion of negative time frequencies. This makes it possible to express an elementary wave in the mathemati- cally convenient form

n(x,y,t) - a Exp (i2Tr(£x + my + ft + a)) (2.7)

where a = 1/2A. In the real world a complex wave of this type implies the existence of another wave n*(x,y,t) which is the complex conjugate of n(x,y,t) above. This complex conjugate is given by

n*(x,y,t) = a Exp (-i2ir(£x + my + ft + a))

= a Exp (12TT.[(HDX + (-m)y + (-f)t + (-«)]). (2.8)

The fact that negative frequencies are considered is explicit in the above relation.

A property of the above model, which will be used later in connec- tion with the directional analysis of waves from measurements obtained from an array of detectors, is expressed by the equation for the phase difference of two measurements made at two different points in space and time. Assume we know the value of n(x,y,t) at the three-dimensional coordinates (x0,y0,t0) and (x0+X, y0+Y, tQ+T), where X, Y, and T are constants. The phases at the two points are given by

<Kx ,y ,t ) « lx + my + ft + a, (2.9) TN o'-'o O 0 o o

4>(x + X, y + Y, t + T) .5= Jl(x + X) + m(y + Y) + f(t + T) + a o ' 'o ' o No o o

(2.10)

This gives a phase difference of

A<j> = (£X + mY + fT). (2.11)

To obtain a more complicated wave system consisting of many waves of various frequencies and directions, we can linearly superimpose (add up) many waves of the form given above. If we do this, we can write

N

n(^t}=2^E**(i**(****^w*+fw*«'*iM. (2.i2)

For this wave system to be real, the terms must occur in complex con- jugate pairs as indicated above.

For completeness, consider a model for an infinite but countable number of distinct (discrete) waves and write

^y.tiaiS^^11*^*4^*^***)). (2.i3) *»>

Again the terms must occur in complex conjugate pairs for the wave system to be real. This will be assumed to be the case in future discussions.

A model for a wave system in the case where energy exists for con- tinuous intervals of frequency and direction should be considered. In particular, consider the general case of continuous direction from 0 to 2TT radians and continuous frequency in the interval (-fn, + fn), or even the interval (-», °°). In theory the above model does not hold for the continuous case. The power spectrum for the infinite but count- able case would be a set of Dirac delta functions of amplitude ar standing on the points (A.n, 1%, fn) of a three-dimensional frequency space. The continuous case produces a power spectrum, S0(£,m,f), which is everywhere nonnegative and in general continuous over the region of three-dimensional frequency space where power is assumed to exist. A reasonable model for n(x,y,t) in the continuous case must be determined. Consider a single wave element

an Exp(i2Tr(£nx + n^y + f„t + an)). .. (2.14)

The energy or mean square in this element is a^. Assume the ele- ment is a part of a continuum of elements for -°° < f < +°> and 0 _< 8 < 2TT. In this case an must be an infinitesimal energy associated with the frequency differential, df, and space frequency differentials, d£, and dm, which are related to the direction 0 of the wave element as before. Let the power spectrum, S(£,m,f), be defined with units of amplitude squared and divided by unit spacial frequency, £, unit spacial frequency, m, and unit time frequency, f. The power spectrum is then a spectral density value at (£,m,f). In this case we must have the infinitesimal energy, a^, defined by

«} - Sfl^tJ AUm *4 (2.i5) The real valued, nonnegative function S(£,m,f) is a power (energy

density) spectrum of the standard type in three-dimensional frequency space (£,m,f). Intuitively, we can write an infinitesimal wave ele- ment as

m (i it (* „x t *^♦i; t <^ys(^vo^*^ > (2.16)

where the positive square root is assumed. To arrive at a model of n(x,y,t) for the continuous case, we need only form a triple "sum"of the infinitesimals or, to be precise, the triple integral T|(x,y,t) =

For different sets of values of the phase relation a(£,m,f), the wave system, n(x,y,t), has a different shape, even when S(ü,,m,f) is fixed. In fact there is a wide range of possible shapes of n(x,y,t) for a given S(A,m,f). The above development is more intuitive than mathematically rigorous. It has been shown by Pierson (1955 - pp. 126- 129) that if 2ira(£,m,f) is a random function such that for fixed (&,m,f) phase values of the form 2ITOI MOD 2TT between 0 and 2ir, are equally probable and all phase values are independent, then Equation (2.17) represents an ensemble (collection of all probable) n(x,y,t) for a given S(£,m,f). The random process represented by the ensemble is then a stationary Gaussian process indexed by the three dimensions (x,y,t). Detail discussions of the above can be found in St. Dennis and Pierson (1953 - pp. 289-386) and Kinsman (1965 - pp. 368-386). The fact that a particular sea-way can be considered as a realization of a stationary three-dimensional Gaussian process has been verified. Refer to- Pierson and Marks (1952).

The model in Equation (2.17) as a stationary random process will be assumed in following discussion.

3. A DIRECTIONAL WAVE SPECTRUM

The power spectrum S(ß,m,f) is the directional wave spectrum of n(x,y,t). If n(x,y,t) is to be real, every infinitesimal of the form given in the continuous case above presumes the existence of its com- plex conjugate. Let us consider the one-sided power spectral density, S'(£0,m0,f ), of a single real wave where 0 <_ f < <=°. For such a real wave element of length XQ, from a direction 6Q, 0 <_ 9 < 2TT, the value S'U0,m0,f0), = Sao,m0,f ) + S(-£0,-mo,-f0), where £Q = KQ cos QQ, m = K sin6Q, and KQ = 1/A0. If the above real wave came from a direction (2TT - 6) , the power density would be S' U0,-m0> f0) = S(&0,-m ,fQ) + S(-Ä0>mo,-f0).

Figure 2 illustrates a real wave of length X0, from a direction 60, in two-dimensional spacial frequency (wave number) space.

Wave Direction

FIGURE 2. REAL WAVE IN WAVE NUMBER SPACE

If the wave number relation

K*2l»tV*T4*H(*KKW) (3.1)

holds, refer to Kinsman (1965 - p. 157) and Munk et al (1963 - p. 527), where h ■ water depth, g = acceleration of gravity, and K = wave number = 1/X, X being the wave length; then a relationship between f and (£,m) is implied that requires a wave frequency f0 to have a unique wave num- ber k . From this we have the general one-sided spectral form for waves where f = fQ of

S' (Ä,m,f_) = (zero where St,2 + m2 f K 2

2 2 a power density >^ 0 for £ + M K.

S'(£,m,f0) thus defines power density at f = f for wave energy over 0 < 9 < 2TT. Figure 3 illustrates this case in wave number space.

S(l, m, fe)

S'(jL, m0, f8)

r + m2 - Kf

FIGURE 3. DIRECTIONAL WAVE SPECTRUM AT A FIXED FREQUENCY, f

We want to estimate the shape of S'(£,m,f0) above the circle I2 + m K in a directional wave train analysis. Remember, the S'(2,,m,f ) above is restricted to f > 0 and is, in fact, equal to

where

if n(x,y,t) is to be real.

(3.2)

Let us see how S' (£,m,f) might be found: we have said that n(x,y,t) can be assumed to be a stationary Gaussian process. One char- acteristic_of._such; _a process is that for fixed_values (xQ, yQ, tQ) of _the_proces9 indices, n(xpt Vo« tQ)is random variable with a Gaussian distribution; i.e.,

i-JA1

?«w(i(*..&.0<n.)» Ipste-** * ' i ■1

2 where y is the arithmetic mean of n and a is the variance. Intui- tively n is as likely to be positive as negative, so let us assume that Prob (n(xQ, y0, t0) < 0) ■ 1/2. Since n is Gaussian distributed, and is thus symmetric about its mean, we have Prob(n(x0, y0, t0) < y) = 1/2 or that u » 0. For a , we have (using expected value notation)

cr'-L^ -/fl- E[J\«..»..0] where we are thinking of n as a random variable.

A Gaussian process is completely defined statistically if we know the term of the mean

fc(n(x>y>t)) and the covariances

where X, Y, and T are space and time separations, respectively. Refer to Parzen (1962 - pages 88-89). We have assumed E(n(x,y,t)) = y = 0 and that the process is stationary (only weakly stationary is necessary). Hence, by definition of weak stationarity, we have for each (x,y,t), and yet independent of the particular x,y,t values, the covariance form

M*XT)sE[n(*,M)V**X>a+Y,t*T)] (3.3)

All of the properties of the stationary Gaussian process n(x,y,t) are implicit in R(X,Y,T), just as a knowledge of y and c for a single Gaussian random variable completely defines such a random variable. Here it is important to understand that we are discussing expected values across all possible realizations at a point (x,y,t); i.e., across the ensemble of all possible sea wave shapes at (x,y,t) for a given S(£,m,f).

There is a simple and unique relationship between R(x,y,t) and S(£,m,f). Consider a single real wave element (from Equation (2.16) and (3.2)) as a random process and write n(x,y,t) = [EXP(i2Tr(£x+my+ft+a))

Form the covariance function

where E is the expected value over the ensemble for any fixed x,y,t. Note that ■* #* •-■_'_.--

is a constant with respect to the expected value.

Consider the following problem. Let u be a random variable with uniform probability density function

» " 6 « U< tV rUia*s

{.' else u> tare

Define a random variable Z = e u. The expected value of Z is defined as

CM

Considering the random variable nature of the phase 2TO as described following Equation (2.17) at the end of Section 2, and applying the above notion to the cross product terms of R(X,Y,T) we obtain

R(XYT) =[£*f»(x*ir(!X + *vY + *T)) +

10

For a real wave element we have, where S(£,,m,f) = S(-Ä,,-m,-f), see

Equation (3.2), R(X,Y,T) =£^ (i t1| (|*+ M Y* f T>5 l^.*)*^*»**

which is simply the sum of the covariance functions of two complex wave elements which are conjugate pairs. It also follows that R(X,Y,T) is real valued.

Reverting to the complex wave element form, and noting that the expected ensemble value of cross products between different wave ele- ments is zero in a manner similar to the case of cross products shown above, we obtain the composite general relationship

Rixyy)=jlj E*Klalr(ix**Y*tT)5(*.*,,f)aw*.a* (3.4)

We have demonstrated, but not rigorously proven, that the covari- ance function R(X,Y,T) is the three-dimensional Fourier transform of the directional power spectrum S(£,m,f).

We cannot hope to be able to estimate R(X,Y,T) for continuous values of X, Y, and T. However, t;here is a way around this problem, we can write the above as Q I^- _, _,*

Ji«^(Ufr<T)|jj^(ittr(lx«^Y)S^)ajJA<

which is in the form of a single dimension (variable f) Fourier trans- form of the term in brackets [ ]'s. Note this term is not a function of T. It depends only on the value of (X,Y). Further, by Fourier transform pairs we can write this expression as

«•

[ ]« = {rR(xy.T)E«K-iw*T)AT (36)

-OP

In general, assuming that the term in [ ]'s is complex, we can write

iCtxy.O-LQCxxfn » (R(x:cr)E*t(r4nLfT).iT . (3.7)

11

To find [C(X,Y,f) - iQ(X,Y,f)] we need only know R(X,Y,T) for continuous T for the given value of X,Y). Further, we have just stated that

This is in the form of a Fourier transform; thus, we can write from transform pairs

(3.9)

As has been stated, we cannot hope to have a continuous set of values of (X,Y). The solution is to find R(X,Y,T) for continuous T and selected values of (X,Y) , and then employ the above to estimate S(JL,m,f). This is described in the next section.

4. CROSS SPECTRAL MATRIX OF AN ARRAY

Let us look at n(x,y,t) at two fixed points in space, say (xo» yo) aQd (xl> yi)• This would give two stationary Gaussian pro- cesses indexed on time alone because (x0, y0) and (x^, y^) are fixed. Thus, we may write

If X = (x^ - x0) and Y = (y^ - y0) Then we can say, since n(x,y,t) is assumed weakly stationary (see Equation (3.3)), that

MXXT) = Efo.(*M.(*+T)] where the expected value is over the ensemble for some specific value of t, where - °» < t < °°. Let us extend this idea by a change of notation and let N(X,Y,t) be a two-dimensional (vector) process, double-indexed on time; i.e., let N be a vector function

N(XXt) = (7*/*M*W)-NM ' for -e»<t< co (4.1)

12

We can then write a generalized covariance function (assumed to be finite) as the matrix equation

R(T) = E.[NV) N(t + T)J

_ [EL (n. (t) 7,(t + r)) £ fa. (t) t, (t + T))

RfO,0,TV (4.2)

Now R(-T) is

R(-T) = fR(^o,-T) R(X,Y,-T)

R(-XrY.-T) R(o,o,-T)_

and R(-T) transpose is

R(O»O,-T)

R(-T) (4.3)

R(-X,-Y,-r>

R(X,Y,-T) Rfo, o, -T^

We have by Equation (3.3) and stationarity that

R(~X,-Y,-T) = E(»l(xlf^%t*T) ■?(*.,s^t))

It then follows that

R(T) = H>T) (4.5)

13

From Equation (3.7),

let

and we get

p*(x.Y.<) = [Cfc.Y.f)- *Q(XXf)]

P*(X,Y,f) = J"R(X,Y,T) L^(-Uit<T)4T . Now R(-X,-Y,T) = R(X,Y,-T) by Equation (4.5). Thus, we have from Equation (3.4)

or, by the same procedure, that Equation (3.6) was obtained from Equation (3.4), we have

P*(X,Y,f\ -[R0C,Y,-TUxb(iEir4T)AT (4.6) -OP

Now, since R(X,Y,-T) is real and Fourier transform pairs are unique, we must have from Equation (4.6) that

J*R(x,Y,-T)E*t,(-iair*T)AT = P(X,Y>*)

a parallel form of Equation (3.6). Therefore, the Fourier transform of R(-X,-Y,T) = R(X,Y,-T) is the complex conjugate of the transform of R(X,Y,T); i.e., (see Equation (3.7))

P*C-x,-Y,f) =[p*(x,Y,f)]H= P(X,Y,rt T In general, we have that the Fourier transform of R(T) = R (-T) is

.P (0,0,0 P#(XXf)

P(X,Y,f) ?(Ö,0^) | (4.7)

Note: P*(0,0,f) - P(0,0,f) is real.

14

By definition it follows that

C(X,Y,f) = Re[P(X,Y,f)]

Q(X,Y,f) - Im[P(X,Y,f)]

where -» < f < °° .

The function C(X,Y,f) is called the cospectrum and Q(X,Y,f) is called the quadrature spectrum. Both are spectral density functions.

Explicitly, we can think of (x0, y0) and (x^, y^) as being the location of elements of a probe-array with space separation (X,Y).

The first step to find (or estimate) S(ü,,m,f) (see Equation (3.9)) is to find P(X,Y,f) related to a pair of array elements. This is a problem of estimating the cospectrum and quadrature spectrum of a two- dimensional (vector) stationary Gaussian process. Goodman (Mar 1967 - Chapter 3) has an excellent treatment of this subject, which we will discuss. Kinsman (1965 - Chapters 7-9) also discusses the subject. The essence of the problem is that if P(X,Y,f) is continuous and negli- gible for |f| > fn, then R(X,Y,T) can be obtained by a time average over a particular realization N(X,Y,t) instead of having to average (find the expected value) over the ensemble. This says that we can find R(X,Y,T) by obtaining two time series (realizations) rQ(t) and r^(t) measured over time at only two points; e.g., (x0, y0) and (x]_, yi) where (xi - x0) = X and (y^ - y0) - Y. The relationship between (r (t), r, (t)) and R(X,Y,T) -» < T < « is °

R(X.Y.T)= LIMIT J \ r.(tV r, (t + T) <Lt 0 'to/ -Va

(4.8) Z

where r0(t) is a realization measured over time at (x , y ) and r,(t) is measured at (x^, y^). We can simplify the notation by an expression for a time average given by

R(X,Y,T) = Roi(T) = r0(t)ri(t+T).

It follows from Equation (3.7) that\C(X,Y,f) = CQ1(f) =

J RQI (T) cos(2TTfT)dt,

15

and

/ Q(X,Y,f) = Qoi(f) = / RQI(T) Sin(2ufT)dT. (4.9)

We also have (see Equations (A.2) and (4.7)) CQQ(f) = C11(f) = P0o(f) = Pll(f),

Qoo(f) - Qn(f) = °; Pg^f) = C01(f) - i Q01(f). (4.10)

The phase of PQ.. (f) is given by

Q01(f) Q01(f) = Arctan c

U±(f) . (4.11)

This is the expected phase lead of the signal at (x , y ) over the signal at (x]_, y^) for f where - °° < f < °° .

For an array of N detectors located at (x^, y^), (x£, y£), (xo, y~) ...(xN, yN) we can find N

2 spectra [Y±*] i = 1, N, j = 1, N. This gives a unique spectrum Pi:L (Pll = ... = P._J and since Pi1 = P* (see Equation (4.6) and 4.7)) J 3

we have

M|zi) (4.12)

unique cross spectra or a total of [N(N-l) + 2]/2 unique spectra. Thus, for real n(x,y,t) we have P±. (f) = P*j,(f) and P±i(f) - PJ.(-f) allowing us to define the information about P\X,Y,f) obtainable from an array by a cross spectral matrix

W>) • • • ' ?w(0 (4.13)

16

This information can be used together with Equation (3.9) to obtain an approximation to S(£.,m,f). Numerical details for finding the spectral matrix are given in Bennett, et al (June 1964).

It should be pointed out that negative frequencies are still con- sidered in the relationships being discussed. We do not know P*(X,Y,f) for continuous values of (X,Y). We do know from the spectral matrix the values of

where X.. = (x. - x.)

We also have from Equation (4.7) that

From Equation (3.9) we get

*' - " ' v ^ ^r ^7 r^ (4.14)

-iP?XXrts»Nt^Ü* + *V)]}4x<SiY

or, since S(i,,m, f) is real

5(*.».-f)=:^ |C(XY*) cos[a.Tf(«X*mY)] — CD "OP

-Q(XY.f)siN[E*»OX + TnY)]} AxiY (4.15)

Let us consider treating the points of (X,Y) where we know C(X,Y,f) and Q(X,Y,f) as weighted Dirac delta functions; e.g., at (X]2, Y..^ we get

C(X,Y,^= tltC(Xlt,X0<) i (x-X«V (Y-Y„>.

17

Reverting to the C-H , Q^ notation of Equation (A.13), we have, where Xji = ~Xij and Yj " " ' " " "" " " " form of Equation Xj. = -Xij and YJJ = -Y-H, Cj^ ■ C-H and Q-H = -Qij• T^e numerical Eorm of Equation (4.15) then becomes

sä.».*?) = b1± Cii(f) + z 2? f\fci(*) cosfat(8X..+ >Of. )]

-Q,W«««Lt1'(l*y+1*Xi)]}. (4.i6) Choice of b^j values is arbitrary. A reasonable choice is b., =

[N(N-l) + l]-1 for all (i,j) (refer to following section). We nowJhave a basis for an approximation of S(£,m,f). Before exploiting this result, we need a few side results.

5. SPECIAL CROSS SPECTRAL MATRICES

Assume that we have a real sea wave of frequency f > 0 moving from direction 90 where SLQ ■ KoCos60 and m = KQSin eo, K0 being the wave number from Equation (3.1). We can write the wave as

n (X%H,D =. ACOS(ZTT(JI.-K + >*I.'SI +Lt +<*)) (5#1)

Since the root-mean-square (rms) value of a cosine wave is A//27 we have for the two-sided (-» < f < ») directional power spectrum of the wave in Equation (5.1)

+ «£(» + M i (*l+ *>-> i ( * "»*-)] «2>

or in polar form where K = vl +m ; S = arctan(-r)

+ i(e-^-iT))J(U+.)]4(K-K.)

18

From Equations (3.8) and (5.2) we have for a single wave of frequency f0 > 0 from a direction 80 that the two-sided

p*m = ■( L^c (i t« (j. *..+«. x,)) i (f -1) + + eKp(itn(-i.Xir™Yii))i(4 + f„)]

Qj(t) = -%(-<•) = - Al6«.(«i (l^ YiS)) (5.3)

describe the elements for the spectral matrix of a single wave. Recall id Qi-i(f) = -Qi-j(rf). Substituting >±i -[NCN-D+ir1 = l/M (5>4)

that, in general, C±j (f) = Cj±(f) and Qij(f) = -Qi-j(7f). Substituting into Equation (4.16), we get where b-n = [N(N-1)+11_1 = l/M

and fQ > 0 is assumed

^Zl [C06 (ail (i-X^M^) 005^(1X^^X0) *

-hSlN(aTT(J.Xi3+>n.Xi))SlN(aTt(J^Xij-»-1viNi;i))] on

(5.5)

•Ö «J» The choice of b-n = l/M = [NtN-D+l]""1 and observing that <> 4 - »(»»-ft

19

gives the convenient result (Note: For -f < 0 we would use (-lQ,-m0) in place of (£0,m0))

which agrees with the values of Equation (5.2) at the points (£0,m ,fQ) and (-Ä0,-m0,-f ). Note that there is not general agreement elsewhere. In fact, Equation (4.16) may (and does) give negative values for the approximation of S(£,m,f). This is a problem of probe array design and is directly related to the directional resolving power of a probe array. This problem is discussed later in another section.

Consider the case of a single frequency, fQ > 0, real sea with equal wave energy from all directions; i.e., isotropic waves. In this case, assuming the wave equation (Equation (3.1)) holds, we have

S(W)= [Ä i (|«'+H')-(lN».1))f(M.)ti(f,y] OR »Ml« K^lW, KsV^ , MO e = A«lTd« (Pf)'

S(K,e,rt»ÄJK-K»)^(f-f.)*i(f4t.)] (56)

as the two-sided directional power spectrum for such an isotropic wave.

Since A = Kcos6 and m = Ksin0, Equation (3.8) can be expressed in polar coordinate form as

Elfi = ((S(KA*)«p (i air K (X^ose +\*\n e))KdKde(5.7) -if o

Letting tyj - ( X^ •+ Y.j )

20

and (j).. — ARCTAM (~&M

we can write

1f

T?j(f)= f (s(K,e,4)EKP(UirKBijCös(e-^KaKdG

Using Equation (5.6) for S(K,0,f) we get

P*(fJ = X K. fE*P (I ait K.D^cos (e - $.;)) U -'-If

= ÄK.Jco6(aiTK.Duco9(e-y)a.G +

(5.8)

die

-f LÄKajsiN(21TKdDiiCos(e-(().i))di0 (5.9)

-It

Now, departing from the above development, consider the following integral where

z - atr K.D4.J ; «| = tg

\cos(n©) cos (* cos(0-a>))de -ir

Let <|> = 8 - i|> and we get

COS fcos(n<|>+»^) co&(z cos <|>) cl<J>

21

Expanding cos (no + nijj) we get

COS

K-9

v» i|) ] COS v» ty cos (x cos 4>) <L$

T.siNYt4>jsiNn<|l cosCz cos $Hf . -K-U>

We have (ir - i|») - (-TT -ij/) ■ 2tr so that the integrands are over 2TT

allowing us to write the above as

JW

cos >j t|J I cos tt <(> cos(2. cos

— SiNtti|»JsiN>i^ cos(* cos 4>) A$ . -ir

From Ryzhik and Gradshteyn (1965 - page 402) we have (since sine is odd and cosine is even) the result

\cos(*e) Cos(i cos(e- tp))de

= cosTr>qj[z««cos(2ür\ J„(xV] . (5.10)

In a similar way we get

,1T

\cos(y>e) SIN (2, cos(e-ijjJ<ie J-tf

a cos m|)[aTt SIN(J^) Jn(z.)] j (5.11)

22

where Jn(Z) is a Bessel function of the first kind of integer order n. Returning to Equation (5.9), the above results give, for n = 0 where fQ > 0, the result

?ki (±fj = %l TT K. J. (2Tt K.Di3) .

Since the above is real

CU(±Ü = Z* K.A

W = 0 (5_12)

describe the elements for the spectral matrix of a single frequency isotropic sea.

The above two special cases for sea waves are the extremes of directionality of real sea waves of frequency f > 0. These results will have important applications later.

6. A MEASURE OF ARRAY DIRECTIONAL RESOLVING POWER

From Equation (3.9) we have the Fourier transform

S(«.>n.^ = frp*(KtY,f)EKP(-iairClX+>»Y^<lxaY (e.D -00-0»

In practice we use a probe array with elements at (x^, yi)..., (xk, yk) to obtain P*(X,Y,f) at the separation points (0.0), (X12, Yl2), (-X12,-Y12)..., (Xk_ljk, Yk_ljk), (-Xk_1>k, -Yk_1)k) .

We do not then know P*(X,Y,f) but the product

[P*(X,Y,f) g(X,Y)] (6.2)

23

where g(X,Y) is a set of Dirac lelta functions standing on the separa- tion points of the probe array, and zero elsewhere.

Thus, we have the estimate

Let

«» a»

G(Jl,m)=J[|(x,Y)Exp(-.izTT(!IX + MY))clxcJiY (6.3)

Using this and Equation (3.8) we have for a given U0>m0,f0) that

S lh\J0 =))[j(s(I M X) ^?(i tir(!x*»iY))ai a J j(KfY) -co-ooKa,.« '

eM>(-it1f(l#X + >O0)<lx<lY

jS^Ktj^X.YlExpt-Ae^

s|[s(i.mit)G(^-»,*ni-in)ajiawi •""" (6.4)

As expected, S(J£,m,f) is a two-dimensional convolution of the true directional spectrum S(£,m,f) with G(£,m) the Fourier transform of j(X,Y). We see then that S(i.,m,f) is a weighted average of S(£,m,f)

24

and that G(£,ra) is a measure of the directional resolving power of the assumed probe array. By the nature of g(X,Y) we have from Equation (6.3)

G(J,»)= l+a42^C05(ET((lX.i+wXj)) (6.5)

\ If we assume that the wave equation (Equation (3.1)) holds, we find that S(£,m,f0) is zero when Ä + m2 ^ kg, the wave number for fQ. From this we have for a given U0m0) that the directional resolving power, DRP, is

DRP(l,*i,t) = jp((l*+»it)-K;)G(l.-l,in.-)n)^Ji-

LET J = K,COS© , m = K. SIN 6,WHICH IMPLIES & ♦ ** - K

and we get, for energy coming from a direction 90 at frequency f as a function of 0 <_ 9 <_ 2TT, that

DRP (e | Q0)f.) =G(k.(cos&-cose)JK.(siN0#rStNey) .

From Equation (6.5) we get

DRP(e|a,Q= l+2^£cos[aTT(xiJK.Ccose.-cose>YijK.

(SIN 0.-Si* 0)1

= * + ^££cos[zir(Ci.-Kwcos e)Xy+ (*.- K. SIN e)y\

NOTE THAT DRP^J^S N(NH)+I (6.6)

Compare Equation (6.6) and (5.5). Except for the amplitude term, A2/4M, in Equation (5.5) the equations give identical results. The choice of ^11 = [N(N-l) + 1] = 1/M is again found convenient.

25

7. DIRECTIONAL ANALYSIS FROM THE CROSS SPECTRAL MATRIX

First, we consider a fundamental approach. We have for a pair of detectors I and J located at (xi,yi) and (x^,y^) respectively, the

cross spectral matrix, P±.(f) = C-y (f) + ±Q±.(f) and more impor- tantly 4>i:i(f) the phase lead of I over J given by, ,

W> — [fgJJ . (7.1)

The actual phase lead may differ from this value since the true phase lead 9 is some one of the values

QJi) + k 21T where

Consider a single wave of frequency f with corresponding wave length A0 and wave number K0 = 1/X0, and find the direction the wave must travel; i.e., fit a single wave to the spectral matrix results for the detectors I and J.

Let D-JJ be the distance between I and J. The distance between I and J in wave lengths is K0D-M . In radians this is 2TTK0D-M • From this relation we get

-Z1T K.D^(M 21f K.DW or that only values of h such that

-att K.D^ (ftjft) + *i ZM $ an K.Dy (7.2)

give physically acceptable candidates for the value of 0. If D-y < X0/2 only one h value is valid. If XQ/2 < D^ < \Q at most two values of h are valid, etc.

The problem is how to find the true direction, 60, of a single wave given the above possible value(s) of $. Consider a given value of $ ln terms of wave length units and we obtain

26

(7.3)

Figure 4 illustrates a case for <f> >_ 0 (and thus L0 ^ 0). From the figure we have, where the true direction is 80 and true phase is <|>, that

0o= *ij+f +*"

WHERE

oil

(7.4)

OC =■ ARCSIH

(x3. y3)

(«i» yt)

FIGURE 4. WAVE DIRECTION ANALYSIS

27

Recall that sln(a) = sin(+rr-a). Thus for a given ij) > 0 we get, since a must be obtained from an arcsin relationship, two possible values of wave direction, the true value 6Q and its image

e'=[fij+f] + [-«-«•]

(7.5)

This is illustrated in Figure 5. In actual practice we do not know the true direction, thus a given value of <j> >_ 0 gives the direction as

*"*j*[¥+*] where 0 <^ a £ TT/2 is obtained from the principle value of the arcsin. If <}> < 0, then I actually lags J by | <J> | > 0 and L0 = L0/Dij < 0, so that arcsin (LQ) gives -TT/2 <_ a < 0.

> 0

True Wave Direction

8.

V 9' (False Direction) FIGURE 5. DIRECTION ANALYSIS FROM A PAIR OF ARRAY ELEMENTS

28

The implied directions, for <f> < 0, are directly opposite from those for <j> >_ 0 so that for a given value of <$> <_ 0 we get directions (a < 0)

and ö. = 4>ij-f -*

or A— IÜ.+ 3L+OC

6 = H^fr **•] as before.

Thus, where the principle value of arcsin is assumed, we get a set of possible directions

where

K-^^T^l

rMkkiE.1 h being constrained by

Mklss l4i

An example of this type analysis, for an array pair, is illustrated in Figure 6. Thus several estimates of 60 are available (at least two).

The estimates of true direction, 0o, often vary from one array element pair to another, making the selection of a true 80 value diffi- cult. The selection is also hindered because half of the estimates of 90 are of the image type; i.e., false estimates.

While the above directional method leaves something to be desired, it does illustrate the basic directional information produced by an array of detectors.

A better method, suggested in Munk et al (April 1963), of using the spectral matrix directional information to fit a single wave at each

29

360

270. co Q)

■U tu §~ •rl CO •M Q) CO fl) w M 180- C (!) o p

•l-l v»/ 4J o a u a 90

■—=- 1 1 1 I 0.0 0.2 0.4 0.6 0.8

Normalized Frequency (Fn =0.5 Hz) CH 2-5 Date 6/4/65 Time 1418-1449 Stage 1 WD 282 WS 14

FIGURE 6. DIRECTIONAL ESTIMATES FOR A PAIR OF ARRAY ELEMENTS

1.0

frequency is given below. It is based on Equation (4.16) in the form

r\ L i=i jti+iv. (7 7)'

where M = [N(N-l) + 1], C(t) sl£c, (f)

and N = number of array elements.

30

Recall that for a single real wave of frequency fQ > 0 and known direction eo, C.jj(f0), Q^.(£0) are known (see Equation (5.3)). These values give

S/K..6..0 = S/K.,k -i3rO = -£ ■ When a single, well-directed swell is expected, it is reasonable to assume a single wave for a given frequency exists, and to select 80 and A0 = A^/4 such that the least square error between the theoretical cross spectral matrix for a single wave (see Equations (5.1) and (5.3)) and an observed cross spectral matrix is a minimum. Accordingly, using the expressions of Equation (5.3) and an observed cross spectral matrix for a given frequency, f0, we can form the squared error

H = (C0 - A0)2

£'£.r<VA.siN ("( Uu+»i.Yy))]8

NOTE: A0= A : Ca =^EC.i

Expanding and collecting terms we get

- aA.[C0 + H.LZ, Cü cos (an (I. X..+ v,#X.))

or using results in Equation (7.7)

N-IJi

+ a

(7.8)

(7.9)

31

10)

To find A that minimizes H, consider o

iä aa[H(K-iVO[A.-S(l.m:.t)]so «fA.

(7.11)

This requires that AQ be of the form AQ = S(£,m,f ) and a resulting value of H of the form

H- C.1 ♦^((C^*^)-CK(N-I)*I][S(I.*»-0J (7.12)

2 2 2 Since C0, C-JJ, and Q^ are all nonnegative, a minimum H results when S(£.,m,f0) is a maximum. A choice of %Q and mQ that maximizes S(£,m,f0) implies a 0O = arctan m0/£,Q which is optimum. Remember that we are assuming K~ = £,2 + M£ holds, along with the wave equation. The results then for each fQ is a two-sided energy spectrum estimate A (f0,6Q). Appendix B contains a listing of a FORTRAN II program for finding A0(f0,80), the least square wave fit, from a set of spectral matrices obtained from the task SWOC data collection and analysis system described in Bennett et al (June 1964).

Examples of least square single-wave train fit analysis from Bennett (March 1968) are shown in Figure 7.

A more complete collection of the directional spectra calculated from data collected off Panama City, Florida, is given in Appendix A, and in Bennett'(November 1967), and Bennett and Austin (September 1968), an unpublished Laboratory„Technical Note TN160.

We would actually like a continuous estimate of S'(£,m,f). Con- sider then a third method. From Equation (3?6) we have for a pair of detectors, as illustrated in Figure 8, that ■%■;.

CO QO ""

P*(x,Y,4>a|[s(U.4)Bxp[iE1t(iX*MY))AUM . -co -OR

32

DATE: I SEPT. IMS TIME. 112«-1152

• STAGE I WNU SPEED: ST KNOTS

IM DEGREES

DATE: 4 JUNE IMS TIME: ISM-ISSO STAGE II HIND SPEED: 14 KNOTS WIND DIRECTION: 2S0 DEGREES

FIGURE 7 DIRECTIONAL SPECTRUM Ao(f.0) FIGURE /£ DIRECTIONAL SPECTRUM Ae(f,9)

DATE: 4 JUNE IMS TIME: 1418-144)

o STAGE I ■- WINDSPEEO: 14 KNOTS

WIND DIRECTION: 212 DEGREES

DATE: 9 SEPT. IMS TIME: IS2S-II53 STAGE II WINDSPEEO: 12 KNOTS WIND DIRECTION: 125 DEGREES

FIGURE 3. DIRECTIONAL SPECTRUM A0(f, 0) FIGURE 70 DIRECTIONAL SPECTRUM Ao(f.0)

FIGURE 7. LEAST SQUARE SINGLE WAVE FIT DIRECTIONAL SPECTRA A (f,9 ) o o

33

(*a> ya) Detector 2

Detector 1

FIGURE 8. DETECTOR GEOMETRY

Let Si = K cos0 and m = K sin6

and

fl 2~ Y D = /X + Y , ^ = Arctan ^

or

X = D cosij; and Y = D sin \\>.

Using the above changes of variable, we get

P"*(x,Y,0 = J jS(KöfUxp[k^(KDc6S8cos<)i+KOsmesiN4))j -H o

-T? © ^ (7.1 3)

34

We have from Equation (3.2) that

s(K,e,f.) =S(K,e-it,-f.) If we think in terms of f0 > 0

s'(K,e,f.)=2S(KM) <J.U> We are assuming that the wave number relation of Equation (3.1) holds. Thus Figure 3 is applicable, and we can write the one-sided spectral density as

s7K,e,0=£Q-(e,t)^(K-K.)

<J(K-K.)=j' *"«•

This allows us to write, — «o ^ T» ^ "i" °° ,

WHCRL

if «o

P>X^)=J{«'(a.*.)«f(K-K,)E*p[UirkDc<)s(e-^K<iKdL9 -1t *

If

p*(xxV« [a(e>O£xp[i8LifK,Dcosfe-(|>iJK0<4e -tt (7.15)

We have reduced the problem to finding [a(6,f0 • K ].

From Equation (7.15) we see that

?*(o%oJ\* P(f)= U(e,UK8de -•V

where P(f0) is the power spectral density of frequency f . For better comparison of cases where P^) = P(fQ), |f,| ^ |fQ|, it is convenient to express a(0,fo) in a normalized form

A(9,f0) = a(6,f0)K0

35

where we get

•ft- p(f) = U(e,f)ie (7-i&)

Thus if the energy distribution as a function of direction is the same for P(fi) = P(f0) then we also get

AO.fj) = A(9,f0).

Consider now (assuming A(9,f) can be so expressed) a Fourier series expansion of A(6,f) for fixed f. Clearly it is periodic in 6 with period 2TT. Thus for any given f = fQ we can write A(6,f) as

*- -nm\ -J

Substituting this expansion into Equation (7.15) we get

(7.17)

?*(*.Y»0 = T° £KP i-2.^Kücos(tf-^)dd + S[o.n \cosr\9 Cm I HtH J

EXP ie.it Kt> cos(d-\|<) die + bJsw Tie E*P UirkD cfls/d-t)ie

oa Pty.Y.f) = |«(cos(att KD ces(e-i|>)We +£(&>, lcos*e 005(2**0

cos (d-Tjr^ie-fb^f&^n6 cös(2TfKDcos(e-i|i))olej| tic «W J |? \stt4 (ztr KD cos (d - ^Jjie +

^(a„(cos>i«si^^Kocos(e-t^e +

[si«0 Tje Si*l(a|HO COS(d-T|r)j4fl)j b^ ) SlM 79© Sitf(aflKI> COS(ö-T|r))deJ( (7.18)

36

Thus we can express P*(X,Y,f) as complex infinite series with unknown coefficients a.Q, ai, a2>... and b-,, b£, ... and constants defined by the integrals (let Z = 2TTKD; n = 0, 1, 2, ...) of the form

jcosrjecos(ÄCw(e-^))Ae (7.19)

(7.20) \siM»öCös(3LCos(e-i|f))Ae

jcosyi*$iN(uo$(0-Y)Jcle

'-1T

.1»

(7.21)

and

[SIN nös»M(acos(e-ii>ftdLe (7.22)

Consider Equation (7.19) where $ = (6-i|>) and d<}) = d6, \p being a constant. We then have on changing variables

\ cos (n cp + y» iji) C05(zCöS(p)d<p

s COS >|\|rJCOS)?<pC0S(2CO5<P)dU>

smn<fcos(u0S<p)4<f

(7.23)

Since the integrands in Equation (7.23) are both of period 2TT and the interval [-Tr-iJ/, ir-ip] is of length 2TT we can write the equivalent of Equation (7.23) as

COSVtl|>(cöS»^CöS(lC0S<P)cl(p J-TI *

— sin r\H* (SIN n4> cos (z cos <P) A<P -Tl 1»

37

From Ryzhik and Gradshteyn (1965 - page 402) and noting that the second integrand is odd we get Equation (7.23) equivalent to

co5>iq»[aT\c0s(^ Jy,(*)l (7.25)

**l

+

CO&i

where Jn(Z) is the Bessel function of the first kind. Employing a simi- lar procedure for Equations (7.20), (7.21), and (7.22) we get

<j£ j>cos ww **($) X fa **) + ]pn St« n \y a* SVM^ J^H K D)1 (7.26)

p*(x,v,f) = §-•[« J.U*KD)]

Now

Thus we get 00

VI.M- (7.27)

38

From a spectral matrix of the form in Equation (4.13), M = N(N-l) +1 different equations can be set up using Equation (7.27). This allows us to get a system of equations for any m of the unknown coefficients aQ, a^, a£, ... ; b^, ^2» bß, ... while assuming the rest of the coefficients are negligible. We can then solve for the m desired coefficient values. This has not worked well in practice for two reasons. The inverse of the matrix of constants obtained is sparse and often ill-conditioned. Further, if the wave energy is from a nar- row beam width (30 degrees or less), the first 100 harmonics in the Fourier series expansion can be significant. There is perhaps a more efficient orthogonal set of functions than the sines and cosines of the standard Fourier expansion. Search for such an orthogonal set should prove fruitful. One might start with Walsh or Haar functions. See Hammond and Johnson (February 1960).

8. SUMMARY

It is believed that the least square method of using the informa- tion in a spectral matrix is the best method presently available. Examples of such analysis can be found in several of the papers in the bibliography. A collection of ocean-wave induced, bottom pressure directional spectra from these papers is given in Appendix A.

An iterative extension of the least square method can be found in an excellent paper by Munk et al (April 1963). Some details of this method are given in Appendix C along with an example result and a FORTRAN program for the method.

There is merit to using the coherency,

to form the weights b.. in Equation (4.16). One idea being explored is

39

~\

REFERENCES

Bennett, C. M., "A Directional Analysis of Sea Waves from Bottom Pressure Measurements," Transactions: Ocean Sciences and Engineering of the Atlantic Shelf, Marine Technology Society, Washington, D. C, pp. 71-87 (March 1968), Unclassified.

U.S. Navy Mine Defense Laboratory Report 344, Power Spectra of Bottom Pressure Fluctuations in the Nearshore Gulf of Mexico During 1962 and 1963, by C. M. Bennett, November 1967, Unclassified.

Bennett, C, M., Pittman, E. P., and Austin, G. B., "A Data Processing System for Multiple Time Series Analysis of Ocean Wave induced Bottom Pressure Fluctuations," Proceedings of the First U.S. Navy Symposium on Military Oceanography, U.S. Navy Oceanographic Office, Washington, D. C, pp. 379-415, June 1964.

David Taylor Model Basin Report No. 8, On the Joint Estimation of the Spectra, Cospectrum and Quadrature Spectra of a Two-Dimensional Stationary Gaussian Process, by N. R. Goodman, March 1957, Unclassified.

N. Y. University Technical Report No. 8, (EES Project No. A-366), Two Orthogonal Classes of Functions and Their Possible Applications, by J. L. Hammond and R. S. Johnson, February 1960, Unclassified.

Kinsman, Blair, Wind Waves - Their Generation and Propagation on the Ocean Surface, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1965 (pp. 368-386).

Munk, W. H., Miller, G. R., Snodgrass, F. E., and Barber, N. F., "Directional Recording of Swell from Distant Storms," Philosophical Transactions of the Royal Society of London, Series A. Vol. 255, No. 1062, pp. 505-584, April 1963.

Parzen, Emanuel, Stochastic Processes, Holden-Day, Inc., San Francisco, California (1962).

Pierson, W. J. Jr., Wind Generated Gravity Waves, Academic Press, Inc., New York (1955), v. 2, pp. 126-129, 1955.

Pierson, W. J. Jr., and Marks, Wilbur, "The Power Spectrum Analysis of Ocean Wave Records," Transactions, American Geophysical Union, v. 36, No. 6 (1952).

Ryzhik, I. M. and Gradshteyn, I. S., Table of Integrals, Series, and Products, Academic Press, New York, 1965.

St. Denis, M., and Pierson, W. J. Jr., "On the Motions of Ships in Confused Seas," Transactions of the Society of Naval Architects and Marine Engineers, v. 61, pp. 280-357 (1953).

40

BIBLIOGRAPHY

Barber, N. F., "Design of 'Optimum1 Arrays for Direction Finding," Electronic and Radio Engineer, New Series 6, v. 36, pp. 222-232, June 1959.

Bennett, CM., "Digital Filtering of Ocean Wave Pressure Records to Records of Prescribed Power Spectral Content," Proceedings of the Fifth Annual Southeastern Regional Meeting'of Association for Computing Machinery, June 1966.

Bennett, CM., "An Annual Distribution of the Power Spectra of Ocean Wave Induced Bottom Pressure Fluctuations in the Near Shore Gulf of Mexico," (abstract), Transactions, American Geophysical Union, v. 48, No. 1, p. 140, April 1967.

Texas A&M College Reference 62-1T, Instrumentation and Data Handling System for Environmental Studies off Panama City, Florida, by Roy D. Gaul, February 1962, Unclassified.

Texas A&M University Reference 66-12T, Automated Environmental Data Collected off Panama City, Florida, January 1965 - April 1966, by A. Kirst, Jr., and C. W. McMath, June 1966, Unclassified.

Scripps Institute of Oceanography Bulletin, Spectra of Low-Frequency Ocean Waves, by W. H. Munk, F. E. Snodgrass, and J. J. Tucker, v. 7, No. 4, pp. 283-362, 1959.

41

APPENDIX A

A COLLECTION OF DIRECTIONAL OCEAN WAVE BOTTOM PRESSURE POWER SPECTRA

This appendix is a collection of the results of a least square, directional, single-wave train analysis of the cross power spectral matrix resulting from the analysis of ocean bottom pressure data. The data were collected at Stages I and II offshore from Panama City, Florida, during 1965. The data collection system and the estimation of the cross power spectral matrix associated with a set of data are described in Bennett, et al (June 1964). Augmented pentagonal arrays, containing six pressure transducers each, were located seaward of each of the stages. Stage I is 11 miles offshore in approximately 103 feet of water, and Stage II is 2 miles offshore in approximately 63 feet of water.

Certain parameter values are pertinent to the directional analyses presented: the number of data points in each pressure data set is N = 1800; the sampling rate is once per second, At = 1 second. On the cylindrical polar plots frequency is the radial variable and compass bearing the angular variable. The vertical axis is log^Q of power spectral density in inches -seconds of water pressure. The frequency axis range is 0 to 0.3 Hz in 0.05 Hz increments. This is illustrated in Figure 7 of the report. In each plot title, the date, time, and location (stage) is indicated. The value WD is wind direction in com- pass degrees, and WS is wind speed in knots. Appendix B gives a listing of the FORTRAN II computer program used to produce the plots.

A-l

8

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a rJ"

s

M

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n -' ■ X'%.''—/l ) )

iX y y y ' -'" /

w ORTOVln/B T1IC1I9M22I JTBZI «1193 ontc&ui/ra 11*1277-1255 STKZI WITH

mTBO/OI/H T1ICW15 IMS 9TOCI KÖB2 DATEO6/04/65 T1MEI4I8-W49 STAGEI Wo2B2

A-2

DBTCOVOI/BS 11*1600-1639 DTBGE1 H03KJ OBTEGK/WK THCIBID-ITBS STDGCI MB»

0firca6/a«/65 TI«ITO-1731 STnat MQ315 mrtoB/M/65 THCZOOT-ZOJS JTBKI w n

A-3

anzavat/n THE21(B-2>S3 STHZI W no OATE06/05/6S rilCOXD-OOK S1SGEI U010D

DATE 06/05/65 THCOB5-01IJI STIttI »I0O DATE06/05/65 TllcmtB-aiJO 3THGII WIDO

A-4

1701-1730 U7ST63 31 «173 HS16 T3NF1 «115-17q *J< 7! 1901-1930 073ET«5 31 W060 K II T32WllOffi-\91 UM J3

2301-Z33D 073ET65 31 KOB «319 T325fimS2-Jll UM S

A-5

oioi-om nares si WXBS «21 T&HIIWE-OT IUI » 0501-O5W 0BJT65 31 MX» H519 T325FIIWrM51 UM Z7

0501-05» mm 31 MOOTS eis TJWIIBB-TH nw a 0701-0730 OBXTfiS 31 M00B3 W3Z2 T375F1KK-691 ftM 39

A-6

0901-0930 ouirra si won MSIB razwincw-oaj «IN SI U0I-1I31 003ET65 31 H0IV1 H516 T326F1MM-213 FUN 31

1501-1S31 QB3EfB5 3: MOCPO «17 TB& 1 rWÄ-<JS3 FUN 33

A-7

iwi-mi orares si wxm «si TTS&IIGU-TTS IUI SI 1901-1931 0B*«5 31 WOll «SS T3Z6FIIB8I-493 IUI 35

3101-21311 asms 3i mini uszs T3Z7FzioB-<m iu< SB 2301-2330 0KEP65 31 N50B3 H5Z6 T3Z7F2ni62-191 ItW 37

A-8

0101-0130 09SEP6S 51 HODBl H528 T327F2fl2S3-311 FUN : O2OHE30 033EP65 SI HSDfl5 H527 T327F2R342-371 FUN 39

0301-0330 09SEP6S 31 WJOBO W5Z7 T337F2fW03-431 FUN UD M01-0430 095EP65 51 H0OB2 H526 T327F2FWF52-<J91 FUN Ul

A-9

0501-0530 D95EP65 51 H00B5 H52B T327F2H522-55] Hi» 42 0601-063Q Q9SEP65 31 MOOBB H52B T327F2RSK-fil 1 FUN 43

Q701-073D D95EP65 51 H0092 H52B T327F2R642-771 FUN 4H OH0J-CB3Q 095EP65 51 H0103 H527 T0B3M fiOCD-032 FUN 45

A-10

0901-0930 093EPB5 51 HO 117 H52B T0B3F1R063-092 FUN »16 1001-1030 095EP65 31 H0123 H533 T0B3f1ftlZ3-152 RUN 17

1101-1131 093EP65 51 W0I23 «534 T0B3f 1RIS3-3I2 HJN HB 1201-1230 09SEP65 31 H0130 W53H TCB3F1A2H3-272 RUN U9

A-ll

1310-1339 D95EP65 31 N01J3 H335 T003FMJ01-O30 PUN 50 mTravtu/ra TIICI20D-I«9 STIEB to s

OtTOE/m/ES T1IC1J0D-1Z29 5THK2 HD 02 ORTtovN/65 nicrai-iro snoo HO US

A-12

0nn05/(N/G5 Tlieigcll-11125 3TSGE2 WI13 anaa/aum Ticvm-Mis JTHGU «in

0BTtO5/tn/65 T1ICU3D-1256 5TSGK W1JC DHTtO&'Ot/eS T)ICll91-l?ro 3THX2 M0165

A-13

OI7106/01/E5TIMEI220-I2453TH22 mil] onTEns/ai/ra TIICIHS-III9 arm» «250

anra/tn/ra nieuao-tzjo JTPCES «en OATE06/04/65 TIMEI300-I330 STAGE2W0280

A-14

looi-ieno 09sns 32 »no «29 Tjisrima-isi mm 1101-119] H9ST65 32 «HS WW T315F1HIB2-311 H*15

1701-1230 09IT» 33 »173 «30 T31VIWU-7J1 ««16 1301-1330 093EFS5 32 »170 HS33 T31Sf 1IO07-331 lUM

A-15

APPENDIX B

A FORTRAN II PROGRAM FOR SINGLE-WAVE TRAIN ANALYSIS

The FORTRAN II listing of an IBM 704 program for the least square single-wave train analysis of a spectral matrix obtained from an array of ocean wave bottom pressure transducers is included. The listing is from the FORTRAN to ALGOL translator of the Burroughs B-5500 and is syntax free at the FORTRAN II level. The mathematics of the single- wave train analysis is described in the body of this report. The data collection system and calculation of the required cross spectral matrix is described in Bennett et al (June 1964). The plotter subroutines GPHPVW and PFB3D are also included. Bennett (March 1968) describes the plotting technique in PFB3D which was used to produce the plots in Appendix A.

B-l

rriKTRAN TO ALGOL TRANSLATOR PHASE 1 FORTRAN STATEMENTS

C LISTING OF AN IBM 704 FORTRAN PROGRAM TOR DIRECTIONAL WAVE ANALYSIS.

C I335O-? C BENNETT JULY 1967 SWOC 78*301-9210

C REQUEST-0268

C LEAST SQUARE SINGLE WAVE TRAIN FIT AFTER MUNK

DIMENSION ID(11)»F9(100)#WN(100)>PERI0D(100)'WAVLGH(100)'THMAXC100

X)#DEpATN(lOO)»AVEP(100)#P(100»6#6)#EMAX(100)»TEM(ia6)

OIMENSION D(6.6).PS!(6,6),E(100,72),THETA(72)*DE(73)#THZRü(106),

XH(100)*H1C100)*H?(100) »0000(6)

DIMENSION DATA (1000)

E2TEN=0,43*29448

TW0PI*6.281853

RTO-57.29578

OTRsO.0174532925

FIV»0,087266A65

REWIND 3

REWIND 9

CALL PLOTS(DATAnOOO)MOOO)

CALL PLOKO.0.-30,0.-3)

CALL PLOT(?.5,2.5.-3)

DO 5 1-1*72

*I-(!-l)*5

5 THETA(I)»ZI*DTR

B-2

DO 10 I=l»6

DO 10 J = l»6

DC I, J)=0.0

10 PSICI»J)=0.0

0(1»2)=100.0

DC1»3)=100,0

D(1#4)=100,0

D(l»5)=100,0

DC1#4)=100.0

0C2*3) = 117,,558

DC2»4)=190.21?

0 C 2 » "5 ) = 1 9 0 . 2 1 2

0C2»6) = 117'.559

DC3»4)=117.55fl

DC3»5)=190.212

0C3#6)=190.212

DC4»5)=117.558

D C 4 » * ) = 1 9 0 . 2 1 2

D(5»«)=ll/.558

PSI is TRIG ANGLE:

D IS DISTANCE

PSI(1»2)=0.0

PSIC1»3)*72.0*DTR

pSICl*4) = Uü.0*DTR

BL3

PSI(l#5)»2l6.0*DTrt

PSI(1*6)=268.0*DTR

PSIC2*3)=l?6.0*DTrt;

PSI(2»A)=162.0*DTR

PSI(2'5)=196.0*DTR

PSI(?'6)=234.0*DTR

PSH3M) = i98.0*0TR

PSI(3*5)=234.0*0TR

PSIC3»6)=270.*DTR

PSI(4>5)=270.0*DTR

PSI(4*6)=306.0*DTR

PSI(S*6)=3«2.0*DTR

GOODCDs 1 FOR USARLE CHANNEL DATA AND 0 FOR BAD CHANNEL

15 READ 2, ISKIP #CG00D(I)#Isl,6)

2 F0RMATCI2 ,6F1.0 )

IFCISKIP)100»30»20

20 DO 25 I*1MS«IP

READ TAPE 3*IO#FQC1)»WN(1),M*K

IFCK)2l,22,21

21 PAUSE 20202

TAPE OUT OF PHASE WITH DATA READ DESIRED

GO TO 15

22 MP=M+1

DO 23 L=2»MP

B-4

23 REAO TAPE: 3

?5 CONTINUE

GO TO 15

C FO(1)=0.0 CAN NOT RE ME AN I NGLYFULLY PKOCESSED

30 READ TAPE 3* I 0»FO(\),WN( 1 ) ,H.K

IF C K >^ 1 .3.1*21

31 MPsM+1

00 32 L=2»MP

32 READ TAPE 3,ID,FQCl_) .*NCL),M,I<,DELTAT,DEPTH,PERIODCD ,WAVLGH(L)>

X0EPATN(L)»AVEP(L).((P(L«I»J)*J=1»6)»I=1»6)

C P(L. I>J)=-0.0 IF CHANNEL I 0« J IS NO GOOD

c WN(L) IS WAVE MJMBEP=2PI/WAVE LENGTH IN FEET

C PI=3.11159?/...

C VALUE K NO LONGER NEEDED ,

GOOOCH=0.0

DO HO I»l#6

GOOO(I)«GO0D(I>*PC?#I*I)

IFCG00D(I))«1»«0.41

41 GOODCHsGOODCH+1,0

G00D(I>=1.0

«0 CONTINUE

TERMS=.1,0*(GOODCH*(GOODCH-1.0'))

00 70 L»2'MP

SUM = 0.0.

B-5

DU 50 1=1*6

!F(G0ÜD(I))b1.50.Si

51 SUM = SUM*P(L» I . I )

50 CONTINUE

AVEP(L)=SUH/GOOOCH

DO 60 K= t » 7i?

SUM=0.0

DO 90 1= 1*5

IFCGOÜDU) )90»90»91

91 IP » 1*1

DO 95 J« IP>6

IF(GOUD(J) )95,95»9?

9? SUM » SUM*P(L»I#J)*COsF(WN(L)*D(I»J)*CasF(THETA(K)-PSI(I*J)))

X •PCL»J#I>*5IMF(«N(LJ*D(I»J)*C0SF(THETA(K)-PSI(I»J))}

95 CONTINUE

90 CONTINUE

60 E(L»K)=(AVEP(L)+2.0*5UM)/TERMS

IFCSENSE SWITCH I>62.63

62 PRINT6*FQ(L)»AVEP(L)#(ID(I)»I»1»11)»CECL#K)»K»1^2)

6 F0RMAT(1XMPE11,«MPE12.*»11A6/(10C1XMPE11.4)))

63 G=0.1/DTR

DELTA THETAe5DEQ UR Hsl/2*DEL THETA

DE(1)BG*CE(L»2)*E(L#72))

DEC72)=G*(E(L#1 )-E(LiM)J

B-6

00 61 ITHETA=2>71

61 OE(ITHETA)rG*(E(LMTHETA*l)-E(L»ITHETA-l))

DE(73)=DE(13

K = l

00 65 I=l»72

IFCDE(I)*0E(I + n)66»67#65

66 71=1-1

FIV IS 5 DEG IN RADIANS

THZR0(K)»FIV*(ZI + C0r(l)/CDECI)-DECI*1 >)))

K = K + 1

• GO TO 65

67 IFCOECI))6fi,69,6«

69 THZR0(K)=FIV*FL0ATF(I-1 )

K = K*1

6fl IFCDECI+1))65»64»65

64 THZR0(K)=FIV*FL0ATF(I)

K=K*t

6|5 CONTINUE

NZER0S=K-1

00 75 K<=1»NZER0S

TERMS SAME AS ABOVE

SUM»0.0

00 76 I=l»5

IFCGOOOfI) )76»76»77

B-7

77 IP=I+1

DO 79 J=IP»6

IF(GGÜDCJ) )79,79»78

78 SUM = SUM + P(L»I#J)*CnSFCWNCL>*OU»J)*COS*"(THZROCK)-PSlCI>J)))

X -PcL»J#l)*SlNr(WN(L)*D(I»J)*COSfcTHZRO(K)-PSl(I»J)))

79 CONTINUE

76 CONTINUE

75 TEM(K) = (AVEP(L)4.?.0*SUM)/TERMS

EMAXCL)=TEM(1)

THMA<CU=*TD*(TWnPl-THZRO<l))

DO 70 K=2»NZERGS

TF(EMAXCL)-IEM(K5)73»74»74

73 EMAX(D=TEM(K)

THMAX(L)=f'TD*(TWOPl-THZRO(K))

THMAX IS BEARING FROM MAGNITIC NORTH

7U CONTINUE

SUM=0.0

00 80 1=1*5

IFCGTÜDCD )80/80#SJ

81 IP=I+1

00 83 J=IP»6

IFCGOÜDCJ) )83>83#82

82 SUMsSUM+PCL/I»J)*PcL»I»J)*PCL»J»I)*P(L#J»I)

83 CONTINUE

B-8

80 CONTINUE

SS0»AVEPCL)*AVEPCL)*2.0*SUM

H(L)*SSQ-TERMS*EMAX(L)*EMAX(L)

*TILOA*AVEP(Ly/TWOPI

HTILOA«SSQ-ATILDA*ATlLDA*TERMS

Hl(L)=1.0-(h(L)/HTlL0A)

H2tL>»(EMAXCL)-ATlLDA}/(AVEP(L)-ATIL0A)

IFCSENSE SWITCH U99»70

99 PRINTS, TERMS»EMAX(1_J,ATIL0A^HTILUA

7 F0RMATMXfF5.1>lP3E11.4 )

70 CONTINUE

PRINT 3»ClDCI)»Ial»U) »CG00DU)»I = 1#6)»M

3 P0RMATC1HI,39MLEAST SÖUARE SINGLE WAVE TRAIN FIT OF ,llAb/

XlX#6F2f0»2X»2HM=I3//

X3X#9HFREQUENCY#5X*4HPERI0D»3X#11HWAVE LENGTH»IX»11HATTESTION»aX»

X5HAVE P,8X,tHA#lOX»5HBRNG ,%%,1HM»12X,2HH1.10X,2HH2 )

PRINT A#(F0(L)»PERlOOCL)»WAVLGH(L>»OEPATN(L)#AVEPCL)»EMAX(L)»

XTHMAX(L)»H(L)»H1CL5,H2CL)#L=2»MP)

4 FORMAT(1X#OPF11.7»3X»QPF9.5»IX»OPF11.4,1X>IPE11.4»

X lX»lPEli.4»lX»lPEtl.A»3X,0PF7.2*3X»lPE11.4#4X,0PF6.3»6X*0PF6.3 )

00 1tO L=2,MP

AVEPCL)aE2TEN*L0GFCA8SF(AvEP(L)))

EMAX<L)aE2TEN*L0GF(ABSFCEMAX£L>)>

11 0 WAVLr,H(L)=EMAX(L5-E2TEN*LOGFCABSF(OEPATN(L)))

B-9

AVEPCl)=AVEPm

EMAX(1)=EMAX(?)

WAVLGHCl)=WAVLGHf2)

CALL SYMB0LC2.0»Ot75»0.1»lMWAVE TRAIN. FN = ,0.0,lü)

CALL NUMBER(3.5»0.75»0.1»FQCMP),0.0,2)

CALL GPHPVW(MP,in,AVEP,E^AX,WAVLGH)

C WAVLGH is SURFACE WAVE STAFF SPECTRUM EST FROM BOTTOM PRE5SUKE.

C AVE=P EMAx=V WAvLGH=rf ON THE PLOTS.

ZM = M

NP=DELTAT*ZM*0.5+1.0

DELX=0.0

00 115 1=1,6

CALL NUMBER(OELX,0.1,0.1,GOOD(I),0.0,-1 )

115 0ELX=l)ELX + 0.1

CALL PFB3DCNP,ID«EMAX,FQ,THMAX)

GO TO 15

100 REWIND 3

CALL LXIT

PAUSE 70707

GO TO 15

C ENDCO,1,1,0,1 )

ENO

B-10

FORTRAN TO ALGOL TRANSLATOR PHASE 1 FORTRAN STATEMENTS

SUBROUTINE GPHPVWCL»ID,ZLQGP.ZLOGV.ZLOGW)

SPECIAL FORM OF GPHPVW FOR 13350-2 AUGUST 1967

DIMENSION ZL0GPm,ZL0GvC2),ZL0GWC2),ID(2)

CALL PLOTCO.O.O.0,3)

CALL PLOT (0.0»|1.S»2)

CALL PLOT (8.0.10,5,?)

CALL PLOT (8.0*0.0,2)

CALL PLOT (o.0,0.0,2)

CALL PLOT (2.0*1.5,-3)

CALL SYMBOL CO,O.'l,0,-.l, 10(1),0.0,66)

CALL AXIS (0.0,0.0,20HNO'RMALIZEO FREQUENC Y , -20,5 .0 , 0 . 0,0 .0,0 . 2 )

ZMAXsZLOGP(n

DO 10 I=2'L

COMPAH=ZLOGP(I)

10 ZMAX»MÄX1F(2MAX#C0MPARI)

MAX=7rtAX+3.0

BE=MAX-8

CALL AxlS(0.0»O.OM7HLOG POWER DENSI TY, 17» 8. 0,90, 0,BE, l. 0)

0X = 5,0/FL0ATF(L-1 )

Y=ZL0GP(1)-BE

x=o.o

CALL PL0T(X,Y,3)

B-ll

00 20 I=2»L

X*X+0X

Y=ZL0GP(I)-bE

20 CALL PLOT (XJ>Y,2)

CALL SYMB0L(X»Y#O.t#lHP»0.0»l)

X = 5.0

Y=ZLOGVCL)-BE

CALL SYMBOL(X»Y,0.tj.lHA,0.0»l)

CALL PL0TCX,Y.3)

M = L-t

DO 21 1=1»M

x=x-ox

II=L-I

Y=ZL0GVCII)-8E

21 CALL PL0TCX/Y.2)

WMAX=MAX

X = 0.0

Y=ZLOGW(l)-bE

CALL PL0TCX.Y.3)

00 22 I=2»L

x=x+ox

IFCZLUGWCI)-UMAX 524,23,23

23 Y=8.0

GO TO 22

B-12

24 Y=ZLOÜW(I)-BE

22 CALL PLOTCX/Y»?)

CALL SYMBOL(X»Y»0.l»lHS»0.0#l)

IF (SENSE LIGHT 1 >40>30

30 CALL PLOT (-2,0#9.0»-3)

SENSE LIGHT 1

GO TO 50

40 CALL PLOT C6.0>-t2.D»-3)

50 RETURN

END(0M>1'0,0)

END

B-13

FORTRAN TO ALGOL TRANSLATOR PHASE 1 FORTRAN STATEMENTS

SUBROUTINE PFB3DCN,IO,P,R,B)

C P=FUNCTI0N OF R a POWER DENSITY

C 8= FUNCTION OF R = COMPASS BEARING

C R= FREQUENCY IN HZ

DIMENSION IDC2)»P{?)»RC2)»BC2),CC360)»SC360)

T = i0.0

0=0.70710676

DTH=0.017453293

C OTH IS ONE DEGREEE IN RADIANS IE RADIANS PER 1 DEGREE

A = 0.0

DO 10 I=l»360

A=A+DTH

CCIJ=CGSF(A)

10 S(I)=SINFCA)

CALL PLOTCO.0.0,0.3)

CALL PLOTCO.0,10.5.?)

CALL PL0TC8.0.10.5,2)

CALL PL0T(6.0,0.0*2)

CALL PLOTCO.0.0.0»2)

CALL SYM80LCl.0»0.5,-0.1»IDC1)»0.0,66)

CALL PLOTCfl,0,3.5,-3)

CALL PL0TC3.0.0.0#2)

B-14

CALL SYMB0L(3.25»0.0»0.1MHN .0,0»i)

CALL AXIS(0.0»1.0»0H »0.5.0.90.0.-1.0.1.0)

CALL PL0TC0.0.1.0.3)

CALL PLOTCO.0.0.0.2)

CALL PL0T(-3.0»0.0>?)

CALL SYMB0L(-3.26.0.0»0.1.1HS .0.0.1)

X=-3.0*D

Y = X

CALL SYMBOLCX-0.1»Y-0.1.0.1.1HE .0.0.1)

CALL PL0TCX/Y.3)

CALL PLOTC-X.-Y.?)

CALL SYMB0L(O.l-V»O.l-Y»0.1»lHrt »0.0»D

CALL PL0T(0.0.0.0»-3)

DO 20 I=l»6

A = 0.5*FL0ATF CI )

CALL PLOTCA/0.0.3)

DO 30 J=l»360

Y=A*S(J)*D

X=A*C(J) + Y

30 CALL PL0TCX/Y.2)

20 CONTINUE

CALL PLOTCO.O.O.OJ-3)

0M0.Q*D

10*0 NEEDED TO SCALE 0,0"0.3 TO 0-3 INCHES ON PLOTS

B-15

00 40 I=2*N

RAD=0TH*B(I)

Y=R<I)*D*SINF(-RA0)

XsR(I)*T*C0Sf"(RAD) ♦ Y

IUDO

CALL PL0T(X,Y,IUD)

AY»P(I) + Y + 1.96

2 IS NOT ADDED SO TQP SYMBOL *ULL BE THE REFERENCE IN .06 SYMBOL

CALL PLOT(X,AY,?)

CALL SYMB0L(X,AY,0.08,2,0.0."2)

CALL PL0T(X,AY,3)

40 CALL PLOT(X,Y,?)

IFCSENSE LIGHT 1 )4t,«2

42 CALL PLOTC-4.0,7.0,-3)

SENSE LIGHT 1

GO TO 50

4t CALL PlOTU,0,-14.0,-3)

50 RETURN

END(0#1»1'0M)

END

B-16

APPENDIX C

A FORTRAN PROGRAM FOR ITERATIVE WAVE TRAIN ANALYSIS

The FORTRAN listing of a Burroughs B-5500 program for the itera- tive least square multiple wave train analysis of a spectral matrix is presented. The mathematics is an iterative utilization of the single wave train analysis described in the body of this report. Following the single wave train analysis of the measured spectral matrix the resulting values of single wave train power, A, and wave bearing, 6, are used to find the spectral matrix that would occur for such a wave. Details for this are given in Section 5 of this report. From the above, a residual measured spectral matrix is formed by sub- tracting a fractional portion of the single wave spectral matrix from the previously used spectral matrix. The residual spectral matrix is then single wave train analyzed. The above procedure is continued iteratively until a specified number of iterations have been com- pleted or the residual spectral matrix total power gets smaller than a specified value. Table Cl shows numerical results for several fre- quencies. Figure Cl is a plot of the bearing, 6, and the ratio of A to the total power available in the original measure spectral matrix for the frequency band 0.00833 to 0.24187 Hz. The iteration parameters were set for a maximum of five iterations, a residual power ratio of 0.1, and a fractional portion value of 0.1. Both results are for data collected for, task SW0C at Stage II between 1220 and 1249 hours on 4 June 1965. The wind speed was 12 knots from a bearing of 280 degrees. In Table Cl A. H, and bearing are as previously defined. AVE P is the average C±±(f) for the residual spectral matrix and P is the AVE P for the first iteration; i.e., the original measured average power. The HI and H2 values are measures of the isotropicity of the energy represented by the residual spectral matrix. HI compares the least square value of H with the value H' that would have been obtained if the wave energy were isotrbpic:

HI = 1 - (H/H') .

H2 compares the power A of a single wave fit to the total power, AVE P, and the value A' that would have been obtained for isotropic energy:

H2 = (A-A')/(AVE P-A').

Both HI and H2 are between 0 and 1, the lower limit is for the isotropic case and the upper for a plane wave from a single direction.

C-l

3

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UNCLASSIFIED Sfrunly CI«i«itir»Hon

DOCUMENT CONTROL DATA R&D * fSATuHly rlmttlllrmltan ol till», hotly at tbttimtt mnj induing mnnoittiinn mm* !■» 0niwitd wh9u "•• 0*01011 twpw it 11*001(100

I OHIO'NAIING »CIIVIH iCoipotml» tulhai)

Naval Coastal Systems Laboratory

Panama City, Florida 32401

I: ntrom T ucu^ir* ciiinricino««

UNCLASSIFIED /D. CMOU**

J HIPQ4 T TllLk

THE DIRECTIONAL ANALYSIS OF OCEAN WAVES: AN INTRODUCTORY DISCUSSION

«. OII£«I»IIVI NOT«» (Typt ol r«p<yf mnd In

Informal I. iUTHOKill (Flttl n*m: mlddtm Initial, tarnt nmm»)

Carl M. Bennett

• »»»or o« IE

December 1972 Ja. TOTAL NO- OF PACE»

83 '». MO. or >rri

12 •#. CONTRACT Ol 6NANT NO.

». »nojccT NO.

•*. omaiNAToM'S n%**onr MUMICRIH

NCSL 144-72

•• Task SWOC SR 004 03 01, Task 0582

d. ZR 000 01 01 (0401-40)

•e. oiHi« «EPOUT NOI»I (Any ottimt iwMn awi awr •• ••••« IM« nporl)

First Edition NSRDL/PC 3472 tO. nilTHIBuT'ON STATEMENT

Approved for Public Release; Distribution Unlimited,

JUP°l.tMHf»»lV NOTII

Second edition. This report was originally issued in September 1971 as NSRDL/PC Report 3472.

II. »PON SO MING MIllTAftV ACTIVITY

Commander Naval Ship Systems Command (OOVlK)

19. AaftTRAC T

An introductory discussion of the mathematics behind the directional analysis of ocean waves is presented. There is sufficient detail for a reader interested in applying the methods; further, the report can serve as an entry into the theory. The presentation is basically tutorial but does require a reasonably advanced mathematical background. Results of a program for the measurement of directional ocean wave bottom pressure spectra are included as an appendix. This second edition makes corrections to the first and adds some details of an iterative directional analysis method.

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KEY WORDS LINK A

ROLE

LINK B

ROLE

LINK C

ROLE WT

Ocean waves Spectrum analysis Fourier analysis Matrix methods Convariance Bibliographies

IDENTIFIERS

Cross spectral analysis Directional analysis SWOC (Shallow Water Oceanography) Ocean bottom pressure data

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Chief of Naval Material (MAT-03L4)

Commander, Naval Ship Systems Command (SHIPS OOVlK) (SHIPS 2052) (SHIPS 03542)

Commander, Naval Ship Engineering Center Assistant Secretary of the Navy Chief of Naval Research

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022600 Director, Naval Research Laboratory (ONR 481) (ONR 483) (ONR 102 OS) Ocean Sciences Division (NRL) Library (NRL)

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007900 Director of Defense, Research and Engineering (Ocean Control)

015400 Director, National Oceanographic Data Center 000600 Director, Advanced Research Projects Agency

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028600 Chief, Oceanographic Branch, CERC (Dr. D. Lee Harris) (Dr. Cyril J. Galvin, Jr.)

034700 Director, Woods Hole Oceanographic Inst. (Dr. John C. Beckerle) (Dr. N. N. Panicker)

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030700 Director, Scripps Institute of Oceanography, University of California

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008500 Department of Geotechnical Engineering, Cornell University

008700 Department of Oceanography, Florida Institute of Technology

008800 Department of Oceanography, Florida State University

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011800 Institute of Geophysics, University of Hawaii

009300 Division of Engineering and Applied Physics, Harvard University, Cambridge

004900 Director, Chesapeake Bay Institute, Johns Hopkins University

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010600 Great Lakes Research Division, University of Michigan

008600 Department of Meteorology and Oceanography, New York University

009500 Environmental Science Center, Nova University (Dr. W. S. Richardson)

008900 Head, Department of Oceanography, Oregon State University

029900 Pell Marine Science Library, University of Rhode Island

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009100 Department of Oceanography and Meteorology, Texas A&M University

002000 Applied Physics Laboratory, University of Washington

009000 Head, Department of Oceanography, University of Washington

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015200 National Institute of Oceanography, Wormley, Godalming, Surrey, England

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National Oceanic and Atmospheric Administra- tion, Boulder, Colorado 80302

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Flinders University of South Australia, Horace Lamb Center for Oceanographic Research, Bedford Park, Adeleide, South Australia 5042

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Chairman, Dept. of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada (Copy 195)

Director, Geophysics Laboratory, University of Toronto, Toronto, Canada (Copy 196)

Director, Institute of Earth and Planetary Physics, University of Alberta, Edmonton 7, Alberta, Canada (Copy 197)

Director, Institute of Oceanography, University of British Columbia, Vancouver 8, British Columbia, Canada (Copy 198)

Chairman, Department of Geophysics and Planetary Physics, The University, Newcastle Upon Tyne, NE 1 7 RU, England (Copy 199)

Cambridge University, Madlingley Rise, Madingley, Cambridge CB3 OEZ, England

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Chairman, Department of Geodesy and Geophysics, Cambridge University, Madlingley Rise, Madingley Road, Cambridge CB3 OEZ England (Copy 201)

Director, Institute for Coastal Oceanography and Tides, Birkenhead, Cheshire, England (Copy 202)

Director, Oceanographic Research Institute of the Defense Dept., Kiel, West Germany (Copy 203)

Abteilung fur Theoretische Geophysik, Universität Hamburg, Hamburg, West Germany (Copy 204)

Institute for Advanced Studies, 64 Merrion Square, Dublin, Ireland

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Director, NATO Saclant ASW Research Centre, La Spezia, Italy , (Copy 206)

Director, Geophysical Institute, University of Tokyo, Tokyo, Japan (Copy 207)

University of Auckland, Dept. of Physics, Auckland, Dept. of Physics, Auckland, New Zealand

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Chairman, Dept. of Marine Science, University of Puerto Rico, Mayaguez, Puerto Rico 00708 (Copy 209)

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