The directional swell spectrum off block island

ARCREF

111

cni*

\

\!-.0. I WO*

STEVENS INSTITUTE

OF TECHNOLOGY

CASTLE POINT STATION

HOBOKEN, NEW JERSEY

DAVIDSON

LABORATORY

Report 1155

THE DIRECTIONAL SWELL SPECTRUM OFF BLOCK ISLAND

by P.

H. Rank

C. E. Grosch

S. J. Lukasik G. W. Zepko June 1967

Lg. v.

Sciti.:cp-Tochnische 14

:(.1..tui

R-t155

Report 1155

June 1967

THE DIRECTIONAL SWELL SPECTRUM OFF BLOCK

ISLAND

by

1".! H. Rank

C. E. trosch

S. J. !Lukas 11(

GI. W. Zepko

Prepared for the Office of Naval Research

Department of the Navy Contract Nonr

263(36), NR 062-254

(DL Project

2638/052)

Distribution of this document. is unlimited. Application for copies

may be made to the Defense Documentation Center, Cameron Station,

5010 Duke Street, Alexandria, Virginia 22314. Reproduction of the

document In whole or in part Is permitted for any purpose of the

United States Government.

Approved,

". "-7" Daniel Savitsky, Manager

v 31 pages

Z tables, 27 figures Applied Mechanics Group,

R-1155

ABSTRACT

An array of four pressure gauges was used to measure the swell off Block Island, Rhode Island. The data were analyzed to obtain the direc-tional wave spectrum. Resulting directional information agreed with visual observations at the time of the experiment.

KEYWORDS Wave Spectrum Ocean Swell Directional Wave

TABLE OF CONTENTS

Abstract s

.

* 4 .

INTRODUCHON A

DISCUSSION AND DEVELOPMENT OF THE STANDARD THEORY FOR COMPUTING POWER-SPECTRUM ESTIMATES

The idea* Ca5e

The Practical Case . . 6 N ), V ke *

!,J

0 0 3

10 A 3

Finite, But Continuous Data . .

Lag and, Spectra* Windows , .! P 4

.

infinite But Discrete Data A a A 0 A A A?

p, A * \,,. , * * *,

.

., 4 A . ;A V '7 , R

Finite and Discrete Data ... ,

The Complete Spectrum, E(2,m,f) . . .

.

10 V

.

0.: , , , VO , 'U

APPLICATION Of THE THEORY IN COMPUTING THE.

DIRECTION OF WAVE PROPAGATION . ;.1 P 1.1 P." 116

Location and Dimensions of Array SpectralAnalysis Equations .

6.. * ri)

4 X. m 16,

RESULTS Of SPECTRAL ANALYSiS, .

Directional Calculations From Individual Pairs

.

,

.

24

A 25

CONCLUSION , AO it7 (P,

REFERENCES , , , , , AI A .

.

m 0 'V

Table 1 to 4 i.o *V ok o *i .al

.

*

t

Et ,uo ...-. .4 8

.

b ,

84

,,

.129

.30

Table 2 . . I, , 1 5 . . 16

... .

. . 28 31

R-1155

INTRODUCTION

This report summarizes part of a continuing program to study, both theoretically and by means of laboratory experiments, the process of energy dissipation in shallow water waves; and to relate this process to the en-ergy dissipation of ocean waves which occurs as a result of their inter-action with the bottom in nearshore regions.

The theoretical calculations and laboratory experiments have been summarized in a report by Lukasik and Grosch. A series of field measure-ments to link theory and laboratory experimeasure-ments with oceanographic observa-tions has also been reported by Lukasik and Grosch.2 On the basis of these field measurements, it was concluded that there existed a viscous boundary

layer on the ocean bottom which could be described qualitatively by a linearized laminar-flow solution of the Navier-Stokes equations; however, there appeared to be quantitative discrepancies between the theory and observations.

In view of this discrepancy and the strong probability that other mechanisms are acting to dissipate the energy of ocean swell, a direct

measurement of the gross energy-dissipation rate of ocean swell would appear to be desirable. A possible scheme for carrying out such a measurement

involves the measurement of the directional spectrum of the swell at one

point in the ocean, a wave-refraction calculation to trace the energy

transport (assuming no dissipation) from this first station to a second measuring station some miles off, and a measurement of the wave spectrum at this second station. The difference between the spectrum predicted

(from the directional spectrum measurement and refraction calculations) and the spectrum observed would be the total amount of energy dissipated.

Although simple in principle, this experiment involves a number of practical difficulties. The earlier work2 had shown that it was possible to make adequate measurements of wave spectra in shallow water by simply anchoring the ship, putting a pressure gauge on the bottom, and taking a

few hours of data. Also, it is known that a number of existing computer-oriented numerical techniques3'4 can be used to carry out the wave-refraction calculations with acceptable accuracy. A number of measurements of

directional spectra have been made.5,6 These have generally involved semi-permanent arrays whose size, directional properties, etc., could be tailored to the wave spectra. For various reasons, the details of which need not be explored here, it would be possible in the present instance to have only a temporary array of pressure gauges. This array would have to be laid from the ship in about one day. The array would not have a regular pattern, nor would the distances between gauges or the angular positions be known very

precisely. (As it turned out, there was probably a 20-percent error in the measured length of the legs of the array.) Because of these diffi-culties it was decided to attempt a field measurement which would involve setting up a temporary pressure-gauge array and the measurement of the directional wave spectrum to test the feasibility of the proposed energy-dissipation experiment.

This report summarizes the results obtained from the feasibility ex-periment. To ensure completeness and to expose the underlying assumptions, the first part of the report systematically develops the standard method for computing power-spectrum estimates from a finite number of discrete data points, starting with the idealized assumption that an infinite amount of continuous data is available. The second part of the report applies the

theory in computing the directional spectrum from data obtained with a

pressure-gauge array off Block Island.

The results of this feasibility experiment can be summarized by stating

that it is possible to obtain the directional spectrum with fair accuracy from a hastily laid and inadequately surveyed array. _{The effects of even} relatively large errors (k-f, 20 percent) in measurements of inter-gauge spacings can be eliminated, provided sufficient redundancy is built into

R-1155

DISCUSSION AND DEVELOPMENT OF THE STANDARD THEORY

FOR COMPUTING POWER-SPECTRUM ESTIMATES

THE IDEAL CASE

Consider a real wave train given by the equation

11(x,y,t) =

2:

a exp[i27(2nx + my + ft + an)]

n=1 n

Forming the conjugate products 11 (x,y,t) 11(x + X, y + Y, t + T), and aver-aging over values of x, y, and t, yields

CO

(x,y,t) 11(x + X, y + Y, t + T) =

an exp[i27(inX + mnY + fnT)]

1

(2)

Here we assume that the cross terms vanish; i.e. terms of the form

anak exp(inX + mnY + fnT)

fff

expri27(in-Ak) x + y + (fn-fk) t +

an-

ak] dx dy dt

are zero. This averaged product is called the correlogram or lag correlo-gram of the wave and will be denoted by p(X,Y,T) . Note that it depends

only upon the difference of the arguments of 11 and 11", The term

an

2

is related to the energy of a wave and is called the wave power.

Equation (2) can be written in a continuous form by introducing a

and

power density, E(i,m,f) , yielding

CO

p(X,Y,T) = _{E(i,m,f) exp(i27(iX + mY + fT)) di dm df.}

(3)

where E(i,m,f) _{is also known as the power spectrum.}

Equation (3) implies

that E(i,m,f) _{is the three-dimensional Fourier Transform of p(X,Y,T)} Rewriting Equation (3) in the form

p(X,Y,T) =

f

_{E(i,m,f) exp[i27(iX + Of)] di dm}

exp(i2ufT) df

(4)

we have, for fixed X and V and variable 1

ff

E(i,m,f) exprin(iX + mY)) di dm as the Fourier transform of p(X,Y,T) . That is,

CO

inP(X,Y,T) exp(-i27Tf) dl = _{E(i,m,f) exp[i27(2X + my)] di dm}

Defining the cross-spectral estimates C and Q as

co

C(X,Y,f) = j p(X,Y,T) cos 27Tf dl

Q(X,Y,f) = _{p(X,Y,T) sin 2111-f dT}

we have

CO

C(X,Y,f) - iQ(X,Y,f) =

55

E(i,m,f) exp[i27(iX + mY)] di dm

(5)

-R-1155

or

E(i,m,f) = fir [c(x,y,f) _ ,Q(x,y,f)] exp,_,27(,x mY)] dX dY (7)

But since E(/,m,f) is real, this reduces to

E(i,m,f)

fir

[C(X,Y,f)

cos

27(/X

mY) -

Q(X,Y,f) sin 27(iX mY)I dX dY

(8)

Hence if one knew p(X,Y,T) , one could then compute its Fourier transform

with respect to T , obtaining C(X,Y,f) - iQ(X,Y,f) . Equation (6) then

shows that C(0,0,f) is the total "power" at a frequency f . To

calcu-late the complete power spectrum E(L,m,f) , one would next compute the

two-dimensional Fourier space transform of C(X,Y,f) - iQ(X,Y,f) . As

Equation (8) shows, this reduces to a cosine transform of C(X,Y,f) and

a sine transform of Q(X,Y,f) . In practice one does not know p(X,Y,T)

for all values of X,Y, and T; and hence the preceding argument must be modified.

THE PRACTICAL CASE

There are these two major differences between the ideal case and the practical case: (1) the data is never infinite in extent; (2) the use of digital computers negates the use of continuous data even if it exists.

The problem is now clear. In what manner shall the available data be treated in order to obtain an approximate power spectrum, and in what degree does this approximation represent the true power spectrum? To answer these questions, the use of convolution is indispensable. It should be recalled that the convolution of the functions g(x) and f(x) is defined as

1:09(x-Of(Od

=

f:g()f(x-)d

and denoted by g(x) * f(x) Furthermore, if G(s) and F(s) are the +

the Fourier transform of g(x) f(x) .

Assuming that the waves have an ergodic property, the correlogram can be obtained by summing over time alone. Hence

*,

(x,y,t) Nx+X, y+Y, t+T) =

Tr(xo' yo' 11(xo+X, yo+Y, t+T)

where the first average is to be taken over x, y, and t, and the second is to be taken over t . Therefore we can express p(X,Y,T) in the

fol-lowing form:

P(X,Y,T) = lim

1

1t/2 T1(

x, y

T)

7(xo+X, yo+Y, T+T)

dT

-t/2 0 o

Here 11 has replaced , since the waves are real. For the present we

are interested only in the time dependence, and therefore rewrite Equation

(10) as

0(T)

=

1

5t/2 1.1 t/2 (T) xY

(T+T) dT

oo

-where the dependence of p on X and V is understood.

FINITE BUT CONTINUOUS DATA

Assume that we know 11 continuously for a time interval

tn

. By

an appropriate shift in time, we can best approximate (11) by

(t - T

1)/2

I;(T) - (T - II.

(T +

dT (12) tn- I T I f(tn- 1)12 00 2 XY for

IT!

< tn n

(9)

R-1155

The term t3(T) is called the apparent correlogram, for the ergodic property makes its ensemble average equal to p(T) .

Since 13(T) is the best possible estimate for p(T) , it would appear

that if one took the Fourier transform of p(T)

po(T) = "f3(T) T _{< tn}

=0

IT I>

tn

one would get the best estimate for C(X,Y,f) and Q(X,Y,f) . The fact

that this is not necessarily the case is discussed in the following section.

LAG AND SPECTRAL WINDOWS

A lag window, D(T) , is defined as an even function such that

D(0) = 1

D(T) = 0 for 1 T > _{Tm ,} with Tm <

tn

Its Fourier transform, B(f) , is called the spectral window.

Consider the function pi(T) defined as

pi(T) = Di(T)P.(T) T

Tm

=0

T > 7m

where the index I is used to denote different lag windows. The expression

p(T)

, defined at the end of the last section, has, for example, a lag

window D0' and

D(T)

= 1

IT I

T

0 rT > Tm

Since

av

43(T)]

= P(T)

we have

av [pi(T)1 = Di(T) p(T) (13)

where "ay" is the ensemble average. Hence

av [Ci(f) -;Qi(f)] = Bi(f)* [C(f) - iQ(f)] (110

where C.(f) - iQ.(f) is the Fourier transform of p (T) , and B.(f) is

1

1

the Fourier transform of

D(T)

, i.e. the spectral window.

Therefore (by neglecting statistical fluctuations), we find that C.(f) - ;(1.(f) is the true spectrum smeared out by the spectral window

B.(f) , or

Ci(f) - iQi(f) =

f

Bi(f-f') [C(fl) - iQ(fi)] 15)

From Equation (15) we see that the more closely B.(f) represents a delta function, the more closely C.(f) - iQ.(f) represents the true spectrum. But since D.(T) is non-zero for only a finite time, the narrower B.

1

becomes the greater its side lobes. We therefore must compromise. Table 1

(p. 30) gives five different lag windows and their corresponding spectral windows. Figures 1 through 5 are the graphs of these window pairs.

Al-though Bo is the narrowest, its first side lobe is 20 percent of its peak in magnitude. All the other B's are wider, but the side lobes are

appre-ciably smaller.

INFINITE BUT DISCRETE DATA

It is now assumed that the time series 71 is infinite in extent but known only at Intervals of At . Since we now have discrete data, a new

situation occurs; for if we sample a pure wave having known frequency f dft

R-1155

at equal intervals At , it fs not clear from

this

data whether the wave

has a, frequency f' or a frequency

f r

'

r = 1, t 2, . . . This

At

phenomenon is cal fed, aliasing and

is

illustrated in Figure 6_ Because of al lasing we cannot expect to resolve, frequencies Offering by mutt iptes of

-'

Hence the best for which one can hope is an aliased spectrum with At

this frequency amb i gu it y.

The correlogramr can only be calculated for multiples of At , and is

given by

im

2n+1 1oo(Mt) IIX'/ f(k+r)

k=-n

r 0, ±

We now define an infinite Dirac comb, Vtlr;At) , as

CO

'9(T t) A t E8 (111 - rA t

r=-0P

fts Fourier transform, A(f

' - j-)'_At it given by CO f

2]

otf r=-co Also, we define, pp(T) as p (rA ) = _p(r6t)

=0

T = rAt T rAt (i9)' (16) (18) = 6t) = 1, ± 2,... = - (17) = r = 0, ± ± 2,...

'CO

If we now take the Fourier transform of 7(T;At) pD(T) , we have

'7(T ;At) p0(T) exp(-127fT) dT = V(T;At) p (T) exp(-i2TrfT) dT

= A(f; * [C(f) iQ(f)] CO =

1]

C(f + yt-) - iQ(f + r=-0. = Ca(f) - iQa(f) (20) where

Ca -

iQa(0

is called the aliased spectrum.

The principal part

1

of the spectrum is defined as

Ca - iQa(f) for I

f

I < 2At

As predicted, this result does not represent the true spectrum. If,

however, it is known that no energy occurs for frequencies greater in

magnitude than

---'

then the true spectrum is represented by the

princi-2At

pal part; that is, we have

C(f) - iQ(f) = C(f) - iQ(f) for 1

f

I < 1

2At

1

The frequency is called the folding or Nyquist frequency. Figure 7

26t

illustrates the case when energy occurs above the Nyquist frequency and the case when there is no energy above this frequency.

The only new difficulty arising from sampling at intervals is

alias-ing. It is to be noted that, once the data are taken, no numerical technique can minimize this problem. Hence the time to face aliasing is in the plan-ning stage.

FINITE AND DISCRETE DATA

The usual situation, as explained earlier, involves data which are finite in duration and taken at discrete times. Using the results of the

previous sections for guidance, we now define the apparent discrete correl-ogram as R-1155 n-T/At 1 t -T

4"4oo(tk)xy(tk+T)

n k=1 av[PD(T)] q(T)

D. I

=

Di(T)

IT

1 Tm ;p T = rAt r = 0, 1,...,m 1

r

t +T T 1100(tk) 11xy(tk+1) T = - rAt n k=i -ET

and the finite aliased spectrum as

Ta(f) - iTia(f) = D.(T)V(T;At) PD(T) exP(-i2rfT) dT

_.

=

2]

Di(rAt) f5D(rAt) exp(-i27frAt) r=-m

The identities below follow from the ergodic property ("av" is an ensemble average).

T = rAt (23)

avrTa(f) + = Bi(f) * [ca(f) - iQa(f)] (24)

Therefore Equation (22) yields (if we neglect statistical fluctuations) the allased spectrum broadened by the spectral window Bi(f)

Let

i'&.a(f)]

(22)

5D(T)

Vm(T;Lt)

=

v(T;60

ITI

Tm

= 0

IT !

>

In (26)

Noting that D. (T) vm(T;60 = D.(T) V(T;At) , we have from Equation (22)

tp

Za(f) - f;%(f) =

jr:

Dip(T) Vm(T;At)po(T) exp(-i27fT) dT

= _BA

(f)r=-m

*2]D(r6t)

exp(-i27frAt)

ip

where B.

(0

is the Fourier transform of D. (T) .

p ip

Hence Equations (22) and (27) give two ways of evaluating

Ca(f)

-a

. Using Equation (22), we

passD

through the lag window

and then take the finite Fourier transform. In Equation (27) the transform is taken first and then convolved with the spectral window. Equation (22) should be used if it appears that the convolution would be tedious; such a case occurs with the window Di . If, on the other hand, the lag window

consists of a sum of cosines, convolution is then trivial, for the trans-form of a cosine is a delta function.

THE COMPLETE SPECTRUM, E(I,m,f)

As shown in the first section (p. 2), E(i,m,f) is obtained by comput-ing the Fourier space transform of C - IQ. . Again problems arise due to

the fact that data are available at a finite number of discrete points. We first note that if C - iQ is known at (X,Y) it is also known at

(-x, -Y). Hence we will assume that we know C - iQ at 2m 1 values of

(X,Y) (the additional 1 comes from X = Y = 0). Guided by the previous formulation, we define the following functions:

(1)

A two-dimensional lag window, D(1(X,Y) . This is an even function

of both variables and has the properties

(27)

r=-m

- +

R-1155

1:1 2)(0,0) =

D?)(X,Y) = 0 for (X,Y) R

1

where R is a simply connected region containing the origin.

(2) V(3)(X,Y) = 6(X,Y) + 6(x4.,y-y.) + o(X+X.,Y+Y.)

J J J J

j=1

We then define E.(i,m,f) as

CO

E.(i,m,f) = D!2)(X,Y)

v°°(X,Y) [C(X,Y,f)

-

iQ(X,Y,f)] exp[-i27(2X + mY)] dX dY

1 1

CO

= B.(2)(1,m) E(1,m, f)

(28)

where 0)(i,m) is the Fourier transform of

_D?)(X'

Y)

V (X"

Y). and we find

n

that E.(2,m,f) is the true spectrum distorted by the two-dimensional

spectral window

O.

In calculating E.(L,m,f) we are interested in the direction of propagation of the waves. If we assume that the waves are coming from a few well-defined directions, a narrow spectral window should give the best results. Therefore the use of a rectangular lag window is indicated.

Defining D(2)(X,Y) = / X,Y e Ro

ID(a)(X'o Y) = 0 X,Y Ro

where

Ro is simply connected and contains all

2n + 1 values of

X,Y for which C - iQ is known, the spectral window becomes

B(2)(1,,m) = V(2)(X,Y) D(2)(X,Y) exp[-127(2X + mY)] dX dY

.ff40

vn (X,Y)

expr-i27(LX + mY)1 dX dY

, *

(a)

(11b)

The spectral window may be thought of in another way. If all the energy at a given frequency fo were generated by a wave with distinct

wave numberso and

mo , E(2,m,fo) would be a weighted delta function

at ,m . Assuming K to be the weight of the delta function, we have

o o

Ei(2,m,f0) = fil2)(2,,m)*E(A,m,f0)

= K

f

B.(2)(tA

6(i-A -T1)d§d11

K

B2)(2_,

m-m)

o o

Hence B.(2)(2-2 ,m-m ) represents the distortion introduced by the discrete

o o

data and shows how the true spectrum is spread out. For this reason

E3(/,m) is also called the spread pattern.

As seen in Equations (28) and (29), aliasing in the wave-number plane will also be present; hence care in the choice of

(Xi'Y) should be taken.

To ensure that al/ important wave numbers will be unambiguously resolved, the minimum distance between two detectors should be no greater than half the wavelength of the shortest expected wave of any consequence.

In the beginning of this section we assumed C - iQ was known exactly at a finite number of points, but, as has been shown, this is not the case. Therefore we must take into account the statistical uncertainty and the

finite discrete nature of the data, in obtaining C - iQ . Using the same

techniques as before, we find the results are immediate. They will merely be outlined here.

(1) We define a three-dimensional lag window, D.!3),(X,Y,T), as j

Di3)(X,Y,T) = D?)(X,Y)(2)(X,Y) D.(T) Vm(T)

(Note that the Dirac combs have been incorporated with the lag window.)

R-1155

We then form the appropriate Fourier transform

E..(i,m,f) =

iff

D.."(X,Y,T) f7(X,Y,T) exp [-i2u(/X + mY + fT)]dX dY dT

j j

= B.!3)(2,m,f) * T(i,m,f)

j

where B.!a)s the Fourier transform of D..(W and the ensemble

averages of T, and E equal p and E .

Noting that D.."(X,Y,T) is a product of a function of X and j

Y and a function of T , we finally have

E..(L,m,f) = [(B.(2)(i,m) B.(f) * T(i,m,f)] j

Therefore(by neglecting statistical fluctuations), we find that

Eij(L,m,f) represents the true spectrum distorted by the spectral window B?)(fl,m)B.(f) .

In practice we would tailor

0

B. to fit our need, but in general

j

B(2) (see Barber )and either 132 or B3 (see Blackman and Tukey7) are

appropriate.

(2)

APPLICATION OF THE THEORY IN COMPUTING THE

DIRECTION OF WAVE PROPAGATION

LOCATION AND DIMENSIONS OF ARRAY

The array was located about 1/4-mile south of Block Island. The water depth at each gauge varied from a maximum of 41 ft for gauge 2 to a mini-mum of 35.5 ft for gauge 1. The array (Figure 8)was composed of three outer gauges (I, 2, and 3) and a fourth gauge (4) in the middle. Distance between the pairs of gauges and the angle made with magnetic north is given in Table 2 (p. 31). Due to the difficulties in determining the locations of the gauges from a boat, it is possible that these measurements may be as much as 30 percent in error.

SPECTRAL-ANALYSIS EQUATIONS

The data consisted of a 43-minute simultaneous record of bottom pres-sure from the four gauges. An on-line analogue-to-digital converter was

used to digitize the data on paper tape at time intervals of 2 seconds. Equation (21) was used to compute the correlogram, with the number of

lags equal to fifty.

=

for r = - 50,...,-1,0,1,...,50

and where

tk=

k6t n = number of data points

n-r 1 1 tn-rnt I] 11oo(tk)X.Y.(tk4-1-Lt) k=1 r 0

E

(t ) Th (t +r-At) r < 0 tn+rLt k=1-1. 00 k XiY k (31) (30(X1,Yrr,6t)

-For

r < 0

we may also write

(30 as

n+r-r-)

(X ,Y.,rAt)

-t

+11,

E

T1,o(tk-rAt)71

(t )

rn o g k=1

Substituting

for

r

we finally have

,r

XY.-rAt)

1 1155

n-r

(t. +rAt)II

(t. )

,r

0

t- rAt

oo k X.Y. k k=1 (33)'

Equations (30 and (33) have the immediate consequence

that

J;(0,0,-rAt)

=,

5'0(0,0,tat)

far all i.

Substituting in Equation

22) and noting that

D(rAt)

is an even(

function, we have

50

(X.,

.,f) .E

D(rAt)FON

.,rAt) exlx-i27frAt)

a A a , r

r=-50

50 =

D (0)p0(Xi,Y1,

2E D(rAt)

r=1

(34)

where

k.,rAt) -

0 I

-1 {IP (X

2

.,rAt)

-0

(X.

Y... ,-rA,t) o

XSirn

27frAt}

r <

(32)

cos 27frAt

k -r - > G(Xi,Yi,f,r) = +

-and

An equivalent representation for

Ta

-a

is obtained from Equation

(27). 50 (X. Y.

0 -

(15. (X. Y.,f) = B (f) * (X Y 0) + 2

2]

a a 'o i=1 I (35)

In our analysis we use the lag and spectral windows D2(T) and B2(f) a

Hence for each f the convolution reduces to a weighted average of the bracketed term. The weights are given by

W2(f) 0.5

Wo2(f± th)

= 0.25

W2(f+

)

= 0 I k I 2 (36)

Let if = and denote the bracketed term in (35) as _Uk

-1Vk for

mAt

f = kM . Using (36), we have from (35)

-6a(Xi,Y1,kAf)

-=[(Uk_i + 2Uk + Uk+1) - i

(v

2Vk + Vk4.1)ki

(37)

Due to aliasing, no new information is obtained by taking frequencies

above = 0.25 = 50 if Aliasing implies that

2At Uk = Uk-100 Vk = Vk-100 a + ,

R-1155

Since U is even and V is odd, we have

Uk = U-k Vk = -V-k and uk-100 = u100-k vk-100 = -v100-k We therefore have

=0

0 V, = -V_, = U = U51 51 V49 = -V5] V50 = -V50 = 0 Hence - = (U0 + U1)/2 (38) Za(Xi,Yi,5106f) - = (U49 + U50)/2 (39) V 50 a ,Y. ,50f)

To summarize, we have the following formulas for computing the estimates of C

and Q:

Ca(Xii''0) = +

(u< 2Uk + Uk+1) / 4 k = ,

C (X 50A f ) - + U5),/ 2-a $ Y 0 ' a i' = Tia(Xi,Yi,5cAf) ti (X .,Y.,kAf) 2Vk + Vk+i) k = a where

o(XY,rAt)

i,rAt) o(X.,Yi' rAt)

, rAt)] sin 2urfAt 2

E

(

t)

(t +rAt) t -rAt oo k ,X.Y. k n k=1 ' _I 111

- r

tn-rAt

E

x Y at) 11,00(tk+rAt).11x.y.(tk) k--=1 - E

TO be consistent with other authors we weight the Tast term In

each of the preceding sums by 0.5 .., For lag :windows Which are discontinuous cos 2TrrfAt (40)

(40

Ta(XI,Yt,k6f) (142) Uk Vk =

5 (x

i

5° 2

E

r= 1L Y.,0) + 5c)(Xi

/

2 = + = 0 = / 4 50 "150(Xi,Yi,rt) + -o ₂ r=1 - -= n-r n-r -

-1,2,...,49

at

IT 1

Tm , this improves the spectrum; for lag window 3, for example. But in our case the lag window is continuous, and hence equal to zero at

Tm (i.e., r = 50) Therefore by considering Equation (34) we see that

this added weight is of no consequence. If there were a discontinuity at

IT 1

=

Tm , one would in effect be taking the average value of the

right-and left-hright-and limits for the value of the lag window at 1 T 1 =

Tm .

With four gauges there are six pairs, and hence

Ca-a

is known

at thirteen values of

Let

(f) - =

T (x.,Y.,f)

-

((I (x.,Y.,f)

(44)

Cjk Qjk a a where X. = x -k x. R-1155 and and

(xk'

yk) are the co-ordinates of the kth gauge.

We note that

C.

-iQ.

= C + iQ .

jk jk kj kj

The coherence, Rjk2 , between gauges

j

and k is defined as

2 2

C. + Q

R.2

(f)

R.o

2 t Jk jk

' ' C.. C

kk

The phase angle,

ejk , is given by

e.( f)

= ejk(f) = arg(C + (46) (1+5) . j j

and i and 1 and k are related asvih (44)..

Using Equation (29),, we have for the...estimated Spectrum,

6

f) =

E tc(x

Using (45), we find that this reduces to

6

c(o,m) 4

2

2: C(X.,Y.,

cos 271(Lx. mY.)

s i=1 f) = where - 4.Q(Xi,Yi,f)] exp4-i27(iXi + m

sin, 27.(iX,1 + mY.) 4

6

E

tc(xi,y1,02 +

1=1

.cos

fe.

2r(1x . ra.)] ci

If we further assume that C(0,0.,f) = C..

=1

we have

JJ.

6

1 +2

li:

R. (f) cos [0,. + 2Tr(tX + mY.)1

i=t 1 1 t 6

E

Ili(f) cos ,-. [0. 4 2rkd cos (a i=1 t 1

a

= k sim a di cos Iv. 4 d sin I). 22 (47) (18) i=-6 = - Q(Xi,Yi,f) C(0,0,f) + 2 Q(X.,Y.,f)212 + + , m C(0,0,f = k cos = = = 1 + 2

-R-1155

Assuming a dispersion relationship, we have for each frequency fo a wave

number, ko , associated with it. Hence,

T(1,m,f0) En(a,f0) - _c(0,0,f0) 6 . _ 1 + 2

2:

R.(f ) cos

N

+ 27k d. cos (c,' -1

YI

. o 1 o 1 1=1 (49)

RESULTS OF SPECTRAL ANALYSIS

The frequency spectrum was calculated from the data at each gauge. A typical spectrum is shown in Figure 9. The spectrum is given without units., since there is some ambiguity in the gauge calibrations, due to an apparent' change in the amplifier sensitivity between the time when the gauges were calibrated and the time the data were taken. Calibration signals prior and subsequent to the data-taking indicate that the gauge sensitivity did not change during the period in which the data were taken. Because these gauges

had linear response, the calibrated estimates would have had exactly the same shape, but would have been of different magnitude. The frequency-spectrum estimates show that most of the energy was in a frequency range of 0.08 to 0.13 cps.

The cross-spectral estimates, C and Q , were calculated for each

pair of gauges. The estimates for pairs (1,2) and (1,3) are shown in Figures 10 and 11. From these estimates the coherence, R2 , and phase, e

were calculated and are illustrated in Figures 12 and 13, for all possible pairs of gauges. From this information the normalized directional spectrum

nOM

was computed for various frequencies. Figures 14 through 17 illus-strate four such directional spectra.

In interpreting the directional spectra, we assume that the wave train is coming from one and only one direction. Under this assumption we can see from Figures 14 through 17 that the direction is 190° (from magnetic north

in a counterclockwise direction), for each graph has the common character-istic of a peak at 190°. Assuming this direction, it is now possible to explain the other peaks. In the section dealing with the complete spectrum

(p.12), the theoretical response for a uni-directional wave was given. Modifying this result to take imperfect coherence into account, we have,

for the theoretical normalized spectrum,

R-1155

ET(a'fo) =

1 + 2

E

R. cos {2u((/-L0)X1

(m-mo)YA

i=1

= 1 + 2

/2

R. cos

.{27diko[cos(a -0i)-cos(a0-yl

i=1 (50) ko cos a _to =

ko cosg

o ko sin a ko = ko sin ao d. cos 8. d. sin O. where m = = X. = Y. = Plotted with En on Figures 14 through 17 is ET for ao = 190°.

The good agreement between

ET and Tn gives further evidence that ao is approximately 190°.

Having established the direction of propagation it is interesting to note from Figures 12 and 13 that the coherence between the pairs of gauges which are nearly parallel to the direction of propagation is much higher than the coherence from the pairs of gauges which are nearly perpendicular.

Also

the maximum coherence occurs at lower frequencies if the two gauges are nearly perpendicular to the direction of propagation. Short crestedness of the higher-frequency waves would cause such phenomena.

DIRECTIONAL CALCULATIONS FROM INDIVIDUAL PAIRS

For a given pair of gauges and a given wave, let

ei(f) = phase angle

a. = difference in the angle determined by

the pair of gauges and the direction from which the waves are coming

d. distance between the gauges

+

Then (see Figure 18),

cos a.

J

7 il(f)/ 2TT'kdli

Hence the possible solutions for

a

are given by

± arccos

(f) / 2 TTEkd

(52)

Since

O.

can only be determined to Within a multiple,

of

2TT , we haVe

i

arccos

{[2rj -.ei (f)]/2ukd

(53)

. where

j

must be limited to assure that

I27j

e. (f)

< 27kd.

(54)

?I

Figures

19. through 23 are plots of all the possible

al's

for all

pairs, with the exception of pair (1,4).

Figure 24 is a plot of the most

likely propagation angle if all five, pairs are considered

simultaneously.

That is, all pairs considered have a possible propagation angle at approxi-'

mately 1900.

This further substantiates the angle determined earlier,

Figure 25 is a plot in polar co-ordinates of the range, of the most

likley

angle for each pair of gauges.

The fact that these ranges do not completely

overlap is an indication that the dimensions of the array are not accurate.

This

is further illustrated in the case of the pair (1,4).

Figure 26

is a plot of the possible angles for this pair.

Although the result appears

to be completely inconsistent with the previous calculations, further

anal-ysis indicates that this discrepancy is most likely caused by an inaccuracy

in measuring the length of

d114.

For if we assume the length to be 2G-per-1

cent longer, which is within the accuracy of the measurement of the array,

we obtain, Figure 27 for the possible angles

computed from (1,4).

Hence

(51)

=

=

-R-1155

modifying the length of d14 gives a result consistent with the previous

calculations. This modification will also have an effect on the other dimensions of the array. If the change in the length of d14 is made by moving gauge 4 along the line determined by (1,4), the resulting array is as indicated in Figure 25. The changes in the most likely angles of (2,4) and (3,4) are also indicated, and are in better agreement with the other

CONCLUSION

The results obtained indicate that the directional characteristics of swell can be adequately determined by _{a hastily laid array of pressure} gauges.

The array used in the investigation was excessively large for the wavelengths encountered _{hence the multipeaks due to aliasing observed}

in the directional spectrum. While these extraneous peaks are undesirable for rapid determination of the direction of propagation, the fact that they are predictable gives credence to the mathematical model employed.

R-1155

REFERENCES

LUKASIIK, S. J. and GROSCH, C. E., "Laminar Damping of Oscillatory Waves."' J. Hydraulics Div., Proc. Am. Soc. Civil Engr, 231, January 1963.,

LUKASIK, S. J. and GROSCH, C. E., "Pressure-Velocity Correlations in

Ocean Swell." 68, 5689, 1963.

GRISWOLD, G. Mi., "Numerical Cakulation of Wave Refraction." J. Geophys.

Res 1715, 1963.

INER, EL M. and PUNTON1, V. W., "A Computer Program for Plotting Wave_ fronts and Rays From a Point Source In Dispersive Mediums." J. Geophys.

Res 68, 3473, 1963.

5,

MUNK, V. H., MILLER, G. R., SNODGRASS, F. E. and BARBER, N. F., "Direc=. tional Recording of Swell From Distant Storms." Trans. Royal Society, London, A 255, P 505, 1963,

6.. BARBER, NI.. F. "The Directional Resolving Power of an Array of Wave Detectors," Proc. Conference on Ocean Wave Spectra, May 1-4, 1961, Prentice-Hall, Inc 1963.

7 BLACKMAN', R. B. and TUKEY, J. V., The Measurement of Power Specteak.

Dover Publications, inc., New York, City, New York, 1959,

1

J. Geophys. Res.,

68, 4

Do(T) =

1 1 2 0

D1(T) =

1

-IT

Tm r D2(T) = + cos T

)

2 Tm

=0

D3(T) = 0.54 + 0.46 cos = .04 rrT D4(T) = 0.42 + 0.50 cos Tm 2rT + 0.08 ccs Tm rT Tm

THTm

= Tm T > Tm rn B1(f) Tm (s in rfT rfT m T < T T I < Tm T Tm

THTm

T = Tm T I < Tm

=0

IT

Tm

TABLE 1

Bo(f) =

sin

2rrfTm rf SPECTRAL WINDOW 2 + 0.04 [Bo(f + 1 1 1 B2(f) = Bo(f) + 17. [130(f +

7)+Bo(f

-2T B3(f) = 0.54B0(f)+0.23 [Bo(f + 1'+B (f

2Tmo'

B4(f) = 0.42B0(0+0.25 [Bo(f+ 1 (f 2T o m 1 _7)+130(f

-=0

T Tm

=0

T > Tm

-R-1155

TABLE 2

Pair No. Gauges Degrees With Respect To North Distance Between Gauges 1 1-2 _1790 372 2 1-3

-105°

391 3 2-3

-

54° 461 4 1-4 -1400 240 5

2-4

-

38° 240 6

3-4

lio°

240

0 0.5 I.0 1.0 0.8 Q6

2 o.4

m 02

-0.2 -0.4 0.5 1.0 1.5 2.0 2.5

fTrn

FIGURE 1. LAG AND SPECTRAL WINDOWS FOR i

= 0

o

₀₅

Tm

0.5 1.0 0 8

\\

0.6 0.06 2 cu ca Q4 0.04 R-1155 0.2 0.02 0.5 1.0 1.5 2.0 2.5 f.Tm. 0 0.5 T/T rn. 1.0 1.0

LO

N

0

05

0.5

him

FIGURE 3. LAG AND SPECTRAL WINDOWS FOR i

= 2

1.0

0

0.5 R-1155

T/Trn

0.5 1.0 .0.5 0.02 I 0 2.5

1.0 i-v). 0.6 co 0.4 0.2 0.5 1.0 0.010 0.008 0.006 0.004 0.002 0.002 f.Tm 1.5

FIGURE 5. LAG AND SPECTRAL WINDOWS FOR i = 1

2.5 0.5 1.0

T/Trn

0.8 0

-2.0

i 1 2.5At At L 5 2.5at FIGURE 6.

ALIASING FIVE EQUALLY SPACED DATA POINTS ARE FITTED BY TWO SINE WAVES DIFFERING BY

A. TRUE SPECTRUM PRINCIPAL PART . ALIASED SPECTRUM At 2 1128 PR IN CIP AL PART A ALIASED SPECTRUM At = 11/ 2 A FREQUENCY

FIGURE 7,

THE EFFECT OF THE SAMPLING INTERVAL ON THE

ALIASED SPECTRUM

FREQUENCY

N

0° R- 1 155

FIGURE 9.

FREQUENCY SPECTRUM GAUGE NUMBER 2

0.05

OJO

0.15

0.20

0 0.05 0.10 0.15

020

025

FREQUENCY (CYCLES/SEC0NDS)1

FIGURE la

CROSS-SPECTRAL ESTIMATE

c-IQ

PAIR (ill ,2

0105,

0./00/15

F1REOtJEiNCY t CYCL ES / S ECON D

FIGURE UI.

CROSS - SP ECTR AL ESTIMATE

0 I-4 4 iiJ 0.20 0.25

R-1155 GAUGES 2'AND GAUGES .3 AND 1 :360° ch moo o GAUGES 4 AND If 360° 0 0.05 0.10 0.115 2 1.0 R 0.5 1.0 0.5 1.0 R10.5 0 0 0.10 I

0180°

I 180°

1.0 R2 0.5 1.0 R2 0.5 1.0 RI0.5 360° 4.) 180 GAUGES 4 AND 2 360° ck 180° GAUGES 4 AND 3 0° 0 0.05 0.10 0.15 0 0.05 0.10 0.15

FIGURE 13. COHERENCY R2 AND PHASE cp 360°

f0 =0.06 K =0.00175 0

FIGURE 14.

DIRECTIONAL SPECTRUM VS

-FIGURE 15. DIRECTIONAL SPECTRUM VS Of fo . 0.08 K0 =0.0024

0

10

f

.0

1 0 :0.00311 FIGURE 16. DIRECTIONAL SPECTRUM VS a

fo =0.12 Ko =0.00389 Em

0

ET

a0:190°

FIGURE 17. DIRECTIONAL SPECTRUM VS a

0

0

0

10 5

1

Given a pair of gauges A & B and a pure wave P , the coordinates

are orientated( such that the wave is coming from a direction determined by - 7/2 i.e., the equation for the wave at a fixed time is given

by p, = C cos (27y/A. y).

The phase shift 0 between the wave at A and the wave at B is given by

27a2

_-

27a2

but a2 di cos 10,

The difference c between that angle determined by A and B and the

_

direction, from which the wave is coming is given by cp,i- 11/2

(7 =T) - 7/2

R-1155

=

Ia -

TT

61, 6 = 27d/X cos (a 7) = - 2ndk cos (a)

THE RELATIONSHIP BETWEEN THE PHASE ANGLE AND

,

=

= 7/2 - = -

-240 120 60

-

-

-

-

-0.09 0.10 f (CYL../ SEC I 0.11 0.12 FIGURE 19.

ALL POSSIBLE DIRECTIONS OF PROPAGATION COMPUTED USING

1 - 2 0.13 360 300 0.05 1

FIGURE 20.

ALL PUSS ISLE DIRECTIONS OF PROPAGATION FOR CHANNELS

l,

360 -

300 240 180 120

FIGURE 21.

ALL POSSIBLE DIRECTIONS OF PROPAGATION FOR CHANNELS 2

- 3 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 f (CYL /SEC ) )

240 180 120 0.06 0.07 0.08 0.09 0.10 0.11 0.12 f (CY L / SEC) FIGURE 22.

ALL POSSIBLE DIRECTIONS OF PROPAGATION FOR CHANNELS 2 - 4

0.13

0.14

360 300

FIGURE 23.

ALL POSSIBLE DIRECTIONS OF PROPAGATION

FOR CHANNELS 3 - 4 CD Z 360 300 240 180 120 60

0

0 0.06 0.07 0.08 0.09 0.10 011 0.12 0.13 0.14 f (CYL / SEC)

360 270 180 90 0 360 90 0 CHANNELS 1 AND 2 360 270 180 90 0.02 0.04 0.06 0.08 0.10 f (CYL/ SEC) FIGURE 24.

ANGLE OF PROPAGATION COMPUTED FROM VARIOUS PAIRS OF DETECTORS

0.12 0.14 0 0.02 0.04 0.06 0.08 0.10 f (CYL / SEC) 0.12 0.14 CHANNELS 1 AND 3

^

2 AND 4 360 270 CHANNELS Lii 180 Lii 90 0 360 CHANNELS 2 AND 3 CHANNELS 3 AND 4 270 Lii 180 Lii <C 90 1 80

360 120 60 0 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 f (CY L / SEC) FIGURE 26. ALL POSSIBLE

DIRECT IONS OF PROPAGATION

COMPUTED USING 1 4, WITH D14 =

240'

300 240 180

360 300 240 120 60

- --0.06 0.07 0.08 0.09 0.10 0.11 0.12

f(cvL/ SEC)

FIGURE 27.

ALL POSSIBLE DIRECTIONS OF PROPAGATION COMPUTED USING

I

- 4, WITH MODIFIED

DISTANCE, D14 = 288'

0.13

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UNCLASSIFIED

DOCUMENT CONTROL DATA- R 8, D

!,,orr)t r11,,i fir ntton of WI, both or ohcfrtyr I And irldrvirtil ormotolion no/, he oothred10,0 the over.ift ri,rpire Is rfl,,,111,1,

/ 0 HGINA TING ACTIVITY (Corm/role °who')

Davidson Laboratory

Stevens Institute of Technology

D. RE,OR 1' SECURITY CLASSIFICATION

Unclassified 2b. GROUP

I REPORT TITLE

THE DIRECTIONAL SWELL SPECTRUM OFF BLOCK ISLAND

4 DESCRIPTIVE NOTES (Type of report and(nclusive dares) Final

5 AUTO-10MS/ (First name, middle initial, that nome)

, Rank, Paul, H.

Grosch, Chester, E.

Lukasik, Stephen, J. and Zepko, George, W. 5 REPORT DATE

June 1967

Ta. TOTAL NO DEFACES

31

7b NO OP REPS

7

Ro CONTRACT OR GRANT NO.

Nonr

_263(36),

NR 062-254

b. PROJECT NO.

d.

98. ORIGINATOR'S REPORT NUMBER(S)

R-1155

9b. OTHER REPORT NO(S) (Any other numbers (het rrthy be essigrred

this report)

10 DISTRIBUTION STATEMENT

Distribution of this document is unlimited.

I SUPPLEMENTARY NOTES 12 SPONSORING MILITARY

Office of Department Washington, ACTIVITY Naval Research of the Navy

D. C.

20360

the swell off

obtain the directional with visual

obser-13 ABSTRACT

An array of four pressure gauges was used to measure Block island, Rhode Island. The data were analyzed to wave spectrum. Resulting directional information agreed vations at the time of the experiment.

Ocean Swell

Directional! Wave Spectrum

D D

1473 ( s Aci<

UNCLASSIFIED!