Nceskh I¥4¢-72.. “T } 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 IN ir i QO 0301 NII TABLE OF CONTENTS ENTRODUGITON Sy. to) ey WAVED MODEMS ey fuyrciiet esis) A DIRECTIONAL WAVE SPECTRUM . . CROSS SPECTRAL MATRIX OF AN ARRAY... SPECIAL CROSS SPECTRAL MATRICES .... A MEASURE OF ARRAY DIRECTIONAL RESOLVING POWER. DIRECTIONAL ANALYSIS FROM THE CROSS SPECTRAL MATRIX . SWINE 56 5 6 6 0 oo 4 REFERENCES. . ... - BIBLIOGRAPHY. . . ... APPENDIX A - A COLLECTION OF DIRECTIONAL OCEAN WAVE BOTTOM PRESSURE POWER SPECTRA. ... . APPENDIX B - A FORTRAN II PROGRAM FOR SINGLE-WAVE ABYNION ZANVPNIN GSES) 616 6 6 6 00 6 0) 0 Oo APPENDIX C - A FORTRAN PROGRAM FOR J TERATIVE WAVE TRAIN ANALYSIS. 6.618 Page No. Figure No. il! 2 LIST OF ILLUSTRATIONS Simple Ocean Wave Real Wave in Wave Number Space Directional Wave Spectrum at a Fixed Frequency, £ fe) Wave Direction Analysis Direction Analysis from a Pair of Array Elements Directional Estimates for a Pair of Array Elements Least Square Single Wave Fit Directional Spectra A, (£,8,) Detector Geometry Li Page No. 33 34 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 obtainable from 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. 2. 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 l. Spatial Frequency N(usv, ty) Spatial Frequency Along Along v axis is zero. y axis is m = K sin 8 Direction of @ Wave Travel in Time t u Spatial Frequency along u axis K = 1/r Spatial Frequency along x axis is £=K cos @ FIGURE 1. SIMPLE OCEAN WAVE Assume that the wave is frozen in time over the surface (the Xyy spacial plane). The coordinates (u,v) are a 9 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(2nKu + 279) (2a) where K = 1/\ is the wave number of spacial frequency in cycles per unit length along the u axis, and 27d 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 27) is a fixed time phase shift) n(u,v,t) = cos(2nKu + 276 + 2n ft + 2my). If we combine the effect of the ¢ and wt phase shifts as a = $ + yy, we get n(u,v,t) = cos(2n(Ku + ft + a)) (252) 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 ul =x cos 8) y san) ¢ V = —-x sin 0 + y ‘cos 0: Using the above relations, and letting 2 = 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(2n(2£x + my + ft + a)) @23)) as a model for a wave of height 2A moving from a direction 6 = arctan (m/2) with a phase shift of 2ma. 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 gx + my + ft = a constant = (n - a) where n = QO, -1, tl, -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 V, = -f/2 and Vy = -f/m. This follows from the differential expressions D,@)= D, [aes _ mo -it] -t (2.4) x M-% = LH t DweD[ Ms te ft J.-£ es 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 2n(2£x + my + ft + a) we can write n(x,y,t) = 1/2A Exp(i2n(2x + my + ft + a) + 1/2A Exp (i2n(-2£x - my - ft - a)) @z6) 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 (i27(2x + 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 (-i2n(2£x + my + ft + a)) =a Exp (i2n[(-2)x + (-m)y + (-f)t + (-a)]). (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 (Xo,yo,t,.) and (xo+X, yotY, t +I), where X, Y, and T are constants. The phases at the two points are given by (x5 »¥52t,) = LX, ate Wb ate ft, + a, (2.9) (x, + X, Vg + Y, ty + T) = (x, + X) + m(y, + Y) + F(t) +T) +a (2.10) This gives a phase difference of Ad = (2X + mY + fT). @ way) 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 1 (x.9.t) =>, Exp (i 20 (InX+ nd +f, ¢ +05), (2ea2) NB | 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 n(x,y,t) — S a, Exp (i an (I, K+ mY +f,t + %y)) : (DNs) ne \ 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 27 radians and continuous frequency in the interval (-f,, + f,), 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 a standing on the points (2, mp), f,) of a three-dimensional frequency space. The continuous case produces a power spectrum, S,(2,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 a, Exp(i2m(2px + my + f,t + On)). (Qo) 5 The energy or mean square in this element is a2, Assume the ele- ment is a part of a egntinuum of elements for -~ < f < +~ and 0 < 6 < 27m. In this case a, must be an infinitesimal energy associated with the frequency differential, df, and space frequency differentials, dt, and dm, which are related to the direction 6 of the wave element as before. Let the power spectrum, S(2,m,f), be defined with units of amplitude squared and divided by unit spacial frequency, 2%, unit spacial frequency, m, and unit time frequency, f. The power spectrum is then a spectral density yeas at (2,m,f). In this case we must have the infinitesimal energy, a, defined by ay Ss S (i,m, 4) ad adm AS (2.15) The real valued, nonnegative function S(%,m,f) is a power (energy density) spectrum of the standard type in three-dimensional frequency space (2,m,f). Intuitively, we can write an infinitesimal wave ele- ment as [ext (é am (4x + mY vit t «)) VS(4,. 3.) AN dm 44 x (2516) where the positive square root is assumed. To arrive at a noe 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 N(x.y.t) = Jf (lr Gen army oe earl) S (Rm) dd dn af (69 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(2,m,f) is fixed. In fact there is a wide range of possible shapes of n(x,y,t) for a given S(2,m,f). The above development is more intuitive than mathematically rigorous. ey has been shown by Pierson (1955 - pp. 126- 129) that if 27a(2,m,f) is a random function such that for fixed (2,m,f) phase values of the form 21a MOD 27 between O and 21, 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(2,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(2,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'(L5,M5,£,)5 of a single real wave where 0 < f, < ~. For such a real wave element of length i,, from a direction 9), ORO <2 the value 8" Qeotlontiolo S Si@veamapee nas S(-25,-m,»-fo)> where £, = K, cos 8; Mm, = K -gind,, and Kg = 1/A,. If the above real wave came from a direction (27-6), the power density would be Si Cessip te) = S(£,,-m,»£,) + S(=253Mp»-fo) - Figure 2 illustrates a real wave of length A>, from a direction 6, in two-dimensional spacial frequency (wave number) space. Wave Direction FIGURE 2. REAL WAVE IN WAVE NUMBER SPACE If the wave number relation Ka 27 ¢* /yTanh (at Kh) (Saal) 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/\, being the wave length; then a relationship between f and (%,m) is implied that requires a wave frequency fy to have a unique wave num- ber k,. From this we have the general one-sided spectral form for waves where f = f, of 2 2 i] ta N S' (2,m,f£,) = (zero where g2 + m2 # Ky a power density > 0 for g2 + M on S'(2,m,f,) thus defines power density at f = f, for wave energy over 0 <6 < 2n. Figure 3 illustrates this case in wave number space. S(L, mM; 385) Sis (lapap may see) FIGURE 3. DIRECTIONAL WAVE SPECTRUM AT A FIXED FREQUENCY, fy We want to estimate the shape of S'(2,m,f,) above the circle 22 + m2 = K_ “ in a directional wave train analysis. Remember, the S'(2,m,£.) above is restricted to fo > 0 and is, in fact, equal to t S(Xm4) + S (-R-m, -4.)]) S (2h, m4.) = §(- A,-m,-4,) where (3.2) if n(x,y,t) is to be real. Let us see how S' (2,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 (x9, yo, t=) ota _the process indices, n(x9. Yo, to) is random variable with a Gaussian distribution; i.e., Ne -p\* Preb (n (%0. 4, te) < n)= Sas gt (F*) dy | where yp is the arithmetic mean of n and o@ is the variance. Intui- tively n is as likely to be positive as negative, so let us assume that Prob (n(X9, Yoo to) < 0) = 1/2. Since n is Gaussian distributed, and is thus symmetric about its mean, we have Prob(n(x9, yo, to) < u) = 1/2 or that » = 0. For o%, we have (using expected value notation) gta El(r-a)] = El (4.2) where we are thinking of n as a random variable. A Gaussian process is completely defined statistically if we know the form of the mean E(n(x,y,t)) and the covariances - [x (Kat) - EO (x4, €))] q(x +X, +, t+T)- E (» (xeR yeY,t +T))]} : 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)) = uy = 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 RYT) SE Gst) Qkex yey, teT)] OD 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 uy and o“ 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(2,m,f). There is a simple and unique relationship between R(x,y,t) and S(2,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(i2n7(2x+my+ft+a) ) +Exp(-ian(Lasmy +ft + a))]Ve my AT dm df. Form the covariance function R(XY T) = E}(¥.¥.4) n(s+X.4+Y, t+T)| = Elexp (cen (9 (2x+X)+m(2u+Y)+4(et+T) +20) + Exp (2 2a (M(axeX)am (24+Y) +f @t+T)+20) + Exp(iat ({X+ mY +fT)) + Exp (ian (AX+ mY¥+ #T))] S(t.mf) dt don Af where E is the expected value over the ensemble for any fixed x,y,t. Note that \ S (im. f) dhdm df is a constant with respect to the expected value. Consider the following problem. Let u be a random variable with uniform probability density function Fe K OS US 2 radians fe) else where Define a random variable Z = ell, as The expected value of Z is defined (em f(uydu ott K(edus 6 Ete) Considering the random variable nature of the phase 21a 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) = [ exp (izw(lXe+emy + *T)) + exp (-i2n(I1X+ mY +£T))]S(hmf)ad am de. 10 For a real wave element we have, where S(2,m,f) = S(-%,-m,-f), see Equation (3.2), R(X,Y,T) “Exp (é 2n (IK mY+ £T)S (Son, f) dl dm df + Exb (-izw(QX+ mY + fT )) § (-4-m,- f) ALaAmdf 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 R(XYT\= f(( (cep( 20 (IK + mY + {T)S(dm.f)dtdads C4 eo -63 ~ GD 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(2,m,f). We cannot hope to be able to estimate R(X,Y,T) for continuous values of X, Y, and T. However, there is a way around this problen, we can write the above as R(XN.T) = 2 60 fexp (iawfT) (exp (iate (2X mY) S(im.f)Mddeld$ 5.5) i-@® = 60 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 [ Ie = ( (xysr) exp ten fr) a7 (3.6) In general, assuming that the term in [ ]'s is complex, we can write [C(KY,F) -2 Q (XN F)J = J R(xyr) Exp(-a2ufT)AT | (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 cc(ays)-ioayf)l = [ ].= (Sit.m#) Exp(i2t (AX+mY)) dam f (3.8) This is in the form of a Fourier transform; thus, we can write from transform pairs Stns) = ff[COKY,4)-0 0 (KY F) exp (Lem (1K mY) Axa (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(%,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 (X9 Yo) and (x ,, y,). This would give two stationary Gaussian pro- cesses indexed on time alone because (x), yg) and (xj, yj ) are fixed. Thus, we may write Holt) = 4 (Xo,40,¢) and 4a (t)= 4 (x,y,6) If X = (x, - x9) and Y = (yj - yo) Then we can say, since n(x,y,t) is assumed weakly stationary (see Equation (3.3)), that R(XY,T) = Elv(t)-: % (t+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(XX,t) = (7eft)s 4 (8) = N (€) for —ce f£,, 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) r,(t) and r;(t) measured over time at only two points; e.g., (x9, yo) and (x1, y1) where (x] - Xo) = X and (yj - yo) = Y. The relationship between (x, (t) r, (t)) and R(X,Y,T) -~ < T < ~ is ti RIX.Y. T= Limit & | wlt)en (t+T) dt () ty (4.8) where ro(t) is a realization measured over time at (x , y_) and r4(t) is measured at (x1, y,). We can simplify the notation by an expression for a time average given by ROG) = RG CL) = PACE) GEED) i It follows from Equation (3.7) that C(X,Y,f) = Co,(f) = po | Ro1 (T) cos(2nfT) dt, -co 15 and ios) Q(X,Y,f) = Qo1(£) = il Ro j(T) Sin(2n£T) dT. (4.9) -& We also have (see Equations (4.2) and (4.7)) Co (£) = Cy,(f) = Po (f) = Paice) Qo0 (£) P§, (f) S Coy (£) -i Qo1¢£)- (4.10) Qin Ce) 40); The phase of Po (£) is given by SOLES (4.11) Co Cf) Qq3 (£) = Arctan This is the expected phase lead of the signal at (x sienal at (Gol ya) for £ where) —)o<) f o< 27. °°? We) over the For an array of N depectore located at (xj, yj), Cx y2); (x3, Y3 ) » (xy Yu) we can find N eee te ] 202, N,, die N. This nee a Srone spectrum Pai (Pdi S oo and since Pij = 3, (see Equation (4.6) and 4.7)) we have N@=) (4.12) 2 unique cross spectra or a total of [N(N-1) + 2]/2 unique spectra. Thus, for real n(x,y,t) we have Pij (£) = P¥,(£) and Py (£) = P# (-£) allowing us to define the information” about Pry, ¥0538)) obtainable alln an array by a cross spectral matrix mm) Gr) eo = = Corl) QU) P.(4) oe Can (Sf) wnere 0¢ § < oo Q..(4) eae Pua) (4.13) 16 This information can be used together with Equation (3.9) to obtain an approximation to S(2,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 PT yf) fer Cates be begn where X.. Ge x) at j i (y. - y.) ij j aL We also have from Equation (4.7) that VP Cig tod) = POs ip Yi @a8) « (4.14) From Equation (3.9) we get S(tim.f) = ieee Y, ) cos(2m(&X+my) ~iPTX.Y, 4) sin [211 (K+ mY] PAX AY or, since S(£,m,f) is real S(t.m.¢) = (( te (xy £) cos[e2T(Ax+mY)] -ao —~oO -Q(x Y.)sin[en(1X+mY)]} dxdy (4.15) Let us consider treating the points of (X,Y) where we know C(X,Y,f) and Q(X,Y,£) as weighted Dirac delta functions; e.g., at (Ky o> Y42) we get C (X,Y, 4) = be KarXasf) F (X- Kn) df (Y- Yel - IL7/ Reverting to the Ci, Qi5 notation of Equation (4.13), we have, where Xi5 = -Xj; and Yj; = -Yij, C54 = Cis and Qj4 = -Qij- The numerical form of Equation (4.15) then becomes -| S(£,m,f£) = by, Cyy(£) + a= S bs} Gy cos [en (tx, + mY.) =! J=itl Choice of b;, values is arbitrary. A reasonable choice is b.. = [N(N-1) + 1)71 for all (i,j) (refer to following section). We now have 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 6, where 2, = K,Cos6, and m. = K,Sin 85, Ky being the wave number from Equation (3.1). We can write the wave as n(2%,4,t) = Acos(2w(Lx+mey +f.t +a)) (5.1) Since the root-mean-square (rms) value of a cosine wave is A/v2, we have for the two-sided (-»~ < f < ~) directional power spectrum of the wave in Equation (5.1) S(Xm,f) = A [S(1-2) Sn-m.) SEA) + + §(%40.) §(m+m.) S(f al (5.2) or in polar form where K = /y2 emer Gi= arctan (7) S(k.e.4)= A [S(e-0.) S(#-£.) + + §(0-(e.-m))d (+f) ] §(K-K.) 18 From Equations (3.8) and (5.2) we have for a single wave of frequency f, > 0 from a direction 6, that the two-sided Ret) = A [exe (ien(t 0X, m, Y, NS (F=f) + + exe(iat(-d Kom ¥;))4 (4 + f.) | or wHere ne) = C4) — £Q, (8) that C(t) = Cil-f,) = ne C= C6) = B cos (2m(t, X,0 m.¥,)), Q(t) = - Gy (-4) = — 4 swn(en v( Xm ¥) 6.9 describe the elements for the spectral matrix of a single wave. Recall that, in general, Ci; (£) = C54 (£) and ay (Ge) SO) Ge Substituting into Equation (4.16), we get where bij [N (N- DET 1/M (5.4) AX and f, > 0 is assumed S ) m, £ At A : U mf) = Kl A 2 cos (20 (1. X,+ m%;)eos (21 (LX,+m ¥, )) + +sin(2m(1.X,+ me¥;)) SIN (271( 2X; mY,))] oR S(bmf.)= Mh 02d, 5 cos (2n[ X(t ~{.)+¥, 3 (m -m)] (5.5) seiel N(N 1) 2 >’ w=! a The choice of bij =ely/ Mi [N(N-1)+1]72 and observing mick & 4 Seo Jsley 19 gives the convenient result (Note: in place of (25M) ) S) (L., ma, f ) = S(- JN. ,-me ,— fe) =A{i- Sen = 4 FORps toms 0 we would use (ages) 4m which agrees with the values of Equation (5.2) at the points (%,,m ,f)) and (Boosey Note that there is not general agreement elsewhere. In fact, Equation (4.16) may (and does) give negative values for the approximation of S(2,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, f, > 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(t,m,f)= [A §({0'+ m=) - (97 + m2)) ][s(¢-t.) + 4 (+4)] Savonene Kas Q"+ m?, K=+\k* , ano 6 = aRcran ("%) S(K,0,f)= AS (KEK?) [i (F-f.) +5 (F442) J (5.6) as the two-sided directional power spectrum for such an isotropic wave. Since % = Kcos® and m = KsinO, Equation (3.8) can be expressed in polar coordinate form as WV oo * Bids | IS(K.6,4) exe (i 2m K (X,cos 8 +Y,5IN 8) KAKA® 5. 7) “nu Dace 2 Letting Djj= (Xi + XG 20 and g.. = ARCTAN (| 4j Xi we can write aye W co BUF) = § [S(Kye,#) exe (Lem kd,jcos@-Q))Kdkda 5 -—W Oo Using Equation (5.6) for S(K,9,f) we get Bld =A K [exe (Lam K,D,cos (6 - o,.)) de Lat tf = (ih K. [cos (at k,D,jcos (8 —$,\\ de + =“ +iA K. [sin (2% K, D,,cos (@ -4,))d¢ -1 (So) Now, departing from the above development, consider the following integral where z= ew Kk Di, = 9; AT (cos (n9) Cos (z cos(e-w))de -w Let > = 0 — py and we get (cos ing +nw) cos(z cos) do Zi Expanding cos(né + ni) we get Wy cos nW cos np cos (zc0s ) do -W- = — SIN mip {sured cos(zcosh)dd . n= We have (rt - wv) - (-n -¥) = 27 so that the integrands are over 27 allowing us to write the above as mig cos nw cos ng cos(zcos 6) do a aT —sinnp) sin ng cos(zcos >) do . = From Ryzhik and Gradshteyn (1965 - page 402) we have (since sine is odd and cosine is even) the result (cos (ne) Cos(z cos (e- wp))d ") = cosnp|an cos (22\ Jz] , (5.10) In a similar way we get T {cos (n6) SIN(2 Cos (O- p)de <1 = cos np[2t Sin es) Je] : (5.11) 22 where J, (4) is a Bessel function of the first kind of integer order n. Returning to Equation (5.9), the above results give, for n = 0 where ee 0, the result R (tt) = Aen, Je (ank.D,) - Since the above is real C,(#$) = 2mkA Cet) = awk AJ,(em kD.) Q,(¢ 4) == 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 (tm. f\=((P'(x,y, eexe(—Lem(Ux+mY)\dxdy «1 In practice we use a probe array with elements at (xj, y;)..-, (X> yx) to obtain P*(X,Y,f) at the separation points (0.0), (Xo, Yy9)> (-X19 »-Y49) 2.900 (X-1,k> Yea ke)? (-Xx-1,k> Ge aL ie) We do not then know P*(X,Y,f) but the product PEC) CGN) (Gm) 23 where g(X,Y) is a set of Dirac delta functions standing on the separa- tion points of the probe array, and zero elsewhere. Thus, we have the estimate S (wi, f= (Pr, Y,4\ at Y)Jexr(- Lan(ix+ my))dudy G (t,m)=f (gx) exe (-ien(4X+mY)) dx dy -@ = (6.3) Using this and Equation (3.8) we have for a given (%,,m),f,) that GO, 62g, & 8 (Lm, f) = f \ (Sm) ene (iem(Ik+mY)) dtd] (ay) exe (-i2m(tX+m¥))dXd = {f(s S(L,m,f.)9(X,¥) exe(-aetr(x (1. ~\)+¥ (mo- -»\)) xd} =60 -@ ii ={(S(0. m. offal 9(X,YexP(-ien(x(2.- -1)+ (m,-m))) axa yfadn = [(S(tm.4)6( Vale m,-m)dddm Brees (6.4) As expected, $(2,m,£) is a two-dimensional convolution of the true directional spectrum S(2,m,f) with G(2,m) the Fourier transform of g(X,Y). We see then that S(2,m,f) is a weighted average of S(,m,f) 24 and that G(%’,m) 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 (im) = 1428, d cos(2n/ (Lx;+ mY,;)) (6.5) ! If we assume that the wave eqyatiog (Equation (3.1)) holds, we find that S(2,m,f,) is zero when 4” + m* # ae the wave number for f, From this we have for a given (29M) that the directional eRilelae power, DRP, is DRP (A, mf.) = (a(n mm) — Ks) G(I.-0,m.-m) Addm -a@ —@ 2 > K rer h=k,.cos@ » M= Ke SIN O, WHICH IMPLIES Vem =K, and we get, for energy coming from a direction 6, at frequency f, as a function of 0 < 6 < 2n, that DRP(@ | @.,£.) = G(k.(cos @,—cos@), K.(sin@,—SIN Q)). From Equation (6.5) we get DRP (e|¢.,f\= = 1425 deos[en( X,jKe (cos@,—cos €)+Y, |Ke d=) grat! (SiN 0, — Sit é))| = 1425 Scosfen (Ch -k,.cos 6) X, + (n,— Ke Sin 6) Y, “I 4=i JzAti Nore THAT DRP(0,| 0, £) = N(N-1) +1 Compare Equation (6.6) and (5.5). Except for the amplitude tern, A2/4M, in Equation (5.5) the equations give identical results. The choice of = [N(N-1) + 1]7~- = 1/M is again found convenient. (6.6) 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 (x;,y;) and (x,,y,) respectively, the cross spectral matrix, P .(f) = Cij (£) + iQ;.(£) and more impor- tantly ¢14j(f£) the phase thad @ie I ae J etal by, O(f) = ARCTAN foun C;; (€) ‘ 1) The actual phase lead may differ from this value since the true phase lead 68 is some one of the values Q(t) +hat where Ri=0,£1,42, Consider a single wave of frequency f, with corresponding wave length \, and wave number K, = 1/\,, and find the direction the wave must travel; ioGog alia 2 etaele wave to the spectral matrix results for the detectors I and J. Let D, ij be the distance between I and J. The distance between I and J in wave lengths is K oDij- In radians this is 2nK.Dij;. From this relation we get —2Tf K. Dy BS d< eT K. Di or that only values of h such that —2uk.D (A +hew S ett K, Dij G2) give physically acceptable candidates for the value of >. If Dij < A/2 only one h value is valid. If Nall 2 < Dij < ro at most two values of h are valid, etc. The problem is how to find the true direction, 9,5, of a single wave given the above possible value(s) of ¢. Consider a given value of » in terms of wave length units and we obtain 26 ink ne “ork, * (7.3) Figure 4 illustrates a case for > 0 (and thus Lg > 0). From the figure we have, where the true direction is 84 and true phase is 9, that 6, = Thank ats +H nol = (7.4) WHERE S\ = ae KD i oR “a= arcein|—_O _ = arcsin| D | ay K, Dij kj FIGURE 4. WAVE DIRECTION ANALYSIS 27 Recall that sin(a) = sin(+7-a). Thus for a given > > 0 we get, since a must be obtained from an arcsin relationship, two possible values of wave direction, the true value 6, and its image es ee at bad = yj- tt a= OG) == Wii- Loe a | (aD) This is illustrated in Figure 5. In actual practice we do not know the true direction, thus a given value of > > 0 gives the direction as a= Wi* ie ec] where 0 < a < 1/2 is obtained from the principle value of the arcsin. If ¢ < 0, then I actually lags J by | ¢ | > 0 and Lg = Lo/Dij 0 so that for a given value of 9» < 0 we get directions (a < 0) or 0,= at = + Ow at a as before. Thus, where the principle value of arcsin is assumed, we get a set of possible directions 8,, = Wi elon + 94] h, = ARCSIN Gil)+ her aw K, Daj (7.6) where h being constrained by Qii( ‘eben an An example of this type analysis, for an array pair, is illustrated in Figure 6. Thus several estimates of 6) are available (at least two). The estimates of true direction, 6,, often vary from one array element pair to another, making the selection of a true 6, value diffi- cult. The selection is also hindered because half of the estimates of 84 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 29 360 270 180 Direction Estimates (Degrees) 90 0.0 0.2 0.4 0.6 0.8 1.0 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 frequency is given below. It is based on Equation (4.16) in the form S(tmf) = £ (CoC) + ao y L[ealtdeosten (Lx4j+ mY) (a7) cae +m Y,, mI where M= [N(N-1) +1], C(f)= = 2 C..(f) =| aA and N = number of array elements. 30 Recall that for a single real wave of frequency £, > 0 and known direction 6, Cy 4(f ), Q;.(£,) are known (see Equation’. (5.3)). These values give j a S{k,.6,.4) = S(k,,f, -W-aiy=2 When a single, well-directed swell is expected, it is reasonable to assume a single wave for a given frequency exists, and to select 6, and Ao = A2/4 such that the least square error between the Haeeraieleaill, 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, f,, we can form the squared error = (C, -'A,)? i 22 [6;- —A,cos(at(h. X;,+ mN;, MN +203 (0, a Assin (an hXpce mY a ist jet isl it (7.8) Expanding and Matte terms we get H= Sone (Cis +Qi3) + [NIN=1) +1] AS 431 Joie ~ ads eye Cis cos(am(te X,,+ m.¥,,)} is year nO jo!N (am ( (4.%; yt Me x) (7.9) AZl Jeaay or using results in Equation (7.7) Syil H= tea Dl Ci +Qi5) +[n( n-i)ei](R-2A.Sllomt] Azi j= Lt To tind A, that minimizes H, consider = = 2(n(n-i\+ iI[A- = SID), ‘| =0 (7.11) ° This requires that A, be of the form A, = S(2,m,£,) and a resulting value of H of the form =) oD) (C +95) ) = [n(n -1) +i)” is) jties (iat) Since Co. Cra and aan are all nonnegative, a minimum H results when Si@vemts ES) is a maximum. A choice of 2, and m, that maximizes S(2,m,f£,) mreiecleens a 0, = arctan my [Xo which is onieinemn. Remember that we are assuming Ko = y2 + Me holds. along with the wave equation. The results then for aeen fo is a two-sided energy spectrum estimate A (£28 =) Appendix B ORIEHINS a listing of a FORTRAN II program for finding ACES so) > 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'(2,m,f). Con- sider then a third method. From Equation (3.8) we have for a pair of detectors, as illustrated in Figure 8, that (Yt) = [(s (L.m.f) exe [ian ({k+mY)] didmn -2 —@ 32 DATE: 9 SEPT. 1966 DATE: 4 JUNE 1965 TIME: 1320-1262 TIME: (300-1330 © STAGE! Z_ STAGE tl <=] WIND SPEED: 37 KNOTS Ed WIND SPEED: (4 KNOTS WIND DIRECTION: (34 DEGREES WIND DIRECTION: 280 DEGREES M ant? : J . i Zi FIGURE 7% DIRECTIONAL SPECTRUM A,(f,0) FIGURE 4% DIRECTIONAL SPECTRUM Ad(f, ce) DATE: 9 SEPT. (965 DATE: 4 JUNE 1965 TIME: 1320-1353 TIME: 1418-1449 STAGE I! g STAGE | WIND SPEED: 32 KNOTS a) WIND SPEED: 14 KNOTS WIND DIRECTION: 282 DEGREES WIND DIRECTION: 125 DEGREES ii H a Swe oe Ll —— Wai FIGURE % DIRECTIONAL SPECTRUM A,(f, 0) FIGURE % DIRECTIONAL SPECTRUM A,(f,0) FIGURE 7. LEAST SQUARE SINGLE WAVE FIT DIRECTIONAL SPECTRA Aj e))) 33 (Xp ’ Yo) Detector 2 ro (X15 Yr) x Detector 1 FIGURE 8. DETECTOR GEOMETRY Let 2 = K cos@ and m = K siné D= ix? at: ye wv = Arctan : and or X = D cosy and Y = D sin bp. Using the above changes of variable, we get M co P"™(x,y,¢) = | [stk af) EXPlien (K D cos 8605 H+ KOSINOSIN 9) -% “Oo Kdkde (7.13) Prixw.s\=| (3 (K0,f) Ex pliant KD cos (4- y)kaK da 34 We have from Equation (3.2). that S(K,e,4) = S(K, 6-1, -f) If we think in terms of fp > 0 a Ss (K,6,,) =e S (k,8,£) a) 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 $'(k,e,f) = 2 (8,8) S (k-K) ich ( K= kK, wHerE (K-K,) hes Baie. This allows us to write, mao<¢ F< +eo , Rye te = { {a (6, f.)d(K-K,)exPliewk Dcos(o- WKdkde - O a PUXN fe (a (o,f) expiant K,D cos(o- yk, de -T (7a) We have reduced the preblem to finding [a(0,f, - K,]. From Equation (7.15) we see that P*/o,0,f)= P(f)= falet)Kde sd where P(f,.) is the power spectral density of frequency £,- For better comparison of cases where P(f;) = P(£,), [£,| # Te, | it is convenient to express a(@,£,) in a normalized form A(6,f,) = a(8,f,)K, 35 where we get a P(£) = JAle,f)de (7.16) -T Thus if the energy distribution as a function of direction is the same for P(f1) = P(f,) then we also get A(@,£1) = A(6,f,)- Consider now (assuming A(6,f) can be so expressed) a Fourier series expansion of A(6,f) for fixed f. Clearly it is periodic in 6 with period 27. Thus for any given f = f, we can write A(6,f) as A(9) = = +2[0,,cos n@ +b sin n6] (7.17) Substituting this expansion into Equation (7.15) we get S Vv * : ie (x,Y,*) = $ {xe Lav KD cos(6-y) de + > [a, feos ne 24 —-% w cs =F EXP LAN KD cos (6-y) de +b [50 ne EXP Lem KD cos(0-¥} 4] on P™ (X,Y, #) =f [ss 2m KD cos(e-) t)) de DAC Jasne cos(2tekD “Tt cos (e- “W)de +b of ne cos(2t KD cos(e-y)}e + ile (os (at kD cos (6- y}de + oy (2, (cosne sin (21 KD cos(e-¥)de > N=! -v an ba [sm ne SIN (24 KD cos(e-¥) 0) (7. 18) 36 Thus we can express P*(X,Y,f) as complex infinite series with unknown CoecEflctents ag), zis) aie and) Dai by, -.- and constants defined by the integrals (tet Y =! 27KD: ny= (05 252,000) Of thertorm a\s J tos n6 Cos (2 Cos(@ -¥)) de (7.19) T (sw no cos(2 Cos (8-y))de (7.20) AE . (cos NO SIN(2 cos (o-y)}de (7.21) “h and sin no Sin(2 cos(6-y)\de (7.22) =m Consider Equation (7.19) where » = (6-) and dd = d6, \ being a constant. We then have on changing variables -\ (ces(nd+ni) cos(z Cos g) Ag ee n- = cos n ¥ Jcos ng cos (2605 6) dg (7.23) =N- 1-4 — SINT y (sin n @ cos(z Cos g)dg =v Since the integrands in Equation (7.23) are both of period 2n and the interval [-7-, 1-W] is of length 27 we can write the equivalent of Equation (7.23) as cos n v (cos nq Cos(% Cos 4) dg — — SIN ny {sin ng cos (z cos g) ad (7.24) 37) From Ryzhik and Gradshteyn (1965 - page 402) and noting that the second integrand is odd we get Equation (7.23) equivalent to cos ny (on cos (2) Je (z) | (7.25) where J,(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 P(X.) = Sef an J, (21 Ko)} + > [ances ny an cos (2) J. (am kD) + ba SiINnY 2% cos (3) Sn (an Kr) +4 [o. Cosny 2m ae (2m kD) nel + by, SINK Y ev sw(nt) J, (2% K 0) | (7.26) Now nn ath a.) n odd cos (11 } = { (-l)"% even and N (3) = n even S\ oy ot gate Thus we get P™(xY,#) = Qe fam J, (en «0)} +2 [en J, (21 «0) (- la Cos nw + b,, SIN ny) 4+6,° +4 (ae (anko)(-!) ye (a, Cos ny+b,snny)] NzV3,8°° (7.27) 38 From a spectral matrix of the form in Equation (4.13), M = N(N-1) +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, 445 425 ++. 3 by, by, b3, ... 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, R.- (+) Bs Msc : : CRS Pee to form the weights Dag in Equation (4.16). Ome idea being explored is L. = hs a Ings 5A ae pee ae +2 ines R 421 42at! 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, Noon GOS 2) 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 'Optimum' Arrays for Direction Finding," Electronic and Radio Engineer, New Series 6, v. 36, pp. 222-232, June 1959. Bennett, C. M., "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, C. M., "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 logjg 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. ; BE 090 11-0530 OSSEPES 3 ROOT? WS 1B TR6F 1 ROB: |-033 FUN 30 110: Fs 1131 OBSEFE! 5 32 WO ™ WS 16 TREF 1AL BL vais FUN 3) ae ae 1 i CZ | ii i ARE LZ Hh ih i ee ll (NNN A-7 8 35 ° S 8 “ nw 8 ° ae 3 | a oY 0301-0330 OSSEPES 5? WOOBO WS27 1327F2AN02-43) AUN 40 0401-0430 O9SEPES S) HOOB2 W526 T327F2AYB2-491 AUN 4) 0701-0730 O3SEP6S S) WOO92 WS26 1327F2AG42-771 AUN 44 0801-0830 OSSEP6S S) WO103 W527 TOS3F1ROU3-032 AUN 4S ORTEDS/OW/65 TIME1Z00-1229 STRGE2 WO B2 Mh i A se il | i! ill } A AVY Z fe q 5 ult ve A aa: i rn Ml Ae 201-1230 OSSEPGS 32 WOIZ3 WS3D T3ISFIAZU2-771 RUNIG 1301-1330 O9SEP6S S2 WO1Z0 WS3S T31SF1IA3O2-331 FUNL7 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. FORTRAN TO ALGOL TRANSLATOR PHASE 1 FORTRAN STATEMENTS C LISTING OF AN IBM 704 FORTRAN PROGRAM FOR DIRECTIONAL WAVE ANALYSIS. C 13350"? C BENNETT JULY 1967 SWOC 78°301°8210 Cc REQUEST=0268 Cc LEAST SQUARE SINGLE WAVE TRAIN FIT AFTER MUNK DIMENSION IDC41)29FQ€100)2WNC100)»PERIODC 100) *WAVLGHCE 100)» THMAX( 100 X)*DEPATN(100)2AVEP(100)2P(100»696)» EMAX(C100)*TEM( 146) DIMENSION 06656) sPS16656)5E(100572),THETA(72) »DEC73) »THZRO(1496)>» XH(100)2H1(€100)2H2(100) »G000¢6) DIMENSION DATA ¢€1000) E2TEN=0,43429448 TWOPT=6,281853 RT0257,29578 DTR=0.0174532925 FIV20.087266465 REWIND 3 REWIND 9 CALL PLOTS(DATA(1000)»1000) CALL PLOT(0.92=*30.09°3) CALL PLOT(C2.592.5293) DO 5 Ieis7e2 ZIsC171)*5 5 THETACI)SZI*0TR 10 DO 10 [=1%6 DO 10 J=196 DC I»J)=0.0 PSI(I2J)=0.0 0¢1»2)=100.0 0¢153)=100,0 0¢194)=100.0 DC125)=100,0 D(1s5)=100,0 0¢223)=117,558 0¢224)=190,212 0(295)=199,212 0¢296)=117.558 0¢394)=117,558 D¢395)=190,212 0¢3952=190.212 0¢425)=117.558 0¢4945)2190,212 0¢596)=117.558 PSY IS TRIG ANGLE D IS DISTANCE PSI¢€122)=0.0 PSI(123)2=72.0#DTR PST(124)=144.0*DT R 15 20 743 22 PS1€125)=216.0*DTR PS1(126)=286.0*DTR PS1€223)=126.9*DTR PS1€224)=162-0*DTR PSI€2#5)=198.0*DTR PST(226)=234.0*%DTR PST€3294)=198.90%DTR © PS1(325)=234.9*%DTR PS1(3%6)=270.*DTR PS1(425)=270.0#DTR PS1(496)=306.0*DTR PS1(6526)=342.9%DTR GONDC]1)= 1 FOR USABLE CHANNEL DATA AND 0 FOR BAD CHANNEL READ 2» ISKIP s(GDODCI)s» 12126) FORMAT(I2 »6F1.0 ) IFCISKIPI100230220 DO 25 J=isISKIP READ TAPE 35TDsFQC 1) 2 WNC 1) 5MoK IF(K)24222021 PAUSE 20202 TAPE OUT OF PHASE WITH DATA READ DESIRED GO T9 15 MP=M+1 00 23 L=2»MP 23 READ TAPE 3 25 CONTINUE GU TO 15 FQC1)=0.0 CAN NOT BE MEANINGLYFULLY PROCESSED 30 READ TAPE 35 1DsFQC 1)» WNC 1) o4eK TF CK)21031921 31 MP=M+1 M0 32 L=2sMP 32 READ TAPE 3,1D,FQCL)»WNCL) sMoK sDELTAT »DEPTH»PERIOD(L) ,WAVLGH(L)» XDEPATNCL) sAVEPCLI 9 COPCLelsJ)sJH196)21=196) P(L»sI#J)="0.0 IF CHANNEL [| OR J IS NO GOOD WNCL) IS WAVE NUMBER=2PI/ZWAVE LENGTH IN FEET PI=3,141592/ 000 VALUE K NO LONGER NEEDED GOONCH=0.0 DO 49 1=196 GOODC1)=GOODCT)#PCAsToT) IF CGOODCI))41 240544 41 GOODCH=GOODCH+1.9 GOOD(1)=1-0 40 CONTINUE TERMS=1,0+(GOODCH*(GO0DCH=1.0)) DO 70 L=2»sMP SUM=0-0 DU 59 J=196 IF CGOUDCI))51*50+51 51 SUM=SUM4#PC(LsTrl) 59 CONTINUE AVEP(L)=SUM/GOODCH DO 69 k=1»%72 SUM=0-0 D0 90 I= 195 IFC(GOUDCI) 2909094 91 1P = I44 DO 95 J= IPs6 IF(GOUDCJ) 295595292 92 SUM = SUM*+P(LeTaJ) *COSFCWNCL) *#0C Is J) *COSFCTHETACK)@PSICI»J))) xX *PCLoJeT*SINFCWN(LI*DCTe J) *COSFCTHETACK )°PSICI9Jd))) 95 CONTINUE 90 CONTINUE 60 ECLsK)=CAVEP(L)4+2e0#SUM)/TERMS IFCSENSE SWITCH 1)52963 62 PRINTOsFQCL) oAVEPCLISCIDCI slEtoit ds CECLsK) sK31072) 6 FORMATC AX 1PE11.4°1PEL2 4s 1 1A6/(1001X91PE11.4))) 63 G=0.1/DTR DELTA THETAES5DEG UR H=1/2*DEL THETA DEC1)EGRCECL®2)°F(L»72)) DEC72I=G¥CE(CLo1 “ECL *71)) 61 66 67 69 68 64 65 DO 61 ITHETA22s71 DECITHETAI=G*(ECLsTTHETA+1 ECL» ITHETA@1)) NEC73I=DEC1) K=4 00 65 I=1972 IF CDEC I) *DECI+1) 166267265 FIV 1S 5 DEG IN RADYANS THZROCKI=FIVECZIFCDECID/C DECI )=DECI+1)))) K=K+]1 GO TO 65 IFCDECI) 16895956" THZROCK)=FIV*FLOATFCI“1) K=K+1 IF CDEC I 41) 96526465 THZROCK DEFIV*eFLOATFCI) K=K+]1 CONTINUE NZEROS=K=1 D0 75 K=ieNZEROS TERMS SAME AS ABOVE SUM2000 DO 76 1=195 IFCGOODCI) 176975977 77 IP=Iel D0 79 J=IP»6 IFCGQUD( I) 279979978 78 SUM = SUMtPCLoT oJ) *#COSFCWNCL)*O0CIsJ)*COSFCTHZROCK)@PSICI»J))) x “P(LeJeTI*SINFCWNCL)*DCI 9d) *COSFCTHZROCKI@PSTC(I9Jd))) 79 CONTINUE 76 CONTINUE 75 TEMCKI=CAVEP (CL) +2 09*SUMI/STERMS EMAX(L)=TEM(1) THMAKCLI=SRTU*CTWOPT@THZRO(1)) NO 74 K=2eNZEROS TFCEMAXCLISTEMCKI)I73974074 73 EMAXC(L)=TEMCK) THMAX(L)=RTD*®CTWOPT=THZROCK)) THMAX JS BEARTNG FROM MAGNITIC NORTH 74 CONTINUE SUM=0.0 00 89 J=1%5 IFCGIUDCI) 80789984 81 IP=[+1 DO 83 J=IPs6 IFCGOUDCJ) 283983982 82 SUM=SUM+P(LoToJ)#PCLoTo J) +PC Lodo eP(Lodel) 83 CONTINUE B-8 80 CONTINUE SSQSAVEPCL)*AVEPCLI*#2.0¥SUM H(L)2SSQ*TERMS#EMAX(L) *EMAXCL) ATILDASAVEP(LY/TWUPI HTILDASSSQ*ATILDA*ATILDA*TERMS H1(L)=1.0°CH(L)/HTILDA) H2CLYSCEMAXCLI=ATILDAJ/CAVEPCLI@ATILDA) IFCSENSE SWITCH 129970 99 PRINT7>» TERMS*EMAX¢CLISATILDASHTILDA 7 FORMATCIXsF5el1s1P3E1164 ) 70 CONTINUE PRINT 35CID(I)»T=le11) »(GOUDCI)»T=196)0mM 3 FORMATCIHL,39HLEAST SQUARE SINGLE WAVE TRAIN FIT OF ,11A6/ X1X96F2,092Xe2HM=I3// X 3X5 DHFREQUENCY 2 5X*S5HPERIOD» 3X91 1HNAVE LENGTH? 1Xs 1 LHATTENUATION? 4X» XSHAVE PsBXotHArlOXeSHBRNG »8Xa1HH912X »2HH1 oe L0X»2HH2) PRINT Gs CFQ(L)sPERTONCL) sWAVLGH(L?) »DEPATN(L) »AVEPCL) »EMAX(L)>» XTHMAXCL) HCL) HL CL) sH2CL) »LS25MP) Q@ FORMATCIX20PF 110679 3XS0PF9eSe1XSOPFItodsiXsiPEli ods X 1X9 1PE110491X91P E14 40 3X2 OP F729 3X0 1PE11 49 4Xs0PF6,356X20PF6.3 ) 0G 110 L=2sMP AVEPCL) =E2TEN*LOGP CABSFCAVEPCL))) EMAX(L)SE2TEN*LOGF CABSFCEMAX(L))) 110 WAVLGHCL)=EMAX(L)SE2TEN*LOGFCABSFCDEPATNCL))) AVEP(1)=AVEP(2) EMAXC1)=EMAX(€2) WAVLGHC1)=WAVLGHC 2) CALL SYMBOL(2.05%.75%0.1214HWAVE TRAIN.FN= 20.014) CALL NUMBER(3.5500752001%FQC(MP) 900092) CALL GPHPVW(MP» IN» AVEPs EMAX»WAVLGH) WAVLGH IS SURFACE WAVE STAFF SPECTRUM EST FROM BOTTOM PRESSUXE. AVE=P EMAy=V WAVLGH=w ON THE PLOTS, ZM=M NP=DELTAT*ZM*0.541.0 DELX=0,.0 90 115 I=156 CALL NUMBER(DELX20.120.01»GO0D(1)»0.0s—1) 115 DELX=DELX+0.1 CALL PFB3DCNP* IDs EMAXsFQ*THMAX ) lei) fa) SS) 100 REWIND 3 CALL EXIT PAUSE 70707 GO 79 15 END(O%15190s1) END B-10 FORTRAN TO ALGOL TRANSLATOR PHASE 1 FORTRAN STATEMENTS SUBROUTINE GPHPVW(L»ID»ZLOGP»ZLOGV.ZLOGW) SPECIAL FORM OF GPHPVW FOR 1335072 AUGUST 1967 DIMENSION ZLOGPC2)57L0GVC2)»ZLOGW(2),10¢€2) CALL PLOTCO0.020.9%3) CALL PLOT (0.0219e592) CALL PLOT (8.091005s2) CALL PLOT (8.090.052) CALL PLOT (0.020,022) CALL PLOT (€2e0s1-59%3) CALL SYMBOL €0.0°71.097.1910(1)50.0%66) CALL AXIS (0.950.0,20HNORMALIZED FREQUENCY 5=20550002050+020.2) ZMAX2ZLOGP(1) DO 10 T=2°L COMPAR=ZLOGPCI) ZMAX=MAX1FCZM4X»CUMPAR) MAX=7MAX+3.0 RE=SMAXK=8 CALL AXISC0O+0%000247HLOG POWER DENSITY172820290¢0sBE5 100) DX=5,0/FLOATFCL=1) Y=ZLOGP(1)=BE X=0.0 CALL PLOTCX»Y»3) B-11 20 21 23 DO 20 I[=2°L X=X+DX Y=ZLOGPC(I)=°B6E CALL PLOT (XsY»s2) CALL SYMBOL(XsY¥sOelelHPs0.001) X=5.0 Y=ZLOGVC(LI=BE CALL SYMBOLCXsYeOolstHAs0.0s1) CALL PLOTCXsY»3) M=L=1 DO 21 T=1»™ X=SKX=NX Itsiel YEZLOGVCII)@BE GALE PLOVGhovoe» WMAX=MAX X=0.0 Y=ZLOGWC(1)°bE CALL PLOTCXsYe3) DO 22 T=2°L X=X+9X TF CZLUGWCI)@WMAX)24523923 Y=8.0 GO TO 22 B-12 24 22 30 40 50 YeZLOGW(I)=BE CALL PLOT(CX»Y»2) CALL SYMBOL (Xs Y¥s%e151HS20,001) IF (SENSE LIGHT 124030 CALL PLOT (©2.099e0»923) SENSE LIGHT } GO TO 50 CALL PLOT (6.02 RETURN END( 991919029) END 12.0973) B-13 FORTRAN TO ALGOL TRANSLATOR PHASE 1 FORTRAN STATEMENTS SUBROUTINE PF33DCNrID»PyRoB) P=FUNCTION OF R = POWER DENSITY B= FUNCTION OF R = COMPASS BEARING R= FREQUENCY IN HZ DIMENSION J[DC2)2PC62)9RC2)5BC2) 966360) 95350) T=10.0 D=0.70710676 DTH=9 2017453293 DTH ITS ONE DEGREEE IN RADIANS IE RADIANS PER 1 DEGREE A=0.9 DO 10 1[=1»%360 A=A+0TH CCI) =COSFCA) SCI)J=SSINFCA) CALL PLOTCO0.950.023) CALL PLOTC0,0%10.5»2) CALL PLOT(B.0910.592) CALL PLOT(8.050.922) CALL PLOTC0.0°0.092) CALL SYMBOL(1.09%.5570.121D(1)90.0266) CALL PLOTC4,053.52=°3) CALL PLOT(3.0°0.092) B-14 30 20 CALL SYMBOL(3.25%0,020.121HN +0,021) CALL AX1ISC0.0°1.0°0H £095,0290.0%71.0%1.0) CALL PLOTC0.091.093) CALL PLOTC0.0»0.092) CALL PLOTC#3.020.092) CALL SYMBOL(23.2690,.0%0e1%1HS »0¢001) CALL SYMBOL(X*0.12Y"0els0e1s1HE » 00001) CALL PLOT(X»Ys3) CALL PLOTC#Xs=Y¥s>) CALL SYMBOL (O.1°XsM.1°Ys0e1ls1HW » 06081) CALL PLOT(0.020.02°3) DO 20 J=1%6 A=0.5*FLOATFCI) CALL PLOTCA20.023) DO 30 J=i»360 Y=A*S(CJ)*D XSA*CCJ) + Y CALL PLOTCXsYe2) CONTINUE CALL PLOTCO.020.99°3) D=10.0#D 10*D0 NEEDED TO SCALE 0.070.3 TO 073 INCHES ON PLOTS B-15 49 42 41 50 DO 40 J=2eN RAD=DTH*B(1) Y=RCT)*O*SINFCeORAD) XERCT)*T*COSFCRAD) + Y TUD=3 CALL PLOTCXsYsIUD) AY=P(I1) + Y + 1.96 2 IS NOT ADLED S9 ToP SYMBOL WILL BE THE REFERENCE IN CALL PLOTCXsAYs2) CALL SYMBOL (XsAYs0.082250,0%°2) CALL PLOTCXs AY» 3) CARE PEOn GX Yi92) IFCSENSE LIGHT 1941542 CALL PLOTC£4.0>7.0,23) SENSE LIGHT 1 GO 79 50 CALL PLOTC4,05°14.0,°3) RETURN END( 0915912091) END B- 16 2-08 SYMBOL 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, 9, 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 Q.1, and a fractional portion value of 0.1. Both results are for data collected for task SWOC 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 Hl and H2 values are measures of the isotropicity of the energy represented by the residual spectral matrix. 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SEc) e POWER DENSITY CIN- WAVE DIRECTION ITERATIVE LEAST SQUARE WAVE FIT DIRECTIONAL ANALYSIS C-3 OOENRCOOL 00eH COOL OOO OO0U 000rC000 006ELOOU VOBECOOO OOLELOOL 009ELOOL 00SECO0U VOHECOOO VOEECOOO voce loou vote coou VOLECOOU 0062U00U 008zbo0U 00226000 009eC00U 0o0S¢eCO0U ooneCood OO£EeCOOO 00226000 oobecoou KOKO AGLOLONG 00614000 0O8TLOOL 00210000 0091000 00ST0000 OOH TCOOu oo€ tc 000 00etCoo0o OS O= GGA IUSid 3INWISIN SI G J1yhy YIal Si 1Sd 'oS* J ,t=(94S)G Cie Oot=(940)G toSt*s ,ba(Se dG 1c *0eT=09 "D0 c1eo*0et=(S4tdC tcS* J T=Cn #t)0G &oS*s ,l=(94¢C)0 Cle’ O06t=(G4d)0 c1c*o0el=(14e)G ecoS*s Tat 6¢)U O'out=a(y4t)U O*OUb=(GET)IU OfouT=(h4b)G O'ouT=a(t eb) otoul=(e41u O'oe(reldISd o°de(( '1)y ge=r Ob Ob 9e1=1 UCT OO alu*l7=CldviaHl GaC bel d=1Z c/Jet=l S OC 1437 3HL Ul BVYTYO YaLLO1d 148 (ee fC UE SG °?)INIAd 11V9 G9bP¥Y9ELAU*U=NTS scéctGnztoeu=elg wlGoe*2G=Uly Fotleyc*Y=ldUmd brrecney* O=NI103 OOTTCOOU/S Hwd TU ZUZbZEnZS°UN3$ LOVG/11G90KS 4194974 DIbwh ~ THINOD V1IVG o000TLOOO 006006000 VOBOCOOU 0020C00U COOL AAV *CUUTINIVdIOSCLOFIHD Iu eC UOTICUTNAd = NOISNAWIU (64) G94 * (8H) TYHLN D9 §CY)GHUN ECELIAGECSLIVIGNL 9 CEL) 960949 1S AC 949)U “(ONVIOU7ZHL SC OntIWal 669 494001)d “COO INMNECOOLIHASCHDIGL NOLSN4KTU Ingnoas su Lyev1ls 00900000 00S0C00U OONOCOOC 00f£0C000 00d0C000 OOVOCOOL OOCOLOG( 696) T1edy YadoVe UNV SSSVHOCUNSSOSTITN ENN Nh B4LAV LT4 NIVUL 3Ave Buvnos iSvil 40 NNLivugl! aw SISATVNY NO G4ISVE *CJ6T KOMWW LLSNNIG b WYVO AA SSNTA ADNAN OAS, JN TWAMSINI Ny uG4 214 NIvdg GAvM didiadaW 40 NOliavesid CEE LAS YOAIUMS CeeghyStdnl/duud=6 Ob=GHNQIFN PH AGVIANELING SEE bIEY PG=ZUHUIIn MIL TEOESIAVS SILTUISZ OI MSHUS=E TOYS Vy 1 IS my Wy fw a W cl i ty 9 Nie Vie ieee cs Ceres OGG 8 GO) alaltel\y/dl oO ot Co 00060000 00680000 00880000 00280000 00980000 00S8C000 00080000 00€8C000 00280000 o0otsco0o 00080000 00620000 00820000 00220000 00920000 00S20000 00720000 00€2C000 00220000 00120000 00020000 0069C000 00890000 0029C000 00990000 00690000 00790000 o00e9C000 00290000 00190000 00090000 00680000 008S0000 00250000 009S0000 00SS0006 00#7SC000 00€S0000 00¢SCo00 001S0000 000S0000 00600000 00870000 002n7C000 009nCOO0U 00SHtCO00 00nvCO00 u@ 3AVheeC1)008 (940° U6H*(TLIGI SH1 Ot 9° be SO) TOUWAS 1109 (0960°06=*(TIGI Sn 20%od* te fS° OI T0HWAS 1109 1°02, 190=A 1730 (Te 0° V6=4(1)GN09420°OSAT3GS T° OI NABWIIN 179 9¢121 ¢h OO ¢°O2=A 130 °CXVWHI®XVW9) JO 107d ADNANONd SA ONTYVGE HOS 198V7 ONY SIXV CCO°FeHDGUUD)*HIGONUY) 40° F=SWYal INNILNOD 0°1=(1)U009 O° T+HIGHUI=HIGAOD VHeOns1UCC1)QNuUY) 41 (1%1¢2)d*(1)0009=(1)0009 9*\21 Ob OG 0°0=HIG009 G30335N y39NQ) ON: ANIWA eo s26cthtl*Eeeld 1334 NI HL9nN37 JAVM/Tde=usewlin 3AWM ST C1)NM Goo GN Si fF YO 1 VANNWHO 4I 0° Oe=(F4141)d € 38019 COSTEL ECO STS ECR EL ETI G)) SC TIdAVEC TI NIVdIO 4 SCTIHOTAVM SC TIGOTUId SHLddUS LV TAG HENS C TINE S (190407 (€) dvau dWwc21 ce OG T+W2dwW MEWS CTINMS(TI04601 (€) Gdv3au O4XU*O0°C=dXG e2o0-a Ost) © 22O- CNW elk ds 9 43S019 CO°94ES9TESUTT SIDI VWHOS OAXGSINADNIA SOT LYN SMLIXWWs Swed TEC 9ebeTeC1IGOND) (es9IGV4y TANNVHD Ov Hod O GNV VWiVO TANNVHO 378VSf Yod F 201) 0009 YpGa0*cre=(92S)1Sd Yid*e0°90FE(9%HIISd Hids0°0L2e(G'n)1Sd ulge*0lZ2=(94E)1Sd BideO°vEc=(G*EIISd BLUeO*boT=aCh*t)1S¢d wiGsO0*heze(9°72)1Sd B1G¥O0°86T=(G92)ISd BiUe0c9t=e( se) 1Sd BideO° 92TH (E4cd(Sd wids0*tez2e(941)1Sd ByGeO0°9b2=(S4I)1Sd BidsO0*tHte(hst)1Sd alu*o°es=(e 4b 1Sad (GQ4NNILNO9) @€9 3ATEVL Ov Vt ce oouoo O0OZE1000 C141‘ V)athwMS=ewns 4S 009E 1000 1S*0G*1S¢(C1)0NUY) 41 00SE 1000 9¢,=1 OS OG Oot 1000 O2 OL OD CHLIXVW °19° YLID4T 2h+ SLT Ell f0°UEnNS O02 OOoEETOOO CO9%64SXT) SucH bh is d/v u t OOZELO000 fu ONL&VGE d AAVal Xbb oud Nuliwvadli AyNgNe 444 wd VWwe0d 6 oote 1000 (7204 £6 IN] Ya 000€ 1000 v=aHll 00621000 gW447=7 ul OU 00g2to0u : CO*L4SusINIId| ow 6 wl (4° 64E fund AOhVE AININO FH da? 002¢1000 “XT SR ZAHM EQTIVY WOWINIW aft i Su=NOllvudli whyixvy wfU°CIG*xt I OO9CIOOU/9VIT én JU L114 NIVHL SAUM 31dTA Wy 4NVHOS 19V57 ALLVE4LI wd VWuOd & 00521000 INd.Ndad * I OOHMZTOOO (C4WIOFECATIE SEOTAVE ULI Xue (Ge belo (CT UULD) Se Ch Te tele CJ] I01)46 IN]Yd 00€ 21000 CELSCTOSC AIUD SH TPO%Ss Ge 40°C) TNEWAS 1709 002ezto0u “ Au=Ct UI Fu LISNAGu=CcUCb fa PAM dw=Cb0008 00Tét100G CREUTUSCTI IE SHIT UFOs 40°C) TN GWAS 1109 00021000 u(ZFIAIW =(E 2008 00611000 wNSP 09M. =00)098 oo8tloou w4 3AWM, =C40G98 00ZT100U CheO'OG=e*C1LIGIASMIL OF CO fGC%CeI TNHWAS 11V9 00911000 a hu=Cb)004 00611000 Ch 40S06e SCL IGIAEML IOS fet GC MOLI WNYNAS 179 00vTTO000 “ dus(1)G0¢ Oo€1TOO0 CTO O0Ge*(bIGIAEDL OFS * Eat GC ORI TWNGWAS 119 00211000 “ Su=(1)008 ooTTt!oou Ch EO*OOe SCL IGIHEME OSE Ge fF GC* CHI INNNAS 1109 000TToOOU i hw=(b)098 0060100U CVO 06S SCL IGIGEU MOF bef GC Um TWENAS 17109 0080100C w Nw=C(b)094 00201000 Coed * deel *Ue) 101d 17199 00901000 (eee? sesC*U Jind 119 00601000 Cosh Ge s0°G 11d 1799 00n0TOOU Cosh * Gest eUe)INId 1799 VOEOTOOL Cosh *oeeG*G yiN1d 119 o00z0toou (eeG*tecO*U JiNTd 1109 00TO0TO00 CosG*test *UedINdd I1Vd 00001000 Ceegt*eee*U YyNI1d WIWV9 0066C000 Coe bee? 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WANE OA Wi (WR) Go SIO GIEVAW dd) HIGUUD/WNSSd3AV S9 09 89 69 9 99 19 9 09 06 S6 c6 16 009 FnNTLnod OS (GANNILNOD) 29 AldVL OOTE COOL OOO0ECO0U 006¢2c00U 0082C00U 602e¢000 0092¢¢00C 00S¢é¢00U OOone ODA ale) aK) 04aXG =» WSs = xo HCOSUSE AXVIWHID x Cele = CVC 11 LVe dav IS CVUTTIVeXYN3) = Ch CVOTILHSZ H)= 9°T = Th SwodiavO lidvevudl 1 ve6SS=vUT1LH Tei Mi/ddAW = VUILIY XvW4exXVh4 *Shtgle OSS =k WiiS¥0°c + Gd4AV * d4AD = OSS AINTLNOD INNILNOD CLeredeci ers Wdtc esl sqea( rfl {1 datwnSenns coftebeteC (f )UNMUY)I 41 ¥Y*q1=f te OU T+l=d] TeseueCeC (€1)0NU9)4I G* y=] Ub OU O° O=SKNS XvVhe/X¥nd = 1S391 WINTLNGD HLYUN JILINOWW hOMS YNIRVGG SI XVWHI COMIONZRLelduNi)* GLY = XVhHI CWC aZHL XvWOd (Adwal = xWwWa plen lie d( (Cy WILexvn49) ddI SUMAZN4c=> nl OU COLIONZ he ToGUM1 FUL XVWHI VidgHL 40 NOTSH4ANNY SSvdwOyU Ul OIL CTICaZHL = XVhOe (bdwal = XVW43 Swdais (hy, SeCto + danvd=(nIW4L FINNILNOD WNILNOD COCR EDITS d= CMI UNZHLISNO# CH ELI GeO LINMIN ISH CL SR SW de x COCE O1IIS de CWIUUZHLISOOECH OE LIGHECTINMISUIEC, $1472 dtwAS = WAS e2fo2e6s (fF IGNLY) ddI Y*d1l=r o£ OU Tt+l=dl LLOES LESTE CU NIGHY) ait Gel=] 92 OU O° L=EKNS = (i) (GANNILNOD) 29 AIaVL Ow te ce be v2 tl Gd 9. 6l el dd _S) IN3WD3S 008tc000 00272000 0092000 0062000 007nc000 oof ’< 000 00cHc000 00Tre 000 00072000 006€2000 GNa dO1S CTuiNC2 ) dIZ WW9 6 3so19 (6664640)10 1d WIV9 oot anNITAngd Od 002 OL 09 JNNILNOD O€2 JNNTLNOD Od IN3W99S 40 LHWIS IN3W93S 008€2000 0022000 009E C000 00SECOOE (KOU AMAOONS) OOeE COON 00ZEcO0U COCOCRETIISd © yyWwuddSod J eCPETIG#CTINMINIS ¥XVWI)4 LNIONAd? + Chop ed sCl ered COOCOPE1I1Sd © XWWUNISOI »(P6T)GeCTINMISOD *XWWI)* LN3IINGd) © epPatelijc 2 CPELe Wd ogd OL ud Cu °b3° CF)AQ0UNI41 £94dT =F Oes OC T+I = dl Oo Oe HS CO CMe ChIUEODIel GIT sl Mee WE (G4ANNILNOS) 29 AIEVL J UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA-R&D (Security classification of title, body of abstract and indexing annotation must be entered when (hs overall repor! ls clasellied 1. ORIGINATING ACTIVITY (Corporate author) 2a, REPORT SECURITY CLASSIFICATION Naval Coastal Systems Laboratory UNCLASSIFIED Panama City, Florida 32401 3. REPORT TITLE 2b. GROUP THE DIRECTIONAL ANALYSIS OF OCEAN WAVES: AN INTRODUCTORY DISCUSSION 4. DESCRIPTIVE NOTES (Type of report and inclusive datea) Informal 8. AUTHOR(S) (First name, middle initial, laet name) Carl M. Bennett #6. REPORT OATE Ja. TOTAL NO. OF PAGES 7b. NO. OF REFS ' December 1972 83 12 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) b. PROJECT NO. NCSL 144-72 ; c: Task SWOC SR 004 03 Ol Task 0582 ob. RY NO(S) (Any other numbers that may be assigned a ZR 000 01 01 (0401-40) First Edition NSRDL/PC 3472 10. DISTRIBUTION STATEMENT Approved for Public Release; Distribution Unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Second edition. This report was Gommander originally issued in September 1971 Naval Ship Systems Command (OOV1K) 13. ABSTRACT 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. DD 2""..1473 | UNCLASSIFIED curity Classification UNCLASSIFIED Security Classification KEY WORDS Ocean waves Spectrum analysis Fourier analysis Matrix methods Convariance Bibliographies IDENTIFIERS Cross spectral analysis Directional analysis SWOC (Shallow Water Oceanography) Ocean bottom pressure data UNCLASS IFLED en EEE aT TnEnEE EIEN EERE Security Classification INITIAL DISTRIBUTION NCSL 144-72 000100 Chief of Naval Material (MAT-03L4) (Copy 1) 000400 Commander, Naval Ship Systems Command (SHIPS OOV1K) (Copies 2-4) (SHIPS 2052) (Copies 5-6) (SHIPS 03542) (Copy 7) 000500 Commander, Naval Ship Engineering Center (Copy 8) 003100 Assistant Secretary of the Navy (Copy 9) 005600 Chief of Naval Research (ONR 410) (Copy 10) (ONR 414) (Copy 11) (ONR 420) (Copy 12) (ONR 438) (Copy 13) (ONR 460T) (Copy 14) (ONR 461) (Copy 15) (ONR 462) (Copy 16) (ONR 463) (Copy 17) (ONR 466) (Copy 18) (ONR 468) (Copy 19) 028900 Director, Office of Naval Research (Boston) (Copy 20) 029000 Director, Office of Naval Research (Chicago) (Copy 21) 029100 Director, Office of Naval Research (Pasadena) (Copy 22) 028500 Oceanographer of the Navy (Copies 23-24) 021000 Commander, Naval Oceanographic Office (Copies 25-26) 023900 Commander, Naval Ship Research and Development Center (Code 561) (Copies 27-29) (Code 01) (Copy 30) (Code O1B) (Copy 31) 027200 Commander, Naval Weapons Center, China Lake (Copy 32) 015900 Commander, Naval Air Development Center (Copy 33) 019800 Commander, Naval Electronics Laboratory Center, San Diego (Copies 34-35) 026900 Commander, Naval Undersea Center, San Diego (Copy 36) (Dr. Robert H. Riffenburg) (Copy 37) (Dr. W. J. MacIntire) (Copy 38) 018800 Commanding Officer, Naval Civil Engineering Laboratory (Copy 39) 021200 Commander, Naval Ordnance Laboratory (Copy 40) 028100 Officer in Charge, New London Laboratory, NUSC (Copy 41) 036100 Officer in Charge, Annapolis Laboratory,NSRDC (Copy 42) 022600 Director, Naval Research Laboratory (ONR 481) (Copy 43) (ONR 483) (Copy 44) (ONR 102 OS) (Copy 45) Ocean Sciences Division (NRL) (Copy 46) Library (NRL) (Copy 47) 029600 Commander, Pacific Missile Range (Copy 48) 027900 Officer in Charge, Environmental Prediction Research Facility (Copy 49) 015600 Superintendent, Naval Academy (Copy 50) (Professor Paul R. Van Mater, Jr.) (Copy 51) 022500 Superintendent, Naval Postgraduate School (Copy 52) 002700 Army Research, Office of the Chief of Research and Development, Dept. of the Army (Copy 53) 002500 Director, Army Engineers Waterways Experiment Station (Copy 54) 002300 Army Coastal Engineering Laboratory (Copy 55) 014900 District Engineer, Mobile District Corps of Engineers (W. W. Burdin) (Copy 56) 007900 Director of Defense, Research and Engineering (Ocean Control) (Copy 57) 015400 Director, National Oceanographic Data Center (Copies 58-59) 000600 Director, Advanced Research Projects Agency (Copy 60) (Nuclear Test Detection Office) (Copy 61) (Dr. R. W. Slocum) (Copy 62) 028600 Chief, Oceanographic Branch, CERC (Copy 63) (Dr. D. Lee Harris) (Copy 64) (die, Cyreatil ajo Ceulyatin., wes) (Copy 65) 034700 Director, Woods Hole Oceanographic Inst. (Copy 66) (Dr. John GC. Beckerle) (Copy 67) (Dr. N. N. Panicker) (Copy 68) 015300 National Oceanic & Atmospheric Administration, U. S. Department of Commerce (Copy 69) (Dr. Moe Ringenbach) (Copy 70) 009600 Environmental Science Services Administration U. S. Department of Commerce (Copy 71) 013400 Chief, Marine Science Center, Coastal Geodetic Survey, U.S. Dept. of Commerce (Copy 72) 004200 Director, Bureau of Commercial Fisheries, U. S. Fish and Wildlife Service ‘ (Copy 73) 000700 Allan Hancock Foundation (Copy 74) 010700 Gulf Coast Research Laboratory, Ocean Springs (Copy 75) 012300 Director, Lamont-Doherty Geological Observatory, Columbia University, Palisades (Copies 76-77) (Dr. Arnold L. Gordon) (Copy 78) (Dr. Leonard E. Alsop) (Copy 79) (Dr. John E. Nafe) (Copy 80) 030700 008500 008700 008800 008200 031800 009200 011800 009300 004900 031200 013500 005700 012000 010600 008600 009500 008900 029900 (Dr. Keith McCamy) (Dr. John T. Kuo) (Dr. Tom Herron) (Dr. Manik Talwani) Director, Scripps Institute of Oceanography, University of California (Dr. Walter H. Munk) @r) Diy ee Enman)) (John D. Isaacs) Department of Geotechnical Engineering, Cornell University Department of Oceanography, Florida Institute of Technology Department of Oceanography, Florida State University (Dr. K. Warsh) Chairman, Department of Coastal Engineering, University of Florida University of West Florida (Dr. A. Chaet) Department of Physics, Georgia Southern College (Dr. Arthur Woodrum) Institute of Geophysics, University of Hawaii Division of Engineering and Applied Physics, Harvard University, Cambridge Director, Chesapeake Bay Institute, Johns Hopkins University (W. Stanley Wilson) Officer in Charge, Applied Physics Laboratory, Johns Hopkins University Director, Marine Science Center, Lehigh Univ. Coastal Studies Institute, Louisiana State University Institute of Marine Sciences, University of Miami (Dr. W. Duing) Great Lakes Research Division, University of Michigan Department of Meteorology and Oceanography, New York University Environmental Science Center, Nova University (Dr. W. S. Richardson) Head, Department of Oceanography, Oregon State University Pell Marine Science Library, University of Rhode Island (Copy 81) (Copy 82) (Copy 83) (Copy 84) (Copy 85) (Copy 86) (Copy 87) (Copy 88) (Copy 89) (Copy 90) (Copy 91) (Copy 92) (Copy 93) (Copy 94) (Copy 95) (Copy 96) (Copy 97) (Copy 98) (Copy 99) (Copy 100) (Copies 101-102) (Copy 103) (Copy 104) (Copy 105) (Copy 106) (Copy 107) (Copy 108) 009100 002000 009000 010500 015200 011700 007700 Department of Oceanography and Meteorology, Texas ASM University Applied Physics Laboratory, University of Washington Head, Department of Oceanography, University of Washington Chairman, Department of Oceanography, University of South Florida National Institute of Oceanography, Wormley, Godalming, Surrey, England (Director) (J. Ewing) (Dr. L. Draper) Institute fur Meereskunde Under Universitat, West Germany (Dr. Wolfgang Krauss) (Dr. F. Schott) Director, Defense Documentation Center Dr. William P. Raney, Special Assistant for Research, Navy Department, Washington, DG. 20350 National Oceanic and Atmospheric Administra- tion, Boulder, Colorado 80302 (Earth Sciences Lab) (Wave Propagation Lab) Director, National Weather Service, NOAA, 8060 13th Street, Silver Spring, MD 20910 (Dr. William Kline, Sys Dev 0) Director, National Ocean Survey, NOAA, Rockville, MD 20852 Director, Pacific Marine Laboratory, NOAA Seattle, Washington 98102 Earthquake Mechanism Laboratory, NOAA, 390 Main Street, San Francisco, CA 94105 Ports and Waterways Staff, Office of Marine Environment and Systems, U. S. Coast Guard Headquarters, 400 7th St., SW, Washington, DoGo ZOSEI Director, National Center for Earthquake Research, U. S. Geological Survey, Menlo Park, CA 94025 Director, Seismological Laboratory, California Institute of Technology, Pasadena, CA 91109 (Copies 109-110) (Copy 111) (Copy 112) (Copy 113) (Copy 114) (Copy 115) (Copy 116) (Copy 117) (Copy 118) (Copies 119-130) (Copy 131) (Copy 132) (Copy 133) (Copy 134) (Copy 135) (Copy 136) (Copy 137) (Copy 138) (Copy 139) (Copy 140) (Copy 141) Director, Seismic Data Laboratory, Geotech- Teledyne, 314 Montgomery St., Alexandria, VA 22314 (Copy Director, Thomas J. Watson Research Center, Yorktown Heights, NY 10598 (Copy Director, Institute for Storm Research, Houston, TX 77006 (Copy The Offshore Company, P.O. Box 2765, Houston, TX 77001 (Crane E. Zumwalt) (Copy Tetra Tech, Inc., 630 N. Rosemead Blvd., Pasadena, CA 91107 (Wye5 Wo ilo Colilatjeys))) (Copy (Dr. Bernard LeMehaute) (Copy Director, Geophysical Institute, University of Alaska, College Br., Fairbanks, Alaska 99701 (Copy Chairman, Department of Geological Sciences, Brown University, Providence, RI 02912 (Copy Chairman, Dept. of Geophysics, University of California, Berkeley, CA 94720 (Copy Director, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90024 (Copy Director, Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92502 (Copy University of California, Hydraulic Engineering Division, Berkeley, CA 94720 (Dr. R. L. Wiegel) (Copy Chairman, Division of Fluid, Thermal, Aerospace Sciences, Case Western Reserve University, Cleveland, Ohio 44106 (Copy Central Michigan University, Brooks Science Hall, Box 12, Mt. Pleasant, MI 48858 (Dr. Kenneth Uglum) (Copy Chairman, Department of Geophysical Sciences The University of Chicago, Chicago IL 60637 (Copy Columbia University, Dept. of Physics, New York City, NY 10027 (Dr. Gerald Feinberg) (Copy 142) 143) 144) 145) 146) 147) 148) 149) 150) IL5}10)) 152) 153) 154) 15D) 156) 157) Columbia University, 202 Haskell Hall, BASR, 605 W. 115th St., New York City, NY 10025 (QDye5 /NLem, Go TétILily) Chairman, Department of Geological Sciences, Cornell University, Ithaca, NY 14850 Chairman, College of Marine Studies, Univ. of Delaware, Newark, DE 19711 Chairman, Dept. of Oceanography, Duke Univ., Durham, NC 27706 Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee, FL 32306 (Dr. Ivan Tolstoy) (Dr. Joe Lau) Chairman, Center for Earth and Planetary Physics, Harvard University, Cambridge, MA 02138 University of Hawaii, Department of Ocean Engineering, Honolulu, Hawaii 96822 (Dr. Charles L. Bretschneider) University of Idaho, Department of Physics, Moscow, Idaho 83843 (Dr. Michael E. Browne) University of Iowa, Institute of Hydraulic Research, Iowa City, Iowa 52240 (Dr. Hunter Rouse) Chairman, Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, MD 21218 Chairman, Department of Earth and Planetary (Copy (Copy (Copy (Copy (Copy (Copy (Copy (Copy (Copy (Copy (Copy Sciences, Massachusetts Institute of Technology, Boston, MA 02139 Massachusetts Institute of Technology, Dept. of Naval Architecture and Marine Engineering, Boston, MA 02139 (Attn: Dr. J. N. Newman) Division of Physical Oceanography, School of Marine and Atmospheric Science, University of Miami, 10 Rickenbacker Causeway, Miami, FL 33149 (Dr. Christopher N. K. Mooers) (Dr. Claes Rooth) (Copy (Copy (Copy (Copy 158) 159) 160) 161) 162) 163) 164) 165) 166) 167) 168) 169) 170) 171) 172) Chairman, Department of Natural Science, Michigan State University, East Lansing, MI 48823 (Copy New York University, Institute of Mathematical Sciences, New York City, NY 10003 (Dr. J. Ji. Stoker) (Copy Chairman, Dept. of Geosciences, North Carolina State University, Raleigh, NC 27607 (Copy Chairman, Institute of Oceanography, Old Dominion University, Norfolk, VA 23508 (Copy Chairman, Department of Geosciences, Geophysics Section, The Pennsylvania State University, University Park, PA 16802 (Copy Chairman, Dept. of Earth and Planetary Sciences, University of Pittsburg, Pittsburg, PA 15213 (Copy Chairman, Dept. of Geological and Geophysical Sciences, Princeton, New Jersey 08540 (Copy University of Rhode Island, Graduate School of Oceanography, Kingston, RI 02881 (Dr. Kern Kenyon) (Copy Chairman, Dept. of Geology, Rice University, Houston, TX 77001 (Copy Chairman, Dept. of Earth and Atmospheric Sciences, St. Louis University, St. Louis, MO 63103 (Copy Director, Dallas Geophysical Laboratory, Southern Methodist Univ., Dallas, TX 75222 (Copy Director, Center for Radar Astronomy, Stanford University, Stanford, CA 94305 (Copy Chairman, Department of Geophysics, Stanford University, Stanford, CA 94305 (Copy Chairman, Dept. of Geological Sciences, University of Texas, Austin, TX 78712 (Copy Chairman, Dept. of Geological and Geophysical Sciences, University of Utah, Salt Lake City, Utah 84112 (Copy Director, Marine Research Laboratory, University of Wisconsin, Madison, WI 53706 (Copy University of Wisconsin, Center for Great Lakes Studies, Milwaukee, WI 53201 (Dr. David L. Cutchin) (Copy 173) 174) 175) 176) 177) 178) iL7/9))) 180) 181) 182) 183) 184) 185) 186) 187) 188) 189) Chairman, Dept. of Geology and Geophysics, Yale University, New Haven, CT 06520 (Copy Director, Navy Hydrographic Office, Buenos Aires, Argentina (Copy Chairman, Department of Geophysics and Geochemistry, Australian National University, Canberra, 2600, Australia (Copy Monash University, Geophysical Fluid Dynamics Lab., Clayton, Victoria, Australia 3168 (Dr. B. R. Morton) (Copy Flinders University of South Australia, Horace Lamb Center for Oceanographic Research, Bedford Park, Adeleide, South Australia 5042 (Prof. Bye) (Copy Chairman, Dept. of Oceanography, Dalhousie University, Halifax, Nova Scotia, Canada (Copy Director, Geophysics Laboratory, University of Toronto, Toronto, Canada (Copy Director, Institute of Earth and Planetary Physics, University of Alberta, Edmonton 7, Alberta, Canada (Copy Director, Institute of Oceanography, University of British Columbia, Vancouver 8, British Columbia, Canada (Copy Chairman, Department of Geophysics and Planetary Physics, The University, Newcastle Upon Tyne, NE 1 7 RU, England (Copy Cambridge University, Madlingley Rise, Madingley, Cambridge CB3 OEZ, England (Dr. M. S. Longuet-Higgins) (Copy Chairman, Department of Geodesy and Geophysics, Cambridge University, Madlingley Rise, Madingley Road, Cambridge CB3 OEZ England (Copy Director, Institute for Coastal Oceanography and Tides, Birkenhead, Cheshire, England (Copy Director, Oceanographic Research Institute of the Defense Dept., Kiel, West Germany (Copy Abteilung fur Theoretische Geophysik, Universitat Hamburg, Hamburg, West Germany (Copy 190) 191) 192) 193) 194) 195) 196) IL) 7/)) 198) 199) 200) 201) 202) 203) 204) Institute for Advanced Studies, 64 Merrion Square, Dublin, Ireland (Dr. John Lighton Synge) (Copy 205) 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 (Professor A. C. Kibblewhite) (Copy 208) Chairman, Dept. of Marine Science, University of Puerto Rico, Mayaguez, Puerto Rico 00708 (Copy 209) “~* . a ; : 2 i” ‘ ft) 5 5 is _ “g © i hae oe Se me a" a ‘ Hie ay (cay << pales sie wii eam ts ; ae ye ae ys = ie 7 f ‘ 2 pies |, 0 mace | Ge | re + ree) <)) | owe ee: » : " inl) : pp cy tical ae A 7 ag an a ne 4 oe & na aye , We. S33