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DIRECTIONAL RECORDING OF SWELL FROM DISTANT STORMS 515the spectral values equal twice the number of data points divided by m, or about 60. For60 degrees of freedom there is a 95 % probability that the true spectral value lies between0 7 and I *5 times the computed value.5. FIRST INFERENCEIn this section we discuss the elementary interpretation of cross-spectra from a singleinstrument pair. This prepares the way for the general array theory in ? 6. In addition itprovides the basis for the analysis of the earlier records, based on the north-south pair ofinstruments only.(a) Sample spectrumFigure 4 illustrates a sample spectrum. This includes the power spectra for the south andnorth instruments, the relative coherence and phase, and the directions inferred from thephase. The spectra were computed up to 125 c/ks, but the plots extend only to 90 c/ks.Similar plots were prepared for each day. During October 1959 a third instrument wasinstalled, and three such diagrams were prepared for each day of recording, giving the correspondingrelations between the south and north instruments, the north and west instruments,and the west and south instruments, respectively. To keep the reduction of the observationsfrom becoming too much of a chore, all steps in this operation, including the plotting weredone by machines.The plotted spectra are useful for purposes of illustrating the analysis, and they permitsome early inferences to be drawn. In fact it is found that the computed spectra are not theend, but rather a starting point for further analysis, the goal being an estimate of E(f, 0, t),the time-variable distribution of energy as a function of frequency and direction. Accordinglythe machine was programed to provide the spectral matrix on punched cards in atorm suitable for further machine calculations.(b) Power spectraThe spectra of the sample are corrected for water depth in accordance with (4.1). Thepower spectra are alike in all essential details, as they should be. The essential features area flat spectral level of about 0.01 cm2/(c/ks) at frequencies below 30 c/ks; a peak at 38 c/ks,a second peak at 52 c/ks, and a gradual rise to a relatively flat spectral level between 10and 100 cm2/(c/ks) for frequencies above 70 c/ks.(c) Direction findingWe note that for peak A the north and south instruments are virtually in phase. Thiswould indicate that the wave crests are nearly parallel with the line connecting the twoinstruments. The direction normal to the instrument line is 231?T (figure 1, inset), so weconclude that the waves are approaching either from the open ocean to the south-west orfrom the shore of the island to the north-east; one might receive waves from the shore if theshore were a good reflector. For peak B, the records have a phase difference which showsthat wave crests pass over the south instrument first. They may be coming from the oceanto the south or from the shore to the east. Such ambiguities in direction are inherent in anytwo-point array; our choice of a three-point array was precisely to remove this ambiguityand so permit us to estimate thie amounlt of reflected energy.63-2
516 W. H. MUNK, G. R. MILLER, F. E. SNODGRASS AND N. F. BARBERc/ks001 kd o j ~~~~~~~NORTH180?0b900L-00-901-180010270' -WESTX9~~~~~~~~~,(f)00 (f)180' SOUTH A 00 20 40 60 80c/ksFIGURE 4. The upper curves show the surface spectra from the north and south instruments,respectively, for 26 October 1959. The next curve gives coherence, R2, and the third curve, thephase lead, 0, of the north instrument relative to the south instrument. The bottom curvesshow the direction(s) of a single point source on the seaward side of the array consistent with theobserved phase, qS(f)Consider an elementary wave train of frequencyf from direction ca measured clockwiserelative to the instrument normal (231?T). The resulting signal at the south and northinstruments can be writtenUs =A cos (27rft-q0), UN =A cos 2aft, (5.1)respectively, where 5= 2irkD sin a (5.2)
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