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DIRECTIONAL RECORDING OF SWELL FROM DISTANT STORMS 567where c = phase velocity, and W = wind speed. Actually 2,B varies from about 600 to about1500, so there is a factor flfx of between 4 and 10 to take spreading into account, dependingon c/W.(c) AttenuationAt frequencies above 50 c/ks the observed ridge values deviate from Darbyshire's curves.We interpret this departure as the result of attenuation. The break-off point is remarkablysharp in most cases, thus indicating that attenuation is negligible below 50 c/ks and predominantabove 60 c/ks. The result is of interest inasmuch as our observations refer to themost distant sources yet investigated. Clearly it would be preferable to measure the attenuationdirectly from a comparison of associated ridge cuts at various distance, A; for the timebeing we shall have to be content with inferring the attenuation from a comparison of thepredicted spectrum near the source and the observed spectrum at a distance.Empirical knowledge concerning attenuation is virtually non-existent. Darbyshire(1952, p. 320) suggests a relation0-27 =fla aO gto account for attenuation in the Atlantic; here ao is the amplitude at the source, and athe amplitude after the waves have travelled for a time t.For small attenuation, /Ia < 1; for large attenuation fla I 1. Set = 4 x 105 S. f = 0.05 c/sand 2a2= E(f) Af - 1 cm2(c/ks) - (5 c/ks)= 5cm2 for the case of small attenuation. Thisgives /Ia = 0*04, so that the attenuation is small, as expected. We expect the attenuationto be large for t = 4 x105 S, f = 0.07 c/s, 1a2 20 cm2 (c/ks)-1 (Sc/ks) 100 cm2; thesevalues give /Ia = 0-5. The conclusion is that the onset of appreciable attenuation as indicatedon figure 40 is not out of line with previous experience.14. MEAN MONTHLY SPECTRAAverage monthly spectra with great statistical reliability are plotted on figure 41. Thenumber of daily spectra comprising the monthly means and the degrees of freedom, v, of thecombined spectra are as follows:May June Aug. Sept. Oct.record days 12 13 18 23 16v 640 710 990 1250 880The proportional variance is 2/v, and the 95 % uncertainty limits are roughly 1 ? 1(8/v).The most remarkable feature is the flatness of the spectrum at low frequencies, resemblingthe output of a 'white-noise' generator. There is always the suspicion that this might be anoise level introduced into the recording or analysis and therefore carry no geophysicalsignificance. However, the following evidence indicates the low-frequency portion of thespectra to be real: (i) at the high-frequency limit the raw spectrum of bottom pressure islower by an order of magnitude than it is in the flat portion; (ii) previous analyses to frequenciesas low as 0 3 c/ks, with the use of specially designed low-frequency instruments,gave results consistent with the present analyses (Snodgrass et al. I962); (iii) the coherencebetween different instruments is high at the low-frequency limit (figure 4).69-2
568 W. H. MUNK, G. R. MILLER, F. E. SNODGRASS AND N. F. BARBERStill there remains some concern. Consider for example the spreading of the spectrumdue to the limited duration of the record. The time record may be considered as a record ofinfinite length, multiplied by a functionf(t) = 1 between t= --IT to t= +IT,and zero otherwise. The Fourier transform of this function is Tsin q/qS, with0 iTf T - afNAt - jiTNx, X =f/1fNwherefN = (2At) -I is the nyquist frequency for a sampling rate At, and N the number ofobservations in the time series. Then if E(0) is the power spectrum of the infinite record, thespectrum for a record of length T is given by|E(o6-so)12n, -do100_-- X _- - - - - - ~~~~~~~~~~~~~~~~~~~~~OCT_,_=MAY.10- 95% CONFIDENCE LIMIT - - SEPT.............AUG.012 lJUNE010.01 ';-~2z~----..2z-:-T---zzr.~. -0001 L -I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~AUG....* 7 E ...... - * ~~~~~~~~~~~~~~~~~~~~0 20 40 60 80 100 120c/ksFIGURE 41. The average spectra for all records during the months of May, June, August, September,and October 1959, respectively. The 90 percent confidence limits for May are shown; for othermonths the expected variance is even smaller.Suppose E(0) is constant in the frequency interval xi to x2 nyquists, and zero elsewhere.We wish to compute the spectrum at X3, well outside the range xl to x2. This permits us toreplace sin2 b' by its mean value, j. The computed spectrum at xi is then given by1 X'-XXI x-2dx= N( 1 I_ )T2NJX2_X17 X3-X1 X2-X1/The expression in the parenthesis contains a numerical factor that increases with the width,x2-xI, of the primary spectral band, and with the proximity of xl to this band. The 'contamination'varies inversely with the record length. For our analyses N is usually 3000,and the contamination from the energetic band into the low frequencies is of the order 10-4.If the energetic band is coherent at the three instruments, then the contamination should becoherent also, and the foregoing argument (iii) is not convincing.Since the observed energy density at low frequencies is typically 10- times the peakenergy (figure 41) the contamination appears to be barely avoided. To check this conclusion,we have taken the very long record of 16 October (N =9000), split it into three equallengths, and compared the ratio of low-frequency energy density of the long record to the
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Directional Recording of Swell from
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