Directional Recording of Swell from Distant Storms - Department of ...

518 W. H. MUNK, G. R. MILLER, F. E. SNODGRASS AND N. F. BARBERBecause 0 is the mean of the actual phase differences 01 and 02, the direction a that one woulddeduce from 0 lies between the true directions al and a2. But because R is less than unity,one can infer that the waves do not come as a pencil beam from any single direction. In factif R is near to unity, one can write the angular spread asAa -1- (1 R2)1/(irkD cos),sina OS/(2irkD).For peak A, we find R2 0 5, q 30?, 2irkD = 2X2; hence a - 14? and Aoa - 40?.For peak B the coherence is lower, but the wavelength shorter. We find R2 - 0 4, q - 120,a --37, Aa - 320.The interpretation of beam widening from a single pair of instruments is rather hazardous.Loss of coherence can be due to many different circumstances: instrumental noise, faultyrecording, clock errors, and lack of stationarity (such as the variation in depth during a tidalcycle). In this sense the beam width inferred here must be regarded as an upper limit.In any case a rather better understanding of the wave system is reached by considering allthree instruments.(e) Di'spersionComparison of the spectra of three successive days shows a progressive shift of the peakstoward high frequencies (figure 5). The frequency of peak A increases by 5 c/ks in a day;peak B increases by 8 c/ks per day. This progressive increase in frequency is characteristicof all records. On figure 17 peaks A and B appear as slanting ridges on a contour chart ofspectral density as a function of frequency and time. The two ridges slant at different anglesand intersect on 30 October. The ridges are found to be remarkably straight, so that df/dtis nearly constant for any one ridge.The ridges can be accounted for in terms of classical wave theory. Let x be the distancefrom a source to a recorder, to, the time of generation, and t, the time of recording. The sourceis assumed a point in space-time. This implies that linear dimensions are small compared totravel distances, and duration small compared to travel time.The group velocity, V(f), is the speed with which some frequencyf, is propagated fromsource to receiver: V(f) x/(t- to). (5.4)In deep water V(f) = g/47af. (5.5)It follows that f g(t -to) /4irx, (5.6)so that on a plot off against t a single event lies along a straight line with slopedf/dt - g/47Tx.The intercepts withf 0 occurs at t = to. Each of the ridge lines can then be immediatelyassociated with a source of known time and distance. For peaks A and B the results are:A: to = 2300 P.D.T. on 22 October, A = 590, 0 2450 T,B:to 1200P.D.T.onl8October, A 960, 0 1950T.Here A is the distance expressed in terms of the angle subtended at the earth's centre betweensource and receiver. The directions and widths of the sources have previously been estimated.Evidently the two storms are located in widely different areas of the Pacific Ocean.