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

DIRECTIONAL RECORDING OF SWELL FROM DISTANI STORMS 513Pressures at all three instruments were sampled once every 4 s. This was the limiting speedof the punched paper tape equipment. The Nyquist frequency is half of this sampling frequency,or one cycle in 8 s = 125 cycles per kilosecond (c/ks). One must ensure that thereis negligible power at frequencies higher than 125 c/ks. This was accomplished by placingthe instruments on the bottom at a depth of 100 m. The resulting 'low-pass filtering' wasthen adequate for our purpose.tLet pga cos (2irft) designate the pressure fluctuation at a fixed depth just beneath thesurface, resulting from a wave of amplitude a and frequencyf. Then the associated pressurefluctuation on the sea bottom at depth h is rpga cos (2fft), wherer(f, h) os 1 27f2 ggktanh(2Irkh), (4.1)r(,)cos h(2-ffkh)'with k(f, h) denoting the reciprocal of wavelength.4 The following table gives a few selectedvalues:period (s) 100 40 20 13-3 10 8f (c/ks) 10 25 50 75 100 125r2 (f lOOm) 0.99 0-76 0 30 0 04 0-0016 4 x 10-6The attenuation of the frequencies higher than 125 c/ks is therefore sufficient to avoid appreciablealiasing at the frequencies ofinterest, 100 c/ks and less.All bottom spectra have been divided by r2(f) to yield surface spectra.(c) Suppression of tidesTides introduce high energy into low frequencies, and this gives rise to undesirable sidebands. By using a numerical convolution, the tidal energy was reduced by a factor 104before spectral analysis. The effect of the filter at the lowest two computed frequency bandsis as follows:f (c/ks) 1b25 25energy response 0-263 0-998At higher frequencies the effect of filtering is altogether negligible. For further details werefer to Munk, Snodgrass & Tucker (I959).(d) The spectral matrixThe spectral analysis of the edited and filtered records was performed numerically on anIBM 709 at the Western Data Processing Center of the University of California, LosAngeles. The procedure for single records has been discussed by Blackman & Tukey (1958).The cross-spectral analysis between different records follows along quite similar lines, anda collection of pertinent formulae can be found? in Munk et al. (I 959). Here we shall merelyindicate the principles of the analysis.t A depth of this order also satisfied the requirement that the refractive turning of the wave train shouldbe small (see 9(a), Refraction).+ The term wave number means the reciprocal of wavelength throughout the present paper and not27Tk as is the usage in some texts.? One change in notation has been found advisable: the present r corresponds to - T in the reference.63 VOL. 255. A.