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

DIRECTIONAL RECORDING OF SWELL FROM DISTANT STORMS 563Let c(t) -(t) cos2lTfot, s(t) - q (t) sin2lTfot,foao+bot.If we choose the 'heterodyning frequency 'f0 close tof and increasing at approximately thesame rate asf, that is, a l ao and b bo, then a(t) and b(t) contain termsf-f0 andf+f0;the latter are removed by smoothing, and we are left with a difference tone that is not blurredby dispersion.The method has been applied to the very narrow peak of 12'4 May and to the event of8-4 October which was continuously recorded for nearly a day (figure 19). Values of aoand bo are then chosen to givefo(t) along the ridge. Let -c(t), s(t) be smoothed versions ofc(t), s(t); more precisely, frequencies in excess of the width of the spectral peak are removed.It will be shown that the spectral distribution of the record x(t) relative to the sliding peakfrequencyfo(t) is given byS11 ( fo a) = 2SEQ (a) ? 2Q3i (a), (12.1)where SE3 is the spectrum of c(t) and Q5 the quadrature spectrum between -(t) and s(t).Consider the correlation(b) Heterodyne theorypcc( T) = (q(t) q (t- T) cos 2iqfo Tcos 21Tfo(t- T)>j~ W cos 27ffo T- p (T) cos 21Tfo T,where the approximation refers to the smoothing of c(t). Similarly ps( T) = pj(p s(T) = (q(t)q(t -T)cos27Tfotsin27Tfo(t -T)>p 7,1( T) sin 21Tfo T.The Fourier transforms of the autocorrelations pEc( T) or pss ( T) give the spectraT) andSCca) 2 j'P3j(T)cos 2rtaTd T- 2 fjapV,1, ( T) cos 21Tfo Tcos 2ira Td T= p,{p ,(T) [cos2ir(fo+ a) T-Pcos2ir(fo-a) T] dT-1w Sv, (fo +a) + S,(fo a)] (12.2)and similarly C3i (a) p= 2T) cos 2irfaTd T = 0,_00QEQ(a) = 2f pcs(T) sin 21TaTd T= s..C w rfo + ) - S. C fo - a) I (12.3)Equation (12@1)follows from (12-2) and (12*3). We note that the two spectra should beequal and the co-spectrum should vanish, subject of course to the usual statistical fluctuations.