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

DIRECTIONAL RECORDING OF SWELL FROM DISTANT STORMS 517is the phase difference between the two instruments which are separated by a distance D.So there are always two possible wave directions, namely ax and 1800o a that suit ourobserved phase difference 5S.The directions plotted at the bottom of figure 4 were computed from qS in accordancewith (5.2) but adopting in each case only the direction from the seaward side of the instrumentpair. For each computed value of ax there is an additional solution a' 1800 cx,so that for each computed direction, 0 = 231?0 +c, as plotted, there is another possible direction0' - 282? -0, from the shoreward side of the array.A further ambiguity arises when the waves are so short that the difference in phase at thetwo instruments exceeds half a cyclet. In comparing records one can estimate only theprincipal value of 0, that is, its value in the range ?i -. Its actual value may differ from thisby any multiple of 2iT. So there may be numerous possible values of the direction ax, given bysin acj (q + 2ifj)/(2iTkD), (5.3)wherej is 0, ? 1, + 2, etc., up to the largest value for which sin aj still lies between ? 1.The directions plotted at the bottom of figure 4 were computed in accordance with (5.3),ignoring those directions from the landward side of the pair of instruments. The directionis unique for frequencies below 55 c/ks, but two solutions and then three solutions appear asone moves to higher frequencies. For the higher frequencies it seems reasonable to adopt thecurve of directions that is continuous with the unique curve of directions at the lowerfrequencies. In figure 4, 00(f) is the obvious choice between the three possible directions atthe high-frequency limit. In practice we have found it possible to determine wave directionsfor wavelengths as short as !D (as compared to the limit 2D for experiments conducted ata single frequency). This illustrates the benefit of working in a frequency continuum.(d) Beam widthThe coherence R is a guide to the angular spread of the waves. Consider for instance twowave trains having the same amplitude, a, but slightly different frequencies, f1 and f2,coming from directions ac, and c2 respectively, relative to the normal of the pair of instruments.The corresponding phase differences are 01 and 02, where01 = 2rk1 D sin a1, 02qS 21k2D sin a2.Thus the wave signals at the instruments may be written1N= a cos 2iTf t+a cos 2rf2 t,=, = acos (2irf1t-01) +acos (2lTf2t-q02).On computing the quantities defined in ?4(d) one findsPNN(T) = PSS(T) - a2 cos 2Tf]1 r + a2 cos 2irf2 r,PSN(T) -a2coqS X cos 2rfff r + a2 COS 02cos 2irf2 r.If now we ignore the slight difference betweenf1 andf2, writing each as f, we getCSS CNN= 2a2,and in particular R-cos 2CSN a2(cos l + cosS 02), QSN = a2(sin 01 + sinOS2),(q1--q2), = 1(A1 +02)t There is a close analogy with the problem of aliasing (?4 (b)) when the frequency exceeds half the samplingfrequency.