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

524 W. H. MUNK, G. R. MILLER, F. E. SNODG:RASS AND N. F. BARBERThe maximum value of this occurs at 10, mo and is numerically equal to AO, the actualpower of the swell. The choice of the b's as 71 has brought this about. But this does not preventthe spectrum E' from extending repetitively to indefinitely large wave numbers. Thisunconvincing result arises from the unconvincing assumption made at the start.(b) A best-fitting single wave trainFor local geophysical reasons one may expect to receive mainly well-directed swellsfrom distant storms. A reasonable treatment would be to fit the measured C's and Q'sas well as possible in a least-square error sense by one or more such wave trains. Now asingle wave train of power AO) wave numbers, 10, mo, would give the C's and Q's listed in(7.2), and these imply perfect coherence, which is never observed experimentally. But onemay note that the sum of the squares of the differences between the observed C's and Q'sand those expected for a single wave train isn=3 n=3H- (CO-AO)2+2 E [Cn-AOcos21T(lOXn+mO Y)]2+2 I [Qn+AOsin2ff(l0Xn+mO Y)]21 1n-3-C02 + 2 E (Cn2 + Q2) + 7A2n-3 n-3-2AO [C0+2 i Cncos 2r(lO X + nmO Yn)-2 E Qnsin 21T(l0XXn+mO Yn)].We may chose AO, 10 and mo so as to minimize this squared error. The quantity in squarebrackets is merely seven times E'(l1, mo,f0), the power density that would be predicted fromthe observed C's and Q's for wave numbers 10, mo by the conventional treatment of (7.3).The choice of A0 that minimizes H is seen to beA0 = El (l, mi0,f0). (7.5)n=3Then H= C+2 E (C2+Q2)-7 [E'(0,m0,f0)]2. (76)The choice of l0 MO' that minimizes H and so represents a 'best fitting' wave train is merelythat which makes [E'(l0, m0,f0)]2 as great as possible.- So the conventional treatment is useful after all. The indicated procedure is as follows:Compute E'(l, m,) according to (7.3), using the observed seven quantities Cn, Qn and usingtrial values for 1, m. The particular values, lo) MO, for which [E'(l, if)]2 is greatest definethe best fitting single wave train in the least-square sense; and the corresponding valueE'(l1, mo,f) is the power of the wave train. In making this choice one, of course, ignoresany 1, m values for which E' proves to be negative.An example is shown in figure 8. Writing I k sin a, m i k cos a, we have chosen variousvalues of k in the vicinity of the theoretical value of k at the frequency under consideration.For each trial value of k we computed H(k, a) for various a, and determined its minimumvalue, Hmin. (k), corresponding to a- a. The curve Hmin. (k) has a minimum at some valuek - k. In the example shown the value of ko does not differ measurably from ktheo.Furthermore, values of a0 remained unchanged throughout the interval here under con-sideration. This is essentially the result of many other such determinations; k remains closeto ktheory throughout the frequencies of the sea and swell. At the very low frequencies the