Can Sea Waves Cause Microseisms 



81 



A' is very closely related to A; if k is chosen 

 so that 



k= ji/R (19) 



and if R is large compared with the wave- 

 lengths associated with most energy in the 

 spectrum then (see I Section 3.3) 



A'(u,v) = 



J oo J co (u-U,)7t 



sin (v-v 1 )rt -i( _ ^) 



(v- Vi )ti 



du, dv. 



(20) 



where o, = a(u b Vj). In other words A' 

 is the weighted average of values of A over 

 neighboring wavelengths and directions. Since 

 u and v are proportional to the number of wave- 

 lengths intercepted by the x — and y — axis in 

 S, a "neighboring" wave component is one 

 which has nearly the same number of wave- 

 lengths, in each direction, in S. A' gives a 

 "blurred" picture of A; but the larger the side 

 of the square, the less is the blurring. The 

 region Q ' in the (u, v) — plane which corre- 

 sponds to the blurred spectrum will be almost 

 the same as the region Q corresponding to the 

 original spectrum. A' also varies slowly with 

 the time — the waves in S change gradually — 

 but this rate of change is slow compared with 

 the rate of change of the wave profile, or com- 

 pared with A'. 



The energy of the waves is given very 

 simply in terms of the spectrum function A ' ; 

 in fact, if a denotes the amplitude of the single 

 long-crested wave which has the same mean 

 energy inside S, 



p co p o 

 J»cOv»o 



A' A'* du dv 



(21) 



where a star denotes the conjugate complex 

 function (I equation [189]). a may be called 

 the equivalent wave amplitude of the motion. 



9. General conditions for fluctuations in the 

 mean pressure — We shall evaluate the mean 

 pressure p at the base of the gravity-layer, i.e. 

 at a distance of about V2 A g below the free sur- 

 face, over a square of side 2R. (Here A g re- 

 fers to the mean wavelength of the predomi- 

 nant components in the spectrum.) Consider 

 first the two-dimensional case. The mass of 

 water contained between the surfaces z = t, 

 and z = 1/2 lg and the planes x = ± R no long- 

 er consists of the same particles of water; but 

 it is possible to extend the analysis of Section 

 3 so as to take account of the motion across 

 the boundaries (see I Section 2.2). Provided 

 that the horizontal extent 2R of the interval 

 is large compared with l gthe effect of the flow 

 across the vertical boundaries can be neglected 

 (I Section 3.1). Further, since the motion de- 

 creases rapidly with depth the effect of flow 



across the horizontal plane z = 1/2 Ag is small. 

 The expression for the mean pressure variation 

 is therefore the same as if the free surface were 

 the only moving boundary: 



2 r R 



Similarly in the three-dimensional case 

 P-Po 



(22) 



P 



J^gl, 



s Vs l" f"' AC ' dxdy ' 



(23) 



that is 

 P-Po 



XgL = 



CO co 



(24) 



>t 2 4R 2 J J 



l A C 2 dx dy, 



. OO -CO 



since £' vanishes outside the square S. Now 

 the expression on the right-hand side is closely 

 related to the potential energy of the motion 

 X, ', and can be simply expressed in terms of 

 the Fourier spectrum — function A 1 . In fact 

 (I Section 3.2) 



CO < 



/ / 



y 2 C 2 dx dy = 



(25) 



R( n/k) : 



co a 



I I 



(A'A 1 * + A'AL e 2lGt ) du dv 



