80 



Symposium on Microseisms 



— as one might expect, since a disturbance 

 could be propagated almost instantaneously 

 through this layer. At a distance of about % 

 lg from the free surface the first-order pres- 

 sure variations are much attenuated, and the 

 second-order pressure variations are practical- 

 ly those given by the incompressible theory 

 (equation [6] ). Below this level the displace- 

 ments are comparatively small, but, instead of 

 the uniform, unattenuated pressure fluctua- 

 tions in the incompressible fluid, there is now a 

 compression wave, whose planes of equal phase 

 are horizontal : the pressure field in this wave 

 is given by 



P " P„ 



g z 



(12) 



- a 



a 2 o 2 



cos 2o(z-h)/c' 

 cos 2oh/c' 



cos 2at 



very nearly, where z is the vertical coordinate 

 measured downwards from the mean surface 

 level, and C is the velocity of sound in water. 

 This wave can be regarded as being generated 

 by the unattenuated pressure variation (6). 

 There is a resonance, or "organ-pipe," effect: 

 when cos 2 a h/c ■ vanishes, the pressure on 

 the bottom (z = h) becomes infinite. This 

 happens when 



2ah/( 



U+X)ji 



(13) 



C(x, y, t) = 



R 



A (u,v) e i(ukx + vky + at) du d , 



-OO J-OO 



(14) 



where (x, y) are horizontal coordinates, k is a 

 constant and a is a function of (u, v) : 



o 2 = g k (u 2 + v 2 )' A tanh (u 2 + v 2 )' A k h (15) 



A (u, v) is in general complex, and R denotes 

 the real part. The expression under the inte- 

 gral sign represents a long-crested wave with 

 crests parallel to the line 



u x + v y = 



and of wavelength I given by 



(16) 



I = 



2n 

 (u 2 + v 2 )* k 



(17) 



that is, when the depth is (V 2 n + X A) times 

 the length of the compression wave. In gen- 

 eral, however, the displacements in the com- 

 pression wave are small, being only of the order 

 of a 2 /A c ; the displacement of the centre of 

 gravity of the layer at the surface of thickness 

 y% lg is of the order of a /Xg. This explains 

 why the compressibility of the fluid below has 

 little effect on the pressure fluctuations at the 

 base of the surface layer. 



We have then the following picture (see 

 figure 4) : there is a surface-layer, of depth 

 about 1/2 ^g> in which the compressibility of 

 the water is, in general, unimportant : this may 

 be called the "gravity-layer." Below this lay- 

 er there exist only second-order compression 

 waves, generated by the gravity-waves in the 

 surface layer, and of twice their frequency. 



8. Application to sea waves — So far we have 

 considered only the very special cases of per- 

 fectly periodic and two-dimensional waves. 

 Such waves cannot be expected to occur in the 

 ocean, although the sea surface usually shows 

 a certain degree of periodicity. We shall now 

 consider how the sea surface is to be described 

 in this more general case. 



It can be shown (See I Section 3.2) that 

 any free motion of the sea surface can be ex- 

 pressed as a Fourier integral: 



If the point P, = ( - uk, - vk) is plotted in the 

 (x, y) plane (see figure 5) the direction of the 

 vector P is the direction of propagation of 

 the wave-component and the length of P 

 equals 2it divided by the wavelength. Points 

 on a circle centre O correspond to wave com- 

 ponents of the same wavelength ; diametrically 

 opposite points correspond to waves of the same 

 length but travelling in opposite directions. 

 When the energy is mainly grouped about one 

 wavelength and direction, the complex ampli- 

 tude A(u, v) will be appreciably large only in 

 a certain range of values of (u, v), say Q, as in 

 figure 5. The narrower this region, the more 

 regular will be the appearance of the waves. 



The spectrum A(u, v) of the waves is de- 

 termined uniquely by the motion of the free 

 surface, at a particular instant, over the whole 

 plane (see I, Section 3.2) . Since we shall want 

 to consider the wave motion in only a certain 

 part of the plane, say a square S of side 2R, it 

 is convenient to define a motion X, ' which, at 

 any time, has the same value as t inside S but is 

 zero outside. Let A' be the spectrum function 

 of £', so that 



V = 



/•OO /»oO 



R 



/•<*> (18) 



A«(u,v)e i(ukx + vk y + ot) du dv 

 J-00 



