Can Sea Waves Cause Microseisms 



77 



o 2 = 



g k tanh k h 



(7) 



There is a close analogy with the motion 

 of a pendulum (see figure 2). In a complete 

 cycle the bob of the pendulum is raised and 

 lowered twice, through a distance proportional 

 to the square of the amplitude of swing, when 

 this is small. The only forces acting on the 

 pendulum are gravity, which is constant, and 

 the reaction at the support. Hence there must 

 be a second-order fluctuation in the vertical 

 component of the reaction at the support. 

 Furthermore the reaction will be least when 

 the pendulum is at the top of the swing (the 

 potential energy is greatest) and will be great- 

 est when the pendulum is at the bottom of its 

 swing (the potential energy is least) . 



It will be noticed that the above analytical 

 proof does not necessarily involve the idea of 

 the centre of gravity, whose vertical coordinate 

 z is defined by 



(Em) 



= X (m z) 



(8) 



The theorem on the centre of gravity that was 

 used previously is in fact usually derived from 

 equation (1) : but in the present proof we have 

 appealed directly to the original equations of 

 motion for_the individual particles, without in- 

 troducing z. 



4. Two progressive waves — The above proof 

 can easily be extended to the more general 

 case of two waves of equal period but unequal 

 amplitude travelling in opposite directions. 

 For, such a disturbance is exactly periodic in 

 space. Thus we may consider a region one 

 wavelength in extent, as for the standing wave. 

 This will not always contain the same mass of 



Figure 5. The spectrum representation of a 

 wave group. 



water; but, owing to the periodicity, the ver- 

 tical reaction on the bottom due to the flow of 

 water across one vertical boundary will be ex- 

 actly cancelled by that due to the flow across the 

 opposite boundary (see I Section 2.2) ; thus 

 equation (4) is still exactly valid. The wave 

 profile in this case is represented by 



C = a, cos (kx -Ot) + a 



and so 



% (a/ + 

 giving 



2 cos ( kx + at ) 

 (9) 



x I 1 .* < !d * 



+ 2a x 



2 ot) 



(10) 



- g h 



a 2 o 2 cos 2ot (11) 



The mean pressure fluctuation on the bottom 

 is therefore proportional to the product of the 

 two wave amplitudes aj and a 2 . When these 

 two are equal (a, = a 2 = -I a) we have the 

 case of the standing wave, and when one is 

 zero (a, = a; a 2 = 0) we have the case of 

 the single progressive wave. 



A physical explanation of this result may 

 be given as follows. Suppose that one of the 

 waves, say the wave of amplitude a,, is re- 

 duced to rest by superposing on the whole sys- 

 tem a velocity - c in the direction of x decreas- 

 ing (this will not affect the pressure distribu- 

 tion on the bottom). The second wave will 

 now travel over the first with a velocity -2c. 

 The crests of the second wave will pass alter- 

 nately the troughs and the crests of the first 

 wave - each twice in a complete period. Fig- 

 ure 3 shows the two phases. One may pass 

 from figure 3(a) to figure 3(b) by transferring 

 a mass of fluid, proportional to a 2 , from a 

 trough to a crest of the original wave, i.e. 

 through a vertical distance proportional to a j 

 (the transferred mass does not of course con- 

 sist of identically the same particles of water). 

 The vertical displacement of the centre of grav- 

 ity of the whole mass is therefor shifted by an 

 amount proportional to a , a 2 ; and hence the 

 fluctuation in p is also proportional to a, a 2 . 



5. Attenuation of the particle motion — The 



fact that there is a pressure fluctuation on the 

 bottom even in deep water does not, however, 

 mean that there is movement at those depths. 

 In fact it may be shown (Longuet- 

 Higgins 1953) that in exactly space -periodic 

 motion, whether in a simple progressive wave 

 or a combination of such waves, the particle 

 motion decreases exponentially with the depth, 

 apart from a possible steady current. Now 

 if the velocities at great depths are zero, or 

 steady, it follows from the equations of mo- 

 tion that the pressure-gradient must be inde- 

 pendent of the time. Thus if there is a pres- 



