76 



Symposium on Microseisms 



(a). 



lb) 



Figure 3. Two phases of the interference 

 between two waves of equal length but dif- 

 ferent amplitudes a, and a, travelling in 

 opposite directions. The profile of the 

 first wave (dashed line) is reduced to rest 

 by superposing on the system a velocity -c; 

 the second wave appears to travel over the 

 first with velocity -2c. The full line shows 

 the final wave form. 



the external forces acting on the mass are, 

 first, that due to gravity, which is constant, 

 (the total mass being constant) ; secondly the 

 force from the atmosphere, which is also con- 

 stant, since the pressure p at the free sur- 

 face, if constant, will produce a constant down- 

 wards force lp Q ; thirdly the forces across the 

 vertical planes, which must have zero vertical 

 component, the motion being symmetrical 

 about these planes; and, lastly, the force on 

 the bottom, which equals I p. Since all the 

 other external forces besides k p are constant 

 it follows that p must fluctuate with the time. 

 In figures 2(a) and 2(c) the mass of water 

 above the mean level is proportional to the 

 wave amplitude a; since it is raised through 

 a distance of the order of a, the displacement 

 of the centre of gravity, and hence the mean 

 pressure fluctuation, is proportional to a 2 . 



An explicit expression for p can easily 

 be derived. Let z denote the vertical coordi- 

 nate of a particular element of fluid of mass 

 m, so that z is a function of the time t and of, 

 say, the position of the fluid element when 

 t = o. If F denotes the vertical component of 

 the external forces acting on the mass of wa- 

 ter, we have, on summing the equations of mo- 



tion for each element of fluid, and cancelling 

 the internal forces : 



F = D(m 



dt 2 



) = 



df 



(£m z) (1) 



the summation being over all the particles. The 

 expression in brackets on the right-hand side 

 will be recognized as g " ' times the potential 

 energy of the waves ; in an incompressible fluid 



e J> 



l A £ dx + constant (2) 



where x is a horizontal coordinate, p is the 

 density, X is the wavelength, and X, (x, t) is the 

 vertical displacement of the free surface. But 

 by our previous remarks 



F = I (p -p - Pg h ) . 



where h is the mean depth of water, 

 equating (1) and (3) we find 



(3) 

 On 



h = 



I Jc 



C 



dx. 



(4) 



p a^ 



Now for a standing wave 



Z, — a cos kx cos o t (5) 



where k = 2 jt/X and a = 2k/x (t being the 

 wave period), and higher-order terms have 

 been omitted. On substituting in (4) we find, 

 after simplification, 



g h = 



M a 2 O 2 



cos 2 at (6) 



This shows that, to the second order, the mean 

 pressure p fluctuates sinusoidally, with twice 

 the frequency of the original wave, and with 

 an amplitude proportional to the square of the 

 wave amplitude. The pressure fluctuation is 

 independent of the depth, for a given wave 

 period, though of course the depth enters into 

 the relation of the wave period to the wave- 

 length, given by 



**»r 



Grovity layer 



Base of gravity 

 layer 



Region of compression waves 



•/////////, 



Figure 4. Waves in a heavy, compressible fluid. 



