Can Sea Waves Cause Microseisms 



79 



front arrived over the hydrophone the appear- 

 ance of the pressure record was changed. At 

 moderate depths there were not only first-order 

 pressure fluctuations from the incident and the 

 reflected wave, but also considerable second- 

 order pressure fluctuations, of twice the funda- 

 mental frequency. At greater depths the first- 

 order pressure fluctuations become negligible 

 and only the pressure fluctuations of double 

 the frequency remained. The amplitude of 

 these was in good agreement with equation (6) . 

 When the barrier was removed, and the rear 

 end of the reflected wave train had passed the 

 hydrophone, the second-order pressure fluctua- 

 tions rapidly died out. 



Interference between waves of unequal 

 amplitude was obtained by placing in the tank 

 a vertical barrier extending only to a certain 

 depth below the free surface, which allowed 

 the waves to be partly reflected and partly 

 transmitted. The coefficient of reflection from 

 such a barrier is known theoretically for dif- 

 ferent ratios of the depth of the barrier to the 

 wavelength of the waves, and it was verified 

 that the amplitude of the second-order pressure 

 fluctuations was proportional to the amplitude 

 of the reflected wave. Indeed this property 

 seems to provide a convenient method of ac- 

 tually measuring the coefficient of reflection 

 from different types of obstacles or from plane 

 beaches. 



Since standing waves produce only second- 

 order pressure fluctuations below moderate 

 depths one would expect that, if pressure fluc- 

 tuations were induced deep in the water, stand- 

 ing waves of half the frequency would be pro- 

 duced at the surface. An experiment of this 



► A 



Figure 10. The spectrum representation of 

 incident and reflected wave-groups. 



kind was in fact performed by Faraday 

 (1831) ; (see Section 13 of the present paper) 

 who produced standing waves, of half the fun- 

 damental frequency, by means of a vibrating 

 lath inserted in a basin of water. Faraday re- 

 marked that the general result was little in- 

 fluenced by the depth of water: "I have seen 

 the water in a barrow, and that on the head 

 of an upright cask in a brewer's van passing 

 over stones, exhibit these elevations." (1831, 

 footnote to p. 334). The present author has 

 observed a similar phenomenon on board ship : 

 a pool of water on deck, when excited by the 

 vibration of the ship's engines, sometimes 

 shows a standing-wave pattern whose ampli- 

 tude gradually builds up to a maximum, and 

 then collapses; the process is repeated indefi- 

 nitely. 



7. Standing waves in a compressible fluid — 



The water has so far been assumed to be in- 

 compressible, and we have seen that in this 

 case the pressure fluctuations below about half 

 a wavelength from the surface occur simul- 

 taneously at all points of the fluid. But this 

 can only be true if the least time taken for a 

 disturbance to be propagated to the bottom 

 and back is small compared with the period 

 of the waves. In the deep oceans, where the 

 speed of sound is about 1.4 km/sec and the 

 depth may be of the order of several kilometers, 

 this time may be several seconds. Thus the 

 compressibility of the water must be con- 

 sidered. 



The first-order theory of waves in a heavy, 

 compressible fluid (in which all squares and 

 products of the displacements are neglected) 

 indicates that water waves of a few seconds' 

 period fall into two classes (Whipple and 

 Lee 1935). On the one hand there are waves 

 approximating very nearly to ordinary surface 

 waves in an incompressible fluid, in which the 

 particle displacement decreases exponentially 

 downwards, to first order; these may be called 

 gravity-waves. On the other hand there are 

 long waves controlled chiefly by the compres- 

 sibility of the medium and hardly attenuated 

 at all with depth ; these may be called compres- 

 sion-waves ; their velocity is nearly the velocity 

 of sound in water. The wavelengths of a grav- 

 ity-wave and a compression wave will be de- 

 noted by I g and A c respectively. For waves 

 of period 10 sec. "kg/Xc is of the order of 10 " 2 . 



However, the pressure variations which 

 are of interest to us at present are of second 

 order. To investigate the effect of the com- 

 pressibility, therefore, a complete _ example, 

 namely a motion which in the first approxima- 

 tion is a standing gravity-wave, has been 

 worked out in full to a second approximation 

 (I Section 4). The result is a*s follows. 



Near the free surface, that is within a dis- 

 tance small compared with A c , the waves are 

 unaffected by the compressibility of the water 



