CAN SEA WAVES CAUSE MICROSEISMS? 



By M. S. Longuet-Higgins 



Trinity College at Cambridge 



Abstract — This paper is an exposition of the 

 "wave interference" theory of microseisms. 

 Simple proofs are given of the existence, in 

 water waves, of second-order pressure fluctua- 

 tions which are not attenuated with depth. 

 Such pressure fluctuations in sea waves may be 

 sufficiently large to cause microseisms. The 

 necessary conditions are the interference of 

 opposite groups of waves, such as may occur in 

 cyclones or by the reflection of waves from a 

 coast. 



Introduction — It has long been known that 

 there is some connection between certain types 

 of microseisms and deep atmospheric depres- 

 sions over the ocean; and the similarity be- 

 tween microseisms and sea waves — their 

 periodic character and the increase of their 

 amplitude during a "storm" — naturally sug- 

 gests some causal relation between them. But 

 until recently there have seemed to be many 

 difficulties, both theoretical and observational, 

 to supposing that sea waves could, by direct 

 action on the sea bed, be the cause of all these 

 microseisms ; for the latter have been recorded 

 while the corresponding sea waves were still in 

 deep water, whereas theory seemed to show 

 that the pressure fluctuations associated with 

 water waves were quite insufficient, at such 

 depths, to produce any appreciable movement 

 of the ground. 



However, recent theoretical work in hydro- 

 dynamics has altered this situation: Miche 

 (1944), in quite another connection, discovered 

 the existence, in a standing wave, of second 

 order pressure variations which are not attenu- 

 ated with the depth; a much shorter demon- 

 stration of this result was given by 

 Longuet-Higgins and Ursell (1948), and the 

 result was extended by the present author 

 (1950) to more general systems of waves. In 

 the latter paper it was shown that such pres- 

 sure variations may be quite sufficient, under 

 certain circumstances, to produce the observed 

 ground movement, the chief conditions re- 

 quired being the interference of waves of the 

 same wavelength, but not necessarily of the 

 same amplitude, travelling in opposite direc- 

 tions. This, then, may be called the "wave in- 

 terference theory." 



In the latter paper (which will be referred 

 to as I) the results on which the theory depends 



were derived in a general and concise form, 

 with detailed proofs. In view of the interest 

 of the subject it seems desirable to clarify the 

 main ideas behind the theory and to discuss 

 further some of the more unexpected results. 

 This will be attempted in the present paper, in 

 which we shall rely as far as possible on physi- 

 cal reasoning, and refer where necessary to the 

 former paper for rigorous proofs of the results 

 quoted. We shall conclude with a brief histori- 

 cal review of the theory. 



1. The importance of the mean pressure — Let 

 us suppose that seismic waves are to be genera- 

 ted by some kind of oscillating pressure distri- 

 bution acting on the surface of the earth or of 

 the sea bed. If the period of the oscillation is 

 T, and the corresponding wavelength of seismic 

 waves is L, then the pressure distribution over 

 an area whose diameter is small compared with 

 L may be regarded as being applied at the same 

 point, so far as the resulting disturbance is 

 concerned; for the time-difference involved in 

 applying any pressure at another point of the 

 area would be small compared with T. Hence 

 the resulting disturbance is of the same order 

 of magnitude as if the mean pressure over the 

 area were applied at the point. Now the wave- 

 length of a seismic wave is many times that of 

 a gravity-wave (sea wave) of the same period. 

 It is therefore appropriate to consider the pro- 

 perties of the mean pressure, over a large num- 

 ber of wavelengths, in different kinds of 

 gravity-wave. We shall first consider some 

 very special but physically interesting cases, 

 when the waves are perfectly periodic and the 

 wave-train is infinite in length. It will be as- 

 sumed for the moment that the water is incom- 

 pressible. 



2. The progressive wave — Consider any peri- 

 odic, progressive disturbance which moves, un- 

 changed in form, with velocity c (see figure 1). 

 Let p (t)' denote the mean pressure on a fixed 

 horizontal plane (say the bottom) between two 

 fixed points, A, B, separated by a wave length 



We may show that p (t) is a constant. 

 Let A and B denote the points, separated from 

 A and B respectively by a distance ct. Then 

 since the motion progresses with velocity c the 

 mean pressure over A'B' at time t equals the 

 mean pressure over A B at time 0, i.e. p (0) ; 



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