Can Sea Waves Cause Microseisms 



93 



fulfils A <f> - and satisfies the boundary 

 condition equation 1, to a first approxima- 

 tion if ka l and v 2 = gk . For a second ap- 

 proximation we put # = # j + y a 2 f where d ~ d i ; 

 neglecting terms of third and higher order in 



k a we obtain a 2 f = a 



sin 2ft 



The corresponding surface elevation £ = 

 C i + £ 2 ,with 



<->2 = - — k 



2 

 and the pressure 



cos 



(k: 



ft) + 



( kx + v t) 



al cos 2 (kx- ft) + 



cos 2 (kx + ft) + 2a 2 a 2 cos 2 kx 



kz 



Po + gP z "gPC' e ' - 'A P v 2 (a, 2 



2t) 



-2k; 



- 2 P aj a 2 v cos 2 ft. 



Obviously at large depths ( k z>> 1) the vary- 

 ing part of the pressure is 



p = 2 Pa 



cos 2ft, 



(2) 



which is the result obtained by Miche for a 

 standing wave system (a, = a, ). 



Considering a rather general irrotational 

 movement LONGUET-HIGGINS (1950) was 

 able to generalize equation (2) and to calculate 

 the amplitude of microseisms caused by an 

 arbitrary wave-like motion of the ocean. His 

 final formula (his equation 198) may be in- 

 terpreted in the following (inexact) way. 



The Miche force of the square 1 2 , where 

 I = the mean wavelength of the interfering 

 progressive waves, is according to (2) equal to 



2 paj a 2 f 2 I 2 



If the microscismic amplitude caused by a con- 

 centrated unit force with frequency 2 v at a 

 distance r is denoted by w (2 v , z) the total 

 amplitude will be 



2 Paj a 



W( 2 



z) 



Supposing the phases of ocean waves at points 

 separated by a distance of a wavelength to be 

 uncorrected the amplitude generated by a 

 storm with an area A will be of the order 



ft)* 



2 pa. 



W (2 f , z) 



With A= 10 J km 2 and 1=0. 25km( v =Y 2 ) the 

 vertical amplitude at a distance of 3000 km. 

 appears to be 9.4ja, which is of the order of the 

 observed amplitudes. The detailed investiga- 

 tion of Longuet-Higgins shows that this has 

 to be multiplied by a factor which depends on 

 the frequency spectrum of the wave system. For 

 instance, if the energy of the movement is uni- 

 formly divided in every direction within a 

 range of wave lengths between Aj, and l 2 this 

 factor is 



(t(^)} 



'/ 2 



the numerical value of this quantity is about 

 0.54 if >. i = 400 meters and 1 2 = 154 meters. 

 The vertical amplitude is then 5\i, and the 

 horizontal 3u. 



This theory undoubtedly explains the phe- 

 nomenon of microseisms in a straightforward 

 way. The only difficulty which it encounters 

 is the fact that microseisms occur very often, 

 while it is a matter of considerable doubt 

 whether standing waves of rather large ampli- 

 tudes are as common. 



REFERENCES 



Longuet-Higgins, M. S., and Ursell, F., Sea waves 

 and microseisms. Nature, v. 162, p. 700, 1948. 



Miche, M., Mouvements ondulatories de la mer en pro- 

 fondeur constante ou decroissante. Ann. Ponts et 

 Chaussees, v. 114, pp. 25-87, 131-164, 270-292, 396- 

 406, 1944. 



Whipple, F. J. W., and Lee, A. W., Notes on the theory 

 of microseisms. Mon. Not. Roy. Astr. Soc, Geophys, 

 SuppL, v. 3, pp. 287-297, 1935. 



Discussion from the Floor 



Haskell. (Questioning Longuet-Higgins.) 

 Ocean waves are coherent over more than just 

 one wave length, so shouldn't the area of gen- 

 eration be subdivided into areas that are larger 

 than one wave length on a side — perhaps the 

 wave lengths? (Longu&t-Higtgins answered* 

 perhaps so.) 



Longuet-Higgins. (In answer to Press's ques- 

 tion, "what if the wave periods on the surface 

 occur off the peak of your resonance curve?") 

 The sea waves must be considered as possessing 

 not a single period, say 12 seconds, but a fre- 

 quency spectrum of a certain width, say 8-16 

 seconds (the pressure fluctuations would then 

 be from 4 to 8 seconds period.) The spectrum 

 of the microseisms should be a combination of 

 the spectrum of the pressure variations and 

 that of a response curve. If the most promin- 

 ent period of the pressure variations occurs off 

 the peak of the resonance curve, the most prom- 

 inent period of the microseisms would be ex- 

 pected to be displaced towards the peak. 



