82 



Symposium on Microseisms 



where A I stands for A' ( — u, — v), and is the 

 amplitude of the wave component opposite to 

 A(u, v). On substituting in (24) we have 



P - P 



T - /2 g A 



p ° e 



oo a 



/ / 



(26) 



R J J 



- oo - oo 



, 2iot 



2 A'AL e du dv 



This shows that fluctuations in the mean pres- 

 sure p arise only from opposite pairs of wave 

 components in the spectrum ; that the contribu- 

 tion to p from any opposite pair of wave com- 

 ponents is of twice their frequency and pro- 

 portional to the product of their amplitudes ; 

 and that the total pressure fluctuation is the 

 integrated sum of the contributions from all 

 opposite pairs of wave components separately. 



The necessary condition for the occurrence 



of second-order pressure fluctuations of this 

 type is, therefore, that the sea disturbance 

 should contain some wave-groups of appre- 

 ciable amplitude which are "opposite," i.e. such 

 that part at least of the corresponding region 

 in the Fourier spectrum is opposite to some 

 other part. For example, if Q lies entirely on 

 one side of a diameter of the (u, v) — plane, 

 the mean pressure fluctuation, to the present 

 order, vanishes. 



An important case is when the disturb- 

 ance consists of just two wave groups, cor- 

 responding to regions Q i and Q 2 , and of equiv- 

 alent amplitudes aj and a 2 (see figure 6). 

 Q i- and Q 2 . » denote the regions opposite to 

 Qi and Q2 and Q12 and Q12 denote the re- 

 gions common to Q 1 and Q 2 - and to Q 1. and 

 Q 2 respectively. Effectively, then, the inte- 

 gration in (26) is carried out over the two 

 regions Q i2 and Q 12- • When the spectrum is 

 narrow an order of magnitude for the integral 

 on the right-hand side of (26) can be obtained. 

 It may be shown (see I Section 5.2) that 



Po 



'2 g ^ 



'12 



Q 12 /&! 



Q 2 ) ; 



ke 2io 12 fc 



(27) 



where a 12 is the mean value of a in Q 12 . Thus 

 the mean pressure on S increases proportion- 

 ately to the square root of the region Q12 of 

 overlap of the wave groups, and inversely as 

 the square root of Q 1 and Q 2 separately, for 

 fixed values of a 1 and a 2 . 



10. Calculation of the ground movement — In 



order to estimate the movement of the ground, 

 at great distances, due to waves in a storm area 

 A, we suppose the storm area to be divided up 

 into a number of squares S of side 2R such that 



S contains many wavelengths Kg of the sea 

 waves, but is only a fraction, say less than 

 half, of the length of a seismic wave Xs in the 

 ocean and sea bed. This we may do, since the 

 wavelengths of seismic waves are of the same 

 order as the wavelengths of compression waves 

 in water ; therefore lg/l s is of the order of 

 10" 2 . The mean pressure or total force on 

 the base the gravity-layer can be calculated as 

 in Section 9; the vertical movement of the 

 ground 5 ' due to the waves in this square is of 

 the same order as if the force were concen- 

 trated to a point at the center of the square, i.e. 



6'~4R 2 



iX 



12 



(Q 12 /Q 1 Q 2 ) /2 k W (2(J 12( r) e 2l0 12 t 



(28) 



where r is the distance from the center of the 

 storm and W (a, r)e 10t is the movement of 

 the ground at distance r due to a unit pressure 

 oscillation eicrt applied at a point in the mean 

 free surface. The pressure can be considered 

 to be applied in the mean free surface rather 

 than at the base of the gravity-layer, since the 

 latter is relatively thin compared with the 

 length of the seismic waves. To find the total 



displacement from the storm we may add the 

 energies from the different squares S, on the 

 assumption that the contributions from the 

 different squares are independent. Since 

 there are A/4R such squares in the whole 

 storm area, this means that the disturbance 8 ' 

 fron^each individual square is to be multiplied 

 by A^ /2R. Hence we have 



-4 it 



12 



2 (AQi 2 /Qi^2^ 2 * (2 ° 12, r) 



2ia 12 t 



(29) 



To calculate W (0, r) we may consider the 

 disturbance due to a force applied at the sur- 

 face of a compressible fluid of depth h (rep- 

 resenting the ocean) overlying a semi-infinite 

 elastic medium (representing the sea bed). Al- 

 though this model takes no account of varia- 



tions in the depth of water, or of the propaga- 

 tion of the waves from the sea bed to the land 

 or across geological discontinuities, it can nev- 

 ertheless be expected to give a reasonable esti- 

 mate of the order of magnitude of the ground 

 movement. 



