On Theories of the Origin of Microseisms 



to the expressions for u and w we obtain the 

 motion at the bottom of the ocean, generated 



119 



by a force Q concentrated in one point of the 

 surface. We readily find 



vO -ivt f sin 2 3 f /b 2 . \ -% 



e I ^cos 2(3 — - sin 2 8 -2 cos 3 



;b3 Pl J Q N | P \a 2 P >/ 



Ji ( v — s i n P] d s i n P 



vO -ivt/ sinB 

 i — e J 



27ib 3 Pl 



JsinB / r \ 

 J Q [ v — sinS I 



d sin 3 



It can be shown that the main value of the zeroes of N. The microseismic movements 

 these types of integrals is for large values of r at large distance from the generating force 

 contributed by the residues of the integrand at Qe" ivt are therefore: 



u = -2 m W m si 



"3 m {c° s 23 m (^ - sin 2 3 m )" 2 - 2 cos3 m | - w = ^ W n 



w m = o 



Pl b^ (27ir)^ 



(sin3 m ) 1/2 



(3N/3 sinp) 



v (5 sin3 m -tj + i ui (4) 



From (3) we see that the equation N = 

 determines for each value of vh/c one or more 

 values 3 m ; writing this equation in the form of 

 the denominator in (1) : 



motion is caused by constructive interference 

 of the plane waves in which the spherical wave 

 originating at the origin can be decomposed 

 (Press and Ewing 1948). 



— R exp ( 2 i q) 



(5) 



it is obvious that the values y m corresponding 

 to the roots ^ determine the directions y for 

 which the reflected elementary wave 4>i is iden- 

 tical to ^i- 2 (the phase shift caused by the 

 two reflections at the boundaries cancels the 

 difference in nhase due to twice transversing 

 the layer). Therefore the main part of the 



It follows from (5) that, as | R ! = 1, y m has 

 to be greater than the angle of total reflection 

 for the tranvserse wave; hence b > c and sin 



3 m > 1. 



Again if sin 3 < W c the quantity q is real ; 

 for each value of sin between 1 and b/c we 

 obtain therefore an infinite series of values 

 vh/b satisfying N = 0. In the diagram 

 (fig. 2) several of these modes are shown (we 



