TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. <) 



force does work in generating the cylindrical waves which travel outwards from the 

 source of disturbance. The formula) (40) give, for the value of dv/dt at the origin, 



This expression is really infinite, but we are only concerned with the part of it in the 

 same phase with the force,* which is finite. Taking this alone, we have 



? <"> 



Discarding imaginary parts, we find that the mean rate, per unit length of the axis 

 of z, at which a force Q cos pi does work is 



, .x 



5. We may proceed to the case of a " semi-infinite " elastic solid, bounded (say) 

 by the plane y = 0, and lying on the positive side of this plane. We examine,, in 

 the first place, the effect of given periodic forces applied to the boundary. 



As a typical distribution of normal force, we take 



........ (4G), 



the factor e'f* being as usual understood. The constants A, B in (30) are determined 

 by means of (34), viz. : 



+ (2f 2 - F) B = 0, 



Hence 



2<r-F_Y B _. (4g) 



~EX?T /*' "F(f) /* 

 where, for shortness, 



F (f ) = (2^ 2 - A 2 ) 2 - 4^/8 ....... (49). 



We shall find it convenient, presently, to write also 

 , .. /(0 = (2^ 2 -* 8 ) s + 4f s a/8 ...... . . (50). 



*i 



* The awkwardness is evaded if (as in a previous instance) we distribute the force uniformly over a length 

 2a of the axis of x. This will introduce a factor I^-S^\ under the integral signs in the second 

 member of (44). 



VOL. CCIII. A. 



