TREMORS OVER THE SURFACE OP AN ELASTIC SOLID. 7 



In the applications which we have in view, the vibrations of the solid are supposed 

 due to prescribed forces acting at or near the plane y 0. We therefore assume as 

 a typical solution of (27), applicable to the region y > 0, 



(30), 



where f is real, and a, /3 are the positive real, or positive imaginary,* quantities 



determined by 



s = P_/,2, & = *-& ..... (31). 



For the region y < 0, the corresponding assumption would be 



4> = A.'e>e* c , i/ = B'e ft "e* ........ (32). 



The time-factor is here (and often in the sequel) temporarily omitted. 



The expressions (30), when substituted in (25) and (29), give for the displacements 

 and stresses at the plane y = 



= (#A-B)e**, i> = (-aA--ifB)c*' .... (33), 

 and 



= t - 2A 2 - *B 



..... (34). 

 8 - F A 



The forms corresponding to (32) would be obtained by affixing accents to A and B, 

 and reversing the signs of a. /3. 



4. In order to illustrate, and at the same time test, our method, it is convenient to 

 begin with the solution of a known problem, viz., where a periodic force acts 

 transversally on a line of matter, in an unlimited elastic solid, f 



Let us imagine, in the first instance, that an extraneous force of amount Ye'*'' per 

 unit area acts parallel to y on a thin stratum coincident with the plane y = 0. The 

 normal stress will then be discontinuous at this plane, viz., 



[PyJ^+o ~ l>.J, = -o = - Y '' fl ( 35 )> 



whilst the tangential stress is continuous. These conditions give, by (34), 



- 2ifa (A + A') + (2f 2 - **) (B - B') = 



Again, the continuity of u and v requires 



= 



* This convention should be carefully attended to ; it runs throughout the paper. 

 t RAYLEIGH, ' Theory of Sound,' 2nd ed., 376. 



