TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 21 



Thus, taking account of (86), the last members of (87) and (88) are equivalent to 



cos KX + 2 v/(2ir) A/( 1 - ~] ie ' 



V \ #7 (kx)* 



and 



27rK sin KO; - V(2,r) f 1 - ff) . *- e -~-!- ( - 



\ */ (to) 1 



7,27.2 

 //r> \ /* *v 6> IP 



- ^(2iO - (TJ g^i -pjjr 



respectively. Substituting in (52), and adding in the system (72) as before, we have, 

 for large positive values of x, 



r> n /9 // j,~\ ^(ft-tx-ii,) 



u , = - Q He'<*-"> + S ,Y/.- V i 1 - P) ' "Trv- 



/x, /u. V TT V \ A;-/ (^x) 5 



Q/ 2 // 3 yL-' 3 //" /- // - ^ it' 1 ( ; " -**-W 

 A / VJ>^ /r / . i .e,,, 



^ V TT (^-2A 8 ) (Ax) 



9' 



,'O 9O /o / 7( 2 \ ,V(i''-^-l") 



t , = _ ? Ke'^'' + ^ A/ - 1 - ';, - i 1^1 



fJ. p V 7T \ A,-/ (/.'X)' 



Q 

 "*- 



The first terms in these expressions have already been interpreted. The residual 

 disturbance constitutes a sort of fringe to the cylindrical elastic waves which are 

 propagated into the interior of the solid, and consists of two parts. In one of these 

 the wa ve- velocity p/k, or b~ l , is that of equivoluminal waves; the vibrations (at the 

 surface) are elliptic, the ratio of the vertical to the horizontal diameter of the orbit 

 being 2^(1 A, 2 //c 2 ), or 1'633 for \ = p.. The remaining part has the wave-velocity 

 p/h, or a" 1 , of irrotational waves ; the surface vibrations which it represents 

 are rectilinear, the ratio of the vertical to the horizontal amplitude being 

 (& 3 2h 2 )/2h (k z /i 2 )*, or "3535 for \ = p.. With increasing distance * the 

 amplitude of each part diminishes as x~' s , whereas in an unlimited solid the law 

 is jc~*, as appears from (42). 



Similar results will obviously hold in the case of the other problems considered in 

 Art. 5. 



9. It has been assumed, up to this stage, that the primary disturbance varies as a 

 simple-harmonic function of the time. It is proposed now to generalize the law 

 of variation, and in particular to examine the effect of a single impulse of short 

 duration. From this the general case can be inferred by superposition. 



