TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 



27 



Fig. 6. 



A similar treatment would apply to the formulae (81), and (with some modification) 

 to (84). 



It remains to justify these approximations by showing that the residual disturbance 

 tends with increasing x to the limit 0. For this purpose we have recourse to the 

 formulae of Art. 8. As a sufficient example, take the second term in the last 

 member of (88). If we multiply by e'' 11 , take the real part, and substitute rj = p<l>, 

 k = 2)b, the corresponding term in the value of v , as given by (52), assumes the form* 



-Q cos/; (t - bx) fF U) e-** d<f> + Q- sin p (t - bx) f f(<f>) e~^ d<j>, 

 fj. Jo U. Jo 



where the functions F (<) and /(<), which do not involve p, are of the order </> l when 

 <j> is large. If we generalize this expression by FOURIER'S Theorem (see equation (99)), 



we obtain, in the case of an impulse Q of short duration, 



-S f F (<f>) d<j) \ e-* 4p cos p (t bx) dp + S f / (<) d<j> [ e-* p sinp(t - bx) dp 



7T/X ^0 Jo TTjLt Jo Jo 



Q f rr / JL \ ^^ "<P i y I f/sL\ (^ to) Cc-p -. ^/,\ 



^^ I H / fi\ \ _ I r I fn I >- ~ . , 11UOJ. 



TTfJ, 



* The symbols <^>, F, / are here used temporarily in new senses, 



E 2 



