Wave- Motion in an Elastic Solid. 483 



into two constituents, each capable of existing without the 

 other, in one of which the displacement is equivoluminal, and 

 in the other it is irrotational. Hence if we denote by 

 (fi> Vu £1) * ne equivoluminal constituent, and by (f 2 , ^ 2 , ? 2 ) 

 the irrotational constituent, the complete solution of (1) may 

 be written as follows : — 



£=&+&; v=vi+v%; ?=?i+r 2 • • • ( n )> 



where £ l5 i^, Ji and f 8 , r) 2 , t 2 fulfil the following conditions, 

 (12) and (13), respectively : — 



flfa? % «fe 



% _dw _ div y _ dw \ 



* 2 ~Zv '> V2 ~ ty ; & ~ dz j 

 t« being any solution of f . . \13). 



9 _ 2 d 2 w 2 A: + f 71 



v 2 S7 2 iv=-— 2 ; v 2 = 2_ i 



or /o j 



The first equation of (12) shows that in the (f 1? 97^ £ x ) 

 constituent of the solution there is essentially no dilatation or 

 condensation in any part of the solid ; that is to say, the 

 displacement is equivoluminal. The first three equations of 

 (13) prove that in the (f 2 , ?? 2 , f 2 ) constituent the displacement 

 is essentially irrotational. 



§ 7. We can now see that the most general irrotational 

 solution fulfilling the conditions of § 5 is 



d* F 9 d 2 F 



giving 



* 2 -^7 ; V2 -d^dyV ; ^-dWdz~V ' (14) ' 



(^2 drj^ d^ = l_d L F 2 

 dx^ dy "*" dz v 2 dx r ' ' ' A 14 '> 



and the most general equivoluminal solution fulfilling the 

 same conditions is 



^F_i_Fi j^F, y -JLh nn 



S 1 - dx 2 r t*V Vl dxdyr' ^~dxdzr ' [0) ' 

 giving 



f +$ + g = ° • • • • ^ 



