Wave-Motion in an Elastic Solid. 

 From the second of equations (53) we therefore have 



235 



and 



^ 1 (t)=hqu 2 G 



^ 2 (t)=hqv 2 K 



M sin (cot + «) — N cos (cot + a) ") 



M 2 + N 2 

 M sin (cot — /3 ) — N cos (cot - /3) 



> • (77), 



M 2 + N 2 



where M 2 and N 2 denote the terms of the denominator of the 

 third of equations (53). By the same method of investigation 

 as that which gave us (69), we now find for the sum of the 

 rates of transmission of energy 



27rp 

 3 



/i 2 </*ffl 



2,., 6 



2»G 2 + i>H 2 



(78). 



,_??£is„a, 



M 2 + N 2 ' 

 Substituting the values of G, H, M, N, we obtain 



n / 9r 4 3« 2 , n / 9u 4 3u* , \ 



■av® — n 2 T2 — H • ( 79 )- 



[-*-* («*" + 2y2 ) - 1 + -TTi ( u + 2v ¥ 

 \_q 2 ar J q a> i v 



This agrees with the value of w given by (73'); and thus 

 the verification is complete. 



§ 31. In (78) the numerator of the last term shows the 

 parts due to the equivoluminal and irrotational waves 

 respectively. Denoting by J the ratio of the energy of the 

 equivoluminal wave to that of the irrotational, we have 



J = 



2»G 2 

 t>H 2 



\u( 



9» 4 Zv* 

 qW + g 2 6) 2 



+ 1 ) 



\q 4 or q l co i J 



• (80) 



§ 32. Consider the following four cases : — 



(a) qco very large in comparison with the larger ^ 

 of u and v. 



J= 



2u 



(5) v = qco. 



(c) u=-qco. 



26 



V 





3?/, 





9u 3 





+ 



— 



+ 



t> 3 



u 





V 





J= 



T 2 (u Sv 9v 



O = 77, - + 1- 



16\v u 



(81) 



9V\ 

 w 3 / 



(d) qco very small in comparison with the 

 smaller of u and v. 



