Wave-Motion in an Elastic Solid. 



233 



per unit of volume: and therefore the energy transmitted per 

 unit of time is equal to the product of the propagational 

 velocity into this surface-integral. Thus we find that the 

 rates per unit of time of the transmission of energy by the 

 two sets of waves, of amplitudes represented by (67), are 

 respectively as follows: — 



""pAVffl^ + L*)!! 



(68). 



(equivoluminal) 

 (ir rotational) 



^pAY« 2 [(3-2K) 2 + 4L 2 ]t; 



§ 28. The sum of these two formulas is the whole rate of 

 transmission of energy per unit of time, and must be equal to 

 the rate of doing work by the vibrating rigid globe upon the 

 surrounding elastic solid. Hence if w denote this rate, we 

 must have 



2tto 



aV« 2 { 2m ( K2 + L2 )+^[( 3 - 2K ) 2 + 4L2 ]}- ( 69 )- 



§ 29. To verify this proposition, let us first find the resultant 

 force, P, with which the globe presses and drags the elastic 

 solid, and then the integral work which P does per period, and 

 thence the average work per unit time. Going back to § 9, we 

 see that P is the surface-integral of X over the spherical sur- 

 face of radius q. Hence by the first of equations (25), which, 

 in virtue of the equations 



k+%n=pv*; n=pu? .... (70), 



we may write as follows : — 



X= p {C 2 W-[2X 2 (A + C 2 ) + 2(2A, 2 + 1)B + (\ 2 + 1)C 1 ]m 2 }(71), 



we find 



p s= i^|V^C 8 i> a -2(A + 5B + 2C 1 + O a )M 2 } 



^.TTff-p 



( qv q* \qu q ) j 



6 ( qu l v q z u l q J 



= ^I[f!lP^2{u-v)coK + 2{u' i -v 2 )- +3ro>~|cos«f 



— 2(i< — r)coL — 2(u 2 — r 9 ) sin at j- (72); 



