Since k 1a and k 2a are never equal, the general solution of equation 1 1-53 is 



P = IP, + iJJ 2 , (M-61) 



where iJj, and ip 2 satisfy the equations 



(V 2 + k 1a *) ip, = (II-62) 



and 



(V 2 + k 2a 2 ) i|i 2 = . (H-63) 



Equations II-37 through M-39 can be written in a form similar to equations II-62 and II-63. Thus 



(V 2 + k 3f 2 ) A = , (II-64) 



and 



where 



and 



(V 2 + k 2( 2 ) P = , (ll " 65) 



(V 2 + k 2w 2 ) P = , (ll- 66 ) 



k 2 = i*2fia_.. (II-67) 



3f lis, 



k 2w 2 =-rr ■ (ii-69) 



Comparisons of the equations for air and water indicate that ip 2 represents a compressional wave. Hence, iJj, 

 represents a thermal wave. 



Equation II-64 can be transformed into a scalar equation. It can be shown that a vector T exists such that 

 [48] 



u = V 2 T (H-70) 



and 



A = Vxf . ("1-71) 



Since the problem is axisymmetric, 3/g<p = u,= 0, which implies that r, = 0. Thus 



vxf=T(0) + e(0) + 97 [fF (rr °)-Te] • 



ar ' ae 



so that 



(II-72) 



A = (PA,, , (II-73) 



where r, 0, and <p are unit vectors. Therefore 



V 2 A = $ \wA v - 2 A 1 Q 1 (H-74) 



^ L * r 2 sin 2 6 J 



13 



