III-5 



The wave internal to the fluid sphere must remain finite at the origin; therefore, 

 it must be expressible in terms of j^^ alone: 



p'(x,t)=y^B P (cose)j (k'r)e"^'"^ (III-9b) 



- / J mm m 



m=0 



Note that k' = — r is the appropriate wave number inside the fluid sphere corre- 



Co 

 sponding to an angular frequency uo . Finally, the incident plane wave may be ex- 

 panded in spherical harnionics as: 



p (x,t) = p Y^ (-i)"^ (2m+ 1)P (cose)j (kr)e"^'^^ (III-9c) 



Mnc ^inc ,^^ m -"m 



m=0 



It remains to apply the appropriate boundary conditions on the surface of the sphere. 

 These are that both the pressure and the normal component of velocity must be con- 

 tinuous across the boundary. If the sphere has radius a, continuity of pressure 

 requires: 



p. (a) + p (a) = p'(a) (III- 10a) 



mc sc 



The continuity of the normal velocity at the boundary is equivalent to the continuity 

 of the normal (i.e. , radial) derivative of the pressure, as may be seen from (III-2a). 

 Hence, we have as the second boundary condition: 



^ p. (a) + ^ p (a) = ^ p' (a) (III- 10b) 



o r inc o r ^sc d r 



We substitute (III-9a) - (III-9c) into (III- 10a) and (III- 10b) and separate* terms for 

 each value of m. From the resulting pairs of simultaneous equations, we solve for 

 Ajjj and find: 



A = - ^i"c (-l)"" (2m + 1) (iji.ii) 



m 1 + i C 



m 



^This is permissible, because the Legendre polynomials are orthogonal. 



%xK\^\xx 21.1UtI«f Jnf. 



S-7001-0307 



