Sound behveen Parallel Walls. 303 



Thus 



u = u' + B x dQi/dx + B 2 dQ 2 /dx, 



v = v' + B 1 dQJdy + B 2 dQ 2y %, 



i<; = w' + B x dQjAfe + B 2 dQ.jdz, 



where B l5 B 2 have the values given above. 

 It appears that 



du dv' dw' 



+ ^ + _ Q (8) 



dx dy dz 



These results were applied by Kirchhoff to the case of 

 plane waves, supposed to be propagated in infinite space in 

 the direction of +#, and it may conduce to clearness to deal 

 first with this case. Here v and w' vanish, while u l , Q u Q 2 are 

 independent of y and z. It follows from (8) that u' also 

 vanishes. The equations for Q 1 and Q 2 are 



^Q 1 /^=X 1 Q 1 , d 2 Q 2 /^ 2 = A 2 Q 2 ; . . (9) 



so that we may take 



Q 1 = e-^\ Q 2 = e- X ^, .... (10) 



where the signs of the square roots are to be so chosen that 

 the real parts are positive. Accordingly 



u =A 1 \ 1 *^ i -v) e-*^+A 2 \f(^ -v) e~^S . (11) 



e , =A 1 e- x ^^+A 2 e- x ^, (12) 



in which the constants A„ A 2 may be regarded as determined 

 by the values of u and 6' when x = 0. 



The solution, as expressed by (11), (12) is too general for 

 our purpose, providing as it does for arbitrary communication 

 of heat at x=0. From the quadratic (4) in X we see that 

 if fx' ', p", v be regarded as small quantities, one of the values 

 of X, say \ l9 is approximately equal to h 2 /a 2 , while the other 

 (X 2 ) is very great. The solution which we now require is 

 that corresponding to Xj simply. The second approximation 

 to X x is by (4) 



' _ h 2 J\ hQJ+fJ '+v) ~\ L vbVi 3 

 so that 



±^=r l -^W+»"+>'a-v/a*)}. . (i3) 



If we now write in for h, we see that the typical solution is 



u = e -m'x e in{t-z/a^ ..... .(14) 



