Sound between Parallel Walls. 305 



On the walls at y = +#]. u, v, & must satisfy certain con- 

 ditions. It will here be supposed that there is neither motion 

 of the gas nor change of temperature ; so that when y — +yu 

 u, v f , 6' vanish. The condition of which we are in search is 

 thus expressed by the evanescence of the determinant of (23), 

 (24), (25), viz. : 



m 2 h n__ l\ dlogQ / h\ dhgQ, 

 h/fjb — m 2 \\j \ 2 ) dy \X l J dy 



-(H^=°' (26) 



which is to be satisfied when y= +#i- 



Since u is an even function of y, we have from (3), {'2'2)^ 



Q=cos^v/(m 2 -/^)},] 

 Q 1 =cos\yy/(m*-X 1 )\, I . . . (27) 

 Q 2 =cos\y^/(m' 2 —\ 2 )\. J 



From (23), (25), and from the fact that u = when y=yi, 

 we get as the general value of u, without regard to the 

 constant multiplier, 



T _ Q(y) i y~^i Qi(y) v-iijx 2 Q 2 (y) m 



QM' i 'h/X 1 -h/\ 2 q i (y 1 ) hl\ l -h/X 2 Q 2 (y i y ^ J 



In equation (26) the values of \ l9 X 2 are independent of y u 

 being determined by (4). In the application to air under 

 normal conditions ///, //' ', v may be regarded as small, and 

 we have approximately 



X 1 =h 2 /a\ X, = haVvb 2 (29) 



A second approximation to the value of X 1 is given in (13). 

 It is here assumed that the velocity of propagation of viscous 

 effects of the pitch in question, viz. \Z(2fju'n), is small in 

 comparison with that of sound, so that in/i'/a 2 , or hfi'/a 2 , is a 

 small quantity — a condition abundantly satisfied in practice. 



In interpreting the solution w^e limit ourselves here to the 

 case which arises when /J, fi", v are treated as very small — 

 so small that the layer of gas immediately affected by the 

 walls is but an insignificant fraction of the whole. When 

 yj &c. vanish, we have 



so that unless y be great y s /(7n 2 — X l ) is small. On the 

 other hand, y x */ (m 2 — h/fi') , yi s/ {m 2 —X 2 ) are large. For the 

 moment we leave the value of y\^{m 2 —X 1 ) open, and merely 



