

Sound between Parallel Walls. 307 



The solution differs from that found by Kirchhoff for a 

 circular tube of radius r merely by the substitution o£ 2y x 

 for r *. 



So far, then, as it depends on t and a., the typical solution 

 is 



Jnt p—m'x—im"x 



or when realized, 



e~ m ' x cos (nt—mCx), (40) 



where m', m' f have the values given in (39). This is for 

 waves travelling in the positive direction. 



As a function of ;/, u is given by (28) ; but this may now 

 be simplified in virtue of the supposition that the layer 

 directly influenced bv the viscosity is but a small fraction. 

 of yi . By (27) 



Q(#i) = cos (yi v 7 "//*' • V— *') = cos \y t V»/2/*'.(l— } 

 = i^(W)|cos( yi ^) + ,- S in( yiV /^ 7 )},.(41) 



use being made of (30), in which rj is large. In consequence 

 of (41), Q(y) -r-Q(yi) vanishes unless y be nearly equal to y ± , 

 viz. unless the point considered be within the frictional layer. 

 Tn like manner, and under the same restriction, 



Qa(y)-*-Q«(yi) 



may be neglected. Except in the immediate neighbourhood 

 of the walls, (28) now reduces to 



«--$^ (42) 



Qife/i) 



In the case considered by Kirchhoff, where the argument 

 of Qi is small, we have from (27) approximately 



QiW = Q.to = i, 



and accordingly u = — l. To this approximation the velocity 

 is uniform across the whole section until it begins to fall off 

 as the walls are closely approached. 



As a first step towards the consideration of what occurs 

 when 3/1 is great, we may proceed to a second approximation. 

 Thus, from (27), 



Q(^) = i-M^-^) = i-^ 2 ^-;. • (43) 



a y x 

 so that a//, 2- 



_ 1 _ (m*-y 2 W n * , ; W-fh ln * ,,*-, 



2a 2 yW2 2a*yW% ' ' [ '^ } 



* ' Theory of Sound,' 2nd edit. § 350. 



