PLANE WAVES OF SOUND 191 



confined, practically, to a very thin layer of air near the 

 surface, and is except in very narrow spaces unimportant. 



The matter may be sufficiently illustrated by a very simple 

 case. Suppose that the fluid above the plane y = is acted on 



by a periodic force 



X = fcosnt, (1) 



per unit mass, parallel to Ox, the plane forming a rigid 

 boundary. The consequent motion being everywhere parallel 

 to Ox and independent of the coordinate x, there is no variation 

 of density, and the deformations which are taking place are of 

 the nature of shearing motions parallel to y = 0. Denoting the 

 velocity f by u, the rate of shear will be 



and the shearing stress on a plane parallel to y is accordingly 

 pdu/dy. The stratum bounded by the planes y and y + By 

 therefore experiences a resultant force 



a 



per unit area, parallel to x, and the equation of motion is of the 

 form 



du d*u 



We have to solve this under the condition that u = Q for 

 y = 0. For conciseness we put X = fe int , and reject (in the 

 end) the imaginary part of our expressions. The equation is 

 then satisfied by 



u = (l n + Ae'y)e i ', .................. (4) 



provided m 2 = in/v, or 



m=(l+0& ..................... (5) 



where = vW 2 ") ......................... ( 6 ) 



Since we are looking for a solution which shall be finite for 

 y = oo we take the lower sign. Also, the condition that a = 

 for y = requires that A = f/in. Hence 



