244 DYNAMICAL THEOEY OF SOUND 



second is the form which the velocity-potential of a double 

 source assumes ( 76) when kr is small. This makes 



- ik r + cos0, ......... (9) 



and the condition of zero normal velocity for r = a is therefore 

 approximately satisfied provided B= l%ika?C. Hence in the 

 neighbourhood of the sphere we have 



(10) 



nearly. The velocities are therefore nearly the same as if the 

 fluid were incompressible. The pressure is given by 



p=p + p<j>=p + inp(f> ............. (11) 



This differs from the pressure (p + inpG) which would obtain 

 at the origin if the obstacle were absent by a term which 

 is small, of the order kr, in comparison. At points whose 

 distance r is a moderate multiple of a, whilst still small 

 compared with X, the pressure approximates even more closely 

 to that due to the incident waves alone. 



82. Transmission of Sound by an Aperture. 



In discussing the transmission of sound waves by an aperture 

 in a thin screen we will suppose, in the first instance, that the 

 dimensions of the aperture are small compared with the wave- 

 length. This is of course the most interesting case from an 

 acoustical point of view. 



The screen being supposed to occupy the plane x 0, and 

 the origin being taken in the aperture (S), let a wave- train 

 represented by 



be incident from the left. If we distinguish the functions 

 relating to the two sides of the screen by the suffixes 1 and 2, 

 we should have, if the screen were complete, 



fr^Ce-^ + Ce^*, < 2 = o, ............ (2) 



the second term in <f> 1} which represents reflected waves, being 

 chosen so as to make dfa/da; = for x = 0. 



In the actual problem the disturbance due to the aperture 

 will be confined mainly to the immediate neighbourhood of S, 

 and may be taken to be very small at distances from which, 



