246 DYNAMICAL THEOEY OF SOUND 



The corresponding velocity-potential on the near side is 

 evidently 



e- ikr . .-.(5) 



The energy ( W) transmitted by the aperture per second is 

 by the above reasoning one-half that due to a simple source 

 at 0, whence, by 76 (15), 



W = fccKWir ...................... (6) 



The energy-flux in the primary waves (1) being %pk*cC' 2 , the 

 ratio of W to this is 2^T 2 / 7r - ^ ^ s t be n ted that this is 

 independent of the wave-length X, so long, of course, as X is 

 large compared with the linear dimensions of S. 



The exact calculation of K for various forms of aperture is 

 naturally a matter of some difficulty. For a circular aperture 

 of radius a it is found that K = 2a ; for other forms differing 

 little from a circle the value is sensibly the same as for a 

 circular aperture of the same area, the circle being evidently 

 a " stationary " form, in the sense in which this term is used 

 in the theory of maxima and minima. It appears then that 

 a circular (or nearly circular) aperture transmits the fraction 

 8/7T 2 , or '816, of the energy propagated across an equal area 

 (?ra 2 ) in the primary waves. This is, under the present 

 limitation as to size, very great compared with the energy 

 intercepted by a disk of the same dimensions ( 81). The 

 figure opposite gives the shapes of the surfaces of equal 

 pressure (< = const.), drawn for equidistant values of </>, in the 

 immediate neighbourhood of a circular aperture, and shews 

 how rapidly these tend to assume the spherical form. The 

 directions of vibration of the air-particles are of course normal 

 to these surfaces. 



With regard to further problems of the kind we must 

 content ourselves with a few statements of results. In the 

 case of an aperture in the shape of a long narrow slit, whose 

 breadth is small compared with X, the energy transmitted is 

 again comparable with, and may even considerably exceed, 

 that corresponding to an equal area of wave- front in the 

 primary waves. In the case of a grating composed of equal, 



