SIMPLE-HARMONIC WAVES. DIFFRACTION 237 



acting on the substituted matter. Since this force has to 

 produce an acceleration of momentum pQdU/dt, where Q is 

 the volume displaced by the solid, as well as to balance the 

 reaction just referred to, its amount would be 



, ...... (3) 



if U=Ae int ......................... (4) 



By 78 (2), the velocity-potential at a great distance r will 

 therefore be 



cos 6 = Ae-^ cos 6. ...(5) 



- 



4-TT/oc r 4-Trr 



Comparing with 76 (24) we see that the effect of the vibrating 

 solid is equivalent to a double source of strength C = (Q+ Q')A, 

 and that the emission of energy is accordingly 



(6) 



by 76 (26). In the case of the sphere we have Q = %7ra?, 

 Q'= iQ, and the result accordingly agrees with 77 (18). It 

 can be shewn that for a circular disk of radius a, moving 

 broadside on, Q' = ^ira 3 , whilst Q of course =0. 



80. Communication of Vibrations to a Gas. 



The circumstances which govern the efficiency of a vibrating 

 body in generating sound waves, and the comparative effects in 

 different gases, were elucidated by Stokes in a classical memoir 

 " On the Communication of Vibrations from a Vibrating Body 

 to a surrounding Gas*." The starting point of the investigation 

 was an observation by Prof. J. Leslie (1837), who found that the 

 sound emitted by a bell vibrating in an atmosphere of hydrogen 

 was extremely feeble as compared with the effect in air. No 

 satisfactory explanation of this phenomenon was forthcoming 

 up to the time of Stokes' paper. The essence of the matter is 

 conveyed in the following quotation : 



" When a body is slowly moved to and fro in any gas, the 

 gas behaves almost exactly like an incompressible fluid, and 



* Phil. Trans. 1868. The passage which follows below is from the 

 '* abstract " in the Proc. Roy. Soc. 



