252 DYNAMICAL THEOKY OF SOUND 



the diffraction effects which are especially studied in the theory 

 of Light. 



The question of the impact of waves on a plane lamina can 

 be treated in a similar manner. For this purpose the formula 

 (3) is most convenient. The lamina being in the plane x = 0, 

 and the primary waves being represented by (8), we may write 

 <t> = Ce~ ik * + x , ..................... (11) 



where % is the velocity-potential due to a vibration of the 

 lamina normal to its plane with the velocity ikC, equal and 

 opposite to that in the primary waves. It is evident that the 

 values of this function at any two points which are symmetric- 

 ally situated with respect to the plane x = will be equal in 

 magnitude but opposite in sign. We have then, to the right 

 of the lamina 



This only requires a knowledge of the value of % at the positive 

 face of the lamina, the value at all other points of the plane 

 x = being obviously zero. The case where the dimensions of 

 the lamina are small compared with X has been noticed in 

 81 ; the scattered waves have then a much smaller intensity 

 than those transmitted by an aperture of the same size and 

 shape. In the opposite extreme, the value of % near the 

 positive face is, except near the edge, the same as in the case 

 of an infinite vibrating plate, viz. % = Ce ikx , so that we have 

 with sufficient accuracy 



A detailed study of this integral would indicate, in the complete 

 solution expressed by (11), the existence of a sound-shadow to 

 the right of the lamina. For large values of kr the formula 

 (13) may be replaced by 



ikr 



............ (14) 



and for small obliquities 6 we may further put cos 6 = 1. The 

 formula then becomes, except as to sign, identical with (10), 

 shewing that the disturbance produced by the lamina is, under 



