ON THE REFLEXION AND REFRACTION OF SOUND. 237 



Since the disturbance of all the particles, in any front of the 

 incident plane wave, is the same at the same instant, we shall 

 have for the incident wave 



retaining b and c unaltered, we may give to the fronts of the 

 reflected and refracted waves, any position by making for them 



=F 



Hence, we have in the upper medium, 



<t>=f(ax+by+ct) + F(a'x + by + ct) ........... (4), 



and in the lower one 



<t><=f,(a t v+fy + ct) ................................ (5). 



These, substituted in the general equations (1) and (2), give 



(6). 



Hence, a = a, wh^re the lower signs must evidently be taken 

 to represent the reflected wave. This value proves, that the 

 angle of incidence is equal to that of reflexion. In like manner, 

 the value of a,, will give the known relation of sines for the inci- 

 dent and refracted wave, as will be seen afterwards. 



Having satisfied the general equations (1) and (2), it only 

 remains to satisfy the conditions (A), due to the surface of sepa- 

 ration of the two media. But these by substitution give 



af (by + ct) - aF (by + ct) = af t (by + ct) , 



because a = a, and x = 0. 



Hence by writing, to abridge, the characteristics only of 

 the functions 



2 VA ' 



y w, 



