PLANE WAVES OF SOUND 169 



quantities relating to the incident and reflected wave by the 

 suffixes 1 and 2, respectively, whilst those relating to the 

 transmitted wave are indicated by (grave) accents. Since the 

 velocity and the pressure must be the same for the two media 

 at the origin, we have 



?i + ? 2 = i\ *s 1 + *s 2 = *Y [>=0], ......... (4) 



the equilibrium pressure p being necessarily the same. Now 

 ^ = 0$!, J 2 = cs 2 , c> whence 



c(s l s 2 ) = cs\ ic (! + * 2 ) = *V [# = 0] ....... (5) 



^ 



Hence S " 



These formulae relate in the first instance to the state of 

 things at the origin, on the two sides; but it is easily seen that 

 they will also represent the ratios of amplitudes at correspond- 

 ing points in the respective waves. If the inertia of the second 

 medium were infinite, we should have c = 0, and therefore 

 $2 = $!, as in the case of reflection at a rigid barrier. On the 

 other hand, if the inertia of the second medium were evanescent, 

 we should have c = oo and s 2 = i, as above. 



The energies of corresponding portions of the various waves 

 are proportional to KS^C, KSC, #W, since the lengths occupied 

 by these portions will vary as the respective wave-velocities. 

 The conservation of energy therefore requires 



KSi*C = KS 2 *C + KS"*C', .................. (7) 



this is easily verified from (6). 



If we put K = p c z , K = poC*, we have, from (6), 



*i A> c + p c 



As an example, take the case of air- waves incident normally 

 on the surface of water. We have p /p x = '00129, c/c' = '222, 

 about; whence s z /s l = '99943. There is therefore almost com- 

 plete reflection, with hardly any transmission. 



In the case of two gaseous media having the same ratio of 

 specific heats, and therefore the same elasticity (K = yp ), the 

 formulae simplify ; thus 



* = ^, S -=^ (9) 



S, C + C 5 X C + C 



