EFFECT OF A BOUNDARY 



29 



the boundary must have the same value in both 

 media at the boundary. In symbols, if (Si,Sy,Si) are 

 the components of particle displacement in the first 

 medium, and (Sii,Si„,Su) are the components of 

 particle displacement in the second medium, 



Sy = S,y at 2/ = 0. (102) 



No restrictions of the form of (102) can be placed on 

 the displacements »Si and S^ because displacements 

 parallel to the boundary will not cause loss of con- 

 tact. 



Since equation (102) holds for all time, the time 

 derivatives of Sy and iSi„ must also be equal at the 

 boundary; in other words, 



dUy dUly 



Uy = uiy] -^ = -^ y = 0. (103) 



Because of equation (17), equation (103) implies 



Idp Idpi 



= sbty = 



p dy pi dy 



dp P dpi 



— = at 2/ = 0. 



dy pi dy 



or 



(104) 



We shall call equations (101) and (104) conditions 

 of transition. In the general case, the propagation in 

 one medium depends on the exact nature of the 

 propagation in the other medium, because of the con- 

 ditions of transition. In the case of the sea, however, 

 conditions are often such that we can ignore the exact 

 propagation in the surrounding medium; the transi- 

 tion conditions of the type (104) may then be re- 

 duced to boundary conditions for the sea itself. In the 

 next section, the conclusion is reached that at the 

 yielding boundary between sea and air the follow- 

 ing condition holds: 



p = (105) 



and that at the solid boundary between sea and rock 

 bottom we always have, approximately, 



dp 



'dy 



= 0. 



(106) 



Relations of the type of (105) and (106) will, in many 

 cases, suffice for calculating the sound field in the 

 medium of interest. By use of such boundary condi- 

 tions explicit consideration of the soimd field beyond 

 the boundaries may be made unnecessary. 



2.6.2 Reflection and Refraction 

 of Plane Waves 



Consider now what happens to a plane wave when 

 it hits the plane boimdary y = between two dis- 



similar media, in one of which the sound velocity is c, 

 and in the other of which it is Ci. For generality, we 

 assume that the direction of propagation of the inci- 

 dent wave is oblique to the boundary, making an 

 angle di with the normal to the plane boundary. We 

 can also assume, without losing generality, that the 

 direction of propagation is parallel to the xy plane; 

 y represents the vertical direction positive upward, x 

 a horizontal direction, and z a front-back direction, as 

 in Figure 10. 



SOUND VELOCITY c, 

 DENSITY^, 



SOUND VELOCITY C 

 DENSITY p 



Figure 10. Splitting of plane wave at boundary be- 

 tween two media. 



Since the incident wave is plane, it may be de- 

 scribed by the equation (72a) with x replaced by 



X sin 6i + y cos 8i, 



in view of the obUque direction of propagation. 

 That is, for the incident wave. 



Pi = Ai e 



2xi/((- 



_xsm0i-{-ycosei 



■■) 



(107) 



where p.- represents the sound pressure of the inci- 

 dent wave, and At its complex amplitude. 



We can consider that the incident wave terminates 

 its existence when it hits the boundary and expends 

 its energy in producing a disturbance of the interface. 

 Thus the boundary will act as a sound source, which 

 vibrates with the frequency / of the incident wave. 

 The vibration of the interface will send out sound 

 waves of the frequency / into both media. We shall 

 assume that these two waves are plane waves; this 



