30 



WAVE ACOUSTICS 



result is intuitively apparent, but can be proved only 

 by a long tedious argument. 



For brevity, the wave propagated by the boundary 

 into the second medium will be called the transmitted 

 wave, and the wave propagated back into the first 

 medium is called the reflected wave. We shall now cal- 

 culate the amplitudes and directions of propagation 

 of the transmitted and reflected waves. 



The pressure and complex amplitude of the trans- 

 mitted wave are denoted by p, and Ar, the same 

 quantities for the reflected wave are denoted by 

 Pr and Ar- Let the transmitted wave have the direc- 

 tion 9t, relative to the normal, and the reflected wave 

 have the direction 6r, as indicated in Figure 10. The 

 angles dt and dr are usually called the angle of refrac- 

 tion and angle of reflection, respectively, and the 

 angle di is called the angle of incidence. 



Because the reflected and transmitted waves are 

 plane, 



Pr = Are 



Pi = Ate 



,i/{t- 



-xsinflr— ycosfl, 



jrt/(( 



_xsin6i+yco80t\ 



(108) 



(109) 



The sign of y is different in equations (108) and (109) 

 because in equation (108) y decreases with the time 

 on the wave front; in equation (109) it increases. 



Equation (109) gives the resultant total pressure 

 in the second medium. The resultant pressure in the 

 first medium is the sum of the pressures of the inci- 

 dent and reflected waves, which is obtained by adding 

 equations (107) and (108). Denoting the resultant 

 pressure in the first medium by p, we obtain 



P = Pi + Pr = AiC 



2i,if{t 



xsinfli+ycosS. 



-\-Are 



2rif(t 



_x sin dr — y coa dr^ 



(110) 



The pressure must be the same on both sides of the 

 boundary. Therefore, pi + pr = Pta,ty = 0; that is. 



^,-('--^)+^,-('- 



'-') 



= A,e 



2.i/(,-££ili') 



or 



2irt/(| 



AiC " x-\- ArC 



-2»i/°J 



-2«/- 



. ...... ^' 1 



X — Ate ci X \ 



= (111) 



for all values of t and x. Therefore, the bracket itself 

 must be zero. Furthermore, the sum of three har- 



monic functions of x can vanish for all values of x 

 only if their periods are the same. It follows that 



sin dt sin di sin dr 



= = (112) 



Ci • c c 



The second equation of (112) implies that 9i = dr; 

 that is, the angle of incidence is equal to the angle of 

 reflection. The first equation may be rewritten as 



sin di c , ^ 



—7 = -' (113) 



sm dt C\ 



a relation which is well known in optics as Snell's law. 

 Because of equation (112), the exponential factor 

 is the same / for all three terms in the bracket of 

 equation (111) and can be divided out, giving 



At = Ai^Ar. (114) 



The individual amplitudes At and Ar are calculated 

 by making use of the transition conditions (104). By 

 calculating dp/dy from equation (110), and dpt/dy 

 from equation (109) and by substituting these values 

 into equation (104), we obtain 



Ai cos di 2ris{t- 

 e 



xsine, 



) 4.C0Sflr2-'/('-i^i^') 



— e ' 



p 4,C0S^, 2.i/(«-^JlEi') 



(115) 



Pi Ci 



In view of equation (112), the exponential factor is 

 the same for all three terms and may be divided out. 

 Also, di = dr. Thus, equation (115) becomes 



cos di. 



-(Ai - Ar) = -A, 



Pi Ci 



(116) 



Equations (114) and (116) are two linear equations 

 in At and Ar. By solving them in terms of At, the 

 amplitude of the incident wave, and by replacing 

 Ci/c in the result with its equivalent from equation 

 (113), 



A-T — A.i' 



At = Ai 



PiCi cos di — pc cos dt 



PiCi cos di + pc cos dt 



2piCi COS di 



(117) 

 (118) 



plCi cos di -\- pc COS dt 



To eliminate the angle dt from equation (117), equa- 

 tion (113) is used which can be transformed by 

 trigonometric identities into 



^' = l/l-Htan^./l-in- 

 OSdi ' \ c / 



COS( 

 COS di 



Thus, equation (117) becomes 



Ar PlCl — pcB 



Ai piCi -|- pcB 



B. 



(119) 



