REFRACTED AND REFLECTED LONGITUDINAL WAVES 21 



3. The coefficients of refraction {P^P-_^j) (Pj+jP,) are determined in the 

 same way as for head waves. Only the ordinates of the curves (see Fig. 3) 



are noAV taken at the points x = sin a, ~ -^^-^ sin a^ where V: ^ is the lesser of 



the velocities v^ and v-_^-^ , that is at points equal to the sines of the lesser 

 of the angles at which the ray under consideration meets the boundary be- 

 tween the ith and the i + 1th layers. The curves are chosen -with the same 

 parameter as corresponds to the boundary between the iih. and the i+lth 

 layers. Then the ordinates are again multiplied by the quantity, 



iQii'i,P+ Qi+iVi+i,pf 

 The product also immediately gives us the value of the coefficient of 

 refraction {P,P,.d {P,^iP^)- 



4. The coefficients of reflection (P^ P^) and (P,„. P^^.) depend both on 

 the properties of the reflecting boundary itself and also on the angle at 

 which the wave strikes it. To determine these we have constructed graphs 

 (Figs. 5, 6 and 7) based on the tables drawn up by the Petrashen' team; 

 the sines of the angles at which the wave under consideration strikes the 

 given reflecting boundary are plotted against the abscissa, and the parame- 

 ters of the reflecting boundary are used as the parameters of the curves. 

 We shall deal ^vdth each of the following cases separately: 



(a) Reflection of a wave from a layer having higher velocity of longitudinal 

 wave propagation than the layer in which the incident ray is travelling; 



(b) Reflection of a wave from a layer having a louver velocity of propagation 

 of longitudinal waves; 



(c) Reflection of a Avave from the free surface of the medium. 



Case a. To determine the coefficient (P^ P^) or {P^.P^) from the curve 

 corresponding to the parameter A (see Fig. 5) which characterizes the reflecting 

 interface under consideration, we take the ordinate value (P^ P^Jj.gj, if 

 ^m, p< V+i, p' or (P^, Pnr)rei ^^ ^mp ^^ < %_i, p where the abscissa is equal to 

 the sine of the angle at which the wave under consideration strikes the 

 given reflecting boundary. The value of the ordinate is again multiplied by 

 the coefficient of reflection of plane Avaves when the incidence is vertical. 



The product gives the value of the co -efficient of reflection. 



^m, p ^ ^\n + l, p' 



that is, if 



(P p \ ^ _ (p p ^ , QmVm,p — Qm + l'^'m + l, p . 



Qm '^m, p + Qm + l '^m+1, p 



