REFRACTED AND REFLECTED LONGITUDINAL WAVES 27 



equal to the coefficient of reflection or refraction of plane waves with vertical 

 incidence of the wave on a given interface, and being determined from 

 formulas (12) and (13). The distance h^ travelled by the multiple wave 

 under consideration in the ith layer is introduced into the denominator. 

 The formulas thus obtained add to the knowledge we already have 

 from the theory of plane waves, about the intensitites of simple and multiple 

 reflected waves over a shot point, the possibiUty of taking into account the 

 weakening of reflected waves due to the divergence of a spherical wave. 



ACCURACY OF THE PROPOSED METHOD OF CALCULATION 



The method proposed is approximate. We shall now evaluate this method 

 by comparing it with accurate solutions, and with the values obtained when 

 the Leningrad tables and methods were used. 



The basic formulas are merely another way of writing out the expressions 

 for intensity which were given in the papers (^'^) ; no new errors are therefore 

 introduced. The values obtained from the tables for the multipliers 

 Cq, r^P{p), {Pq Pq) have been plotted in figures 2, 4 and 7. We may regard 

 their graphical values as being determined with a sufficient degree of accuracy. 

 The coefficients of refraction and the coefficients of reflection on the other 

 hand have been found approximately by means of the graphs shown. 



Let us now evaluate this approximation. The possible error in determining 

 each of the multipliers (Pj-P^-^j) (Pj-^jP^) for interfaces characterized by the 

 quantity lying within the limits 0.7 <a < 0.9 does not exceed 2%; if 

 this quantity lies between the limits 0.5 <a < 0.7 the error is 5% and 

 finally if it Ues within the limits 0.3 < a < 0.5 the error is 10%. This estimate 

 has been made for refracting boundaries with the parameters indicated 

 above. Only for boundaries with y = 0.4 and A = 0.6 are the errors in 

 determining the refraction coefficients slightly higher. The refraction coeffi- 

 cients for such boundaries can nevertheless be calculated by the method 

 referred to, the errors being reduced as the angle of incidence becomes 

 smaller or approximates to critical. For boundaries characterized by the 

 parameters y = A = 0.6, a = 0.7 to 1.0, the error will never exceed 3%. 



We shall estimate the error entailed in determining the coefficient of 

 reflection separately for the following cases. 



Reflection from a layer with high acoustic rigidity — When the sine 

 of the angles of incidence on the reflecting boundary is equal to 0.75a (a being 

 the parameter of the reflecting boundary under consideration), the error 

 in determining the coefficient of reflection does not exceed 10%. The 

 degree of accuracy rises rapidly as the angle of incidence becomes smaller. 



