SEISMIC METHODS 



653 



The angles made by the rays with the normal are related by the equation : 



sin a : sin h : sin c : sin d ^= Vi : V2 : Vi : ?'« 



When the motion is a simple harmonic one, the displacement | at a time t can be 

 expressed by an equation of the form : 



pi ["^ — Cr sin a -\- s cos a)/ Vi 

 I = Me 



where M is the amplitude, (x sin a + 2 cos a)/V is the phase lag, p is equal to Zir 



times the frequency and i as usual denotes the square root of minus 1. The phase 



lag is obtained by substituting an appropriate value for the "initial" time d in the 

 equation of the wave front : 



:r sin a + s cos a =^ V6 



The resultant displacements | of the several waves and their component displace- 

 ments M and zv in the x and 2 directions respectively may be written as follows, 

 provided it is assumed that the incident longitudinal wave and the four waves pro- 

 duced at the boundary between media 1 and 2 are plane waves in the plane of the 

 paper. (Figure 407) : 



|o = Moe 



|i = Mie 



pi [^ — (x sin a -{- z cos a)/Fi] 



pi ['■ — (,x sin a — s cos a)/Vj) 



?< = M, e 



pi ['■ — (x sin c — s cos c)/Vj) 



^L = Ml e 



£r = MtC 



pi E'" — (.xsinb + s cos b)/V2^ 



pi [■^ — (x sin d + 2 cos d)/v2} 



Mo = |o sin a 

 Wo = |o cos a 



Ml = -1- ^i sin a 

 zvi = — |i cos a 



Ut = |( cos c 

 Wt = |» sin c 



ul = It sin b 

 wl = ^L cos b 



Ut — — Sr COS d 



Wt — |r sin d 



(1) 



The component displacements given by the system of equations (1) must satisfy 

 two equations of the form 



where 



{V--v-)— + v^Vu-^ 



V = :r- + -X— and V^ — z^i, + :r-^ 

 ox OS ox- oz- 



In addition, the following boundary conditions must be satisfied : The sum of the 

 components of the displacement on both sides of the boundary must be the same and 

 the sum of the normal and the tangential components of stress must be the same. 

 That is, 



^("0=J(m.) 



9m2 



dw2 

 dx 



) 



(2) 



(2a) 



(3) 



