506 



SEISMIC METHODS 



[Chap. 9 



4. Single horizontal layer. Equations for this case are obtained by 

 considering the possible longitudinal wave paths through two layers. 

 When a charge is fired at A, waves radiate in all directions. For the wave 

 traveling horizontally to location D, the time is 



ti = - 



Vi 



(9-42) 



(see Fig. 9-43), and cotan a = ds/dti = Vi . To find the path of the wave 

 which reaches the receiver through the lower medium, consider the rays 

 impinging on the boundary. If an incident ray subtends the angle <p 

 with the normal to the boundary and the refracted ray subtends the angle 

 \f/, then, according to Snell's law, sin (p/sin yp = (I — Vi/v2 , where q = 



index of refraction. For a given index 

 of refraction there occurs an angle <p, for 

 which sin xj/ = 1; that is, the refracted 

 beam travels along the boundary surface. 

 If <p becomes greater than this "critical" 

 angle i, total reflection takes place. 

 Hence, sin i = Vi/vs = q. Evidenth% 



^ 8' 





Fig. 9-43. Critical-ray paths, 

 single-layer case. 



Fig. 9-44. Wave fronts in upper and lower layer. 



the only ray which can reach the receiver by refraction will travel hori- 

 zontally on the boundary of the lower medium. It strikes the boundary 

 at the angle of total reflection and leaves it at the same angle. This 

 statement involves the application of Huygen's principle, since any point 

 of the underlayer wave may be considered a source of new waves. 



While this wave proceeds with the velocity of the lower layer, impulses 

 are continually sent upward into the upper medium, where they propagate 

 with the velocity of the latter. In Fig. 9-44 locations 1, 2, and C are 

 considered such source points. Their distance is given by the product 

 V2-^, with / as an arbitrary time unit. The "wave fronts" in the lower 

 layer will occupy these positions in successive equal intervals of time. 

 Similar wave fronts may then be drawn for the wave propagating from 



