SEISMIC METHODS 669 



The travel-time of the wave which is refracted near the critical angle, 

 and therefore travels in the lower stratum along a path which is approxi- 

 mately parallel to the boundary, is obtained by adding the travel-times in 



OA + CS 

 the upper and lower strata. The travel-times in the two strata are — 



AC 

 and , respectively. It is evident from Figure 411 that OA — CS^ — 



V2 



h 



also, AC = Xk — 2h tan ai. Hence, the travel-time for the path 



cos ai 

 OACSk is 



_ 2h Xk — 2h tan a\ 



Ik— 77 r 



V\ cos a\ V2 



The equation of the travel-time curve is 



_ 2h X — 2h tan ax 



V\ cos ai V2 



where T and .v denote travel-time and horizontal distance respectively. 



The last equation may be simplified by replacing the trigonometric 

 functions by their equivalents in terms of the velocities. It follows from 



. . / . Fi\ 



the critical condition 1 sin ai = 77- I that 



^sinax = ^^ 



COS ai 

 and 



Fi 



^'-m 



tan ai = 



'1 "" 77 



K 2 cos a\ 



Hence 



T = 2/,V7^-7^+^ (12) 



The travel-time curve for the refracted wave is therefore a straight line 



having a slope of magnitude jz-. (Figure 412.) 



V 2 



Some disagreement exists among investigators employing seismic methods as to 

 whether the refracted wave travels along the slightly curved path shown in Figure 411 

 or along the interface itself (straight line path).t A vertical gradient in velocity 

 would produce a curved path. However, it is not necessary to assume a vertical 

 velocity gradient below the upper layer to explain the emergence of a refracted wave 

 at the surface. Even though no such gradient existed, the refracted wave would be 

 detected at the surface. The wave may be transmitted by the propagation of a strain 

 set up at the lower boundary of the upper layer. See, for example, G. Joos and J. 



t See O. V. Schmidt, "Ueber Kopfwellen in der Seismjk," Zeitschrift fur Geophysik XV, 1939, 

 p. 141; "Ueber Knallwellenausbreitung in Flussigkeiten und festern Korpern," Zeitschrift fUr 

 technische Physik, 12, 1938, p. 554. 



