SEISMIC METHODS 667 



Following the explosion at 0, surface waves and three types of longi- 

 tudinal waves (direct, reflected, and refracted) are received at Sic pro- 

 vided Xk is sufficiently great. The surface wave travels along the surface. 

 The direct wave traverses the path OS^, the reflected wave the path 

 OBSjc, and the refracted wave the path OACSu- The travel-time curves 

 for the three longitudinal waves may be determined as follows : 



The length of the curved path OS^ is approximately equal to Xjc, 

 because the curvature of the path is due solely to the slight increase of 

 velocity with depth. Hence, the travel-time T^ required by the direct 

 wave to traverse the distance O^^ = x^ is equal to the quotient of the 

 horizontal distance Xu divided by the velocity V\. That is, 



Tjc=-fy- and Xk = TkVi 



The equation of the travel-time curve of the direct wave may be obtained 

 from the last equation by replacing x^ by x and T^ by T, where x denotes 

 the distance between the shot-point and any one of several seismometers 

 located in a straight line through and T denotes the travel-time over 

 the distance x. Thus, the equation of the travel-time curve is 



r=^ (11) 



The curve is therefore a straight line which passes through the origin and 

 has a slope of magnitude -r^ (Figure 41-2.) 



From the geometry of Figure 411, it is seen that the distance from the 

 point to the point of reflection B is 



=V(f)'+'>= = V^+'>^ 



=v 



x^ + 4/i2 



= K V^^ + 4/^2 



when the interface between medium 1 and 2 is parallel to the surface, a 

 condition of symmetry exists and the distance OB = BSk, the total travel 

 distance of the reflection is then OB -{- BSk. 



OB -t- BSk = V'-^^ + 4^2 

 Since the distance 05 + BSk is also equal to TVi , we find that 



Vi 



This is the equation for a rectangular hyperbola, as plotted in Figure 

 412. 



