REFRACTED AND REFLECTED LONGITUDINAL WAVES 29 



the increase in intensity of the refracted waves with decrease in the drop 

 in velocities of propagation of the longitudinal waves at the refracting 

 boundary occurs still more rapidly than the growth of the coefficient F^^ {p). 

 Even if the comparison is made at equal distances from the point of emergence, 

 then when r— Tq > O.SAq refracted waves with higher amplitude will correspond 

 to boundaries with less difference in the velocities of propagation, although 

 for these the points of comparison are at a greater distance from the shot 

 point. For boundaries where the difference in propagation velocities is 

 slight, the intensity of a head wave at some distance from its point of 

 origin will be greater than for a sharp interface at the same distance from 

 its point of origin. The curves shown in Fig. 4 show how the intensity of 

 head waves depends on the values of the parameters y, A and a at the 

 refracting boundary. We can however choose these parameters to be such 

 that when the drop in the propagation velocities of longitudinal waves is 

 slight, the head waves -will have a lower intensity than in a case of greater 

 difference in the velocities of propagation at the interface (but with other 

 parameters y and A). It can be seen that the intensity of the head waves 

 increases in direct proportion to A and in inverse proportion to y. 



The density ratio at the interface also affects the intensity of the head 

 waves. For boundaries A = 0.4 and A = 0.5 the head wave intensity 

 increases as the difference in densities decreases, while for boundaries where 

 A — 0.6 and a ^ 0.35 it decreases. 



The damping of the head waves with distance is determined by the multi- 

 pHer r~'''(r — Tq)"'^'. The influence of the depth of the refracting boundary on 

 the intensity of the head waves has a substantial effect only at distances r 

 comparable with Tq. If the comparison is made for several two-layered media 

 which differ from one another only by the parameter h^ {r being fixed and 

 the same for all the media), we arrive at what seems to be a contradictory 

 conclusion: namely that the greater the bedding depth of the interface 

 the greater the intensity of the primary waves. If however we compare the 

 intensity of the head waves at uniform distances from their points of 

 origin, everything becomes clear. We find that to get head waves of the 

 same intensity at the same distance from their respective points of origin 

 in the case of much deeper interfaces, a much more violent cause of excitation 

 is required. At distances /" ^ /"o the bedding depth of the interface does 

 not influence the intensity of a head wave. The head waves will dampen 

 with distance as r~^. 



Multi-layered Media — Of the many problems connected with the origin and 

 propagation of head waves in multi -layered media, we shall here treat 

 only the following: 



