REFRACTED AND REFLECTED LONGITUDINAL WAVES 33 



proportionality is constant for each curve but is different for different 

 curves. 



From an analysis of the curves shown we may conclude that if we take 

 (/•— Tq')''" = (r— Tq)''' where Tq is the abscissa of the point of origin of the 

 head wave under consideration when there is no ith layer, and Tq is the 

 same when there is such a layer, we shall obtain the following. 



1. The addition of an ith layer characterized by a velocity v^^ ^ lying within 

 the interval ^j_i n < ^i, n < ^i+i. p' '^l^ cause the intensity of the head 

 waves from the underlying horizons to increase. 



2. As v^ increases from v^_^ to v^j^^ the intensity of the head waves at 

 first increases and then diminishes. When v^^ ^ = Vi_^^ ^ or v^j^^^ ^ the intensity 

 of the head waves is of course the same in each case and the same as when 

 there is no ith layer. The bigger the difference between the velocities 'v^_^^ ^ 

 and t'i+i n that is the greater the interval t;j_j ^ < f,- ^ < f,.|.i p given 

 a constant t;,_j Jv^ the greater will be the variations in the intensity 

 values of the head waves excited in the low-lying layers. With a constant 

 value of 6 = r,_j Jvi+i^ „ the intensity of the head waves when v,-^ p changes 

 may show greater variations the higher the value of the parameter ^,_i, pl^n, p* 

 The intensity reaches its peak value at values of v- near to the mean value 

 of the interval Vi_^ ^^i, p ^j+i, p- 



3. The addition of an ith layer with a longitudinal wave travel velocity 

 lower than the lower of the velocities v-_^^ p, z;,.,.^^ p, or greater than the greater 

 of the velocities v^_j^ , i;,-+j , leads to a reduction in the intensity of the 

 head waves formed in the low-lying layers. The reason for this is that 

 we now have an even sharper interface than where there was no ith 

 layer. 



(c) We shall now assume that the value of the longitudinal boundary 

 velocity of the head wave changes, and that the density and velocity 

 throughout the whole covering layer, as well as the density ratio and the 

 density between the transverse and longitudinal velocities on both sides of 

 the interface where the wave is excited remain constant. Under these conditions 

 the angles at which the head wave strikes the intermediate interfaces 

 will vary and this will cause variations in the values of the coefficients of 

 refraction and in the multiplier Cq. Furthermore there mil be a change in 

 the value of the discontinuity in the longitudinal velocities and the principal 

 refracting boundary and on F^^ (p), the coefficient of head wave formation, 

 which depends on it. Lastly the abscissa of the point of emergence of the 

 head wave will also vary. 



We shall first assume that the intensity of the head waves is being 

 compared at distances from Tq such that we can assume yr (r — Tq)'- = r^. 



Applied geophysics 3 



