SEISMIC METHODS 657 



where the left-hand member represents the energy of the incident longitudinal wave 

 and the right-hand member the energies of the reflected and refracted waves. Fur- 

 thermore, if the energy of the incident longitudinal wave be denoted by £0, the 

 energy of the refracted longitudinal wave by £l, and the energy of the reflected 

 longitudinal wave by Ei, 



Eo CdxA' Eo Cd^A'- A' ^ ' 



Obviously, if the ratios —r and — p of the energy factors and the ratio -j- of the densi- 

 ties are known. Equations 7 may be used to obtain the ratio of the energy of the refracted 

 wave to that of the incident wave ( -^ \ and the ratio of the energy of the reflected 



wave to that of the incident wave ( —^ \ . But for a given value of the angle of incidence, 



the ratio of the energy factors can be expressed as a function of the velocities and the 

 moduli of rigidity by solving the systems of equations (5) and (6). Furthermore, if the 

 ratio of the velocities and the ratio of the densities are specified, the ratio of the moduli 

 of rigidity are determined. (Compare p. 658, Equation 9.) Hence, for a particular 



El El 



angle of incidence, the only data necessary to determine the quotients — and ^ are (1) 



the ratio of the velocities and (2) the ratio of the densities. 



In particular, if the densities of the two layers are the same and if the velocities 

 have the ratios 



Fi : F2 : z'l : % = 1.82 : 2.05 : 1.00 : 1.31 



the following results are obtained. 



The reflected longitudinal wave receives 4 per cent of the energy of the incident 

 wave when the latter arrives at the surface normally (a = 0). As the angle of incidence 

 increases, the energy of the reflected longitudinal wave at first decreases until at a 

 value equal to approximately 15° it reaches a minimum value of 0.2 per cent of the 

 incident energy. Thereafter the per cent energy is small until a = 60°. For values 

 of a greater than 62^° (the critical angle for the longitudinal wave), the reflected 

 longitudinal wave receives more than two-thirds of the energy.* For values of a less 

 than the critical angle 62j4°, the refracted longitudinal wave carries by far the 

 greater part of the energy. (The transverse wave, largely because of the minuteness 

 of the energy which it represents in field operations, is not generally considered in 

 seismic prospecting and will be neglected in the present treatment.) 



If it is assumed as before that the longitudinal wave strikes the surface almost 

 vertically, it may be shown that the systems of equations (6) and (7) become 



and 



Also, if 7 = 2, 



On comparing this result with that obtained for the_ amplitudes on p. 656, one 

 verifies the well-known fact that the energy is proportional to the square of the 

 amplitude. 



* The critical angle is defined by the relation : flcnticai = sin ^-— i 



K 2 



