(4) 



656 EXPLORATION GEOPHYSICS 



Summarizing the results just obtained, the four boundary conditions are : 

 Mo sin a + Mi sin a + Mt cos c — Ml sin b + Mr cos d = 

 Mo cos a — Ml cos a + Mt sin c — Ml cos b — Mr sin rf = 



— Mo sin 2a -{-Ml sin 2a + FM, cos 2c + Gil/z, sin 2b — J Mr cos 2d = 



— iWo cos 2c — Ml cos 2c + HM, sin 2c + /Afi cos 2d + KMt sin 2d = 



The 4 boundary conditions and the relation previously stated between the angles and 

 the velocities (sin a : sin b : sin c : sin d = Vi : Vs : Vi : z'2} allow us to calculate 

 the ratios of the amplitudes of the 4 waves produced at the boundary to the ampli- 

 tude of the incident longitudinal wave for any angle of incidence a. 



When the distances between the point of disturbance and the seismometer sta- 

 tions are small relative to the depths of the subsurface boundaries, the reflected 

 and refracted waves strike the surface of the earth almost vertically. In this case, all 

 the angles in the system (4) are approximately zero, and it follows from the second 

 and fourth equations of this system that 



Mo 7 + 1 ' Afo 7 + 1 ^^^''^■' diVi 



If, for example, the densities in the two media are equal (ds = di) and if Vo = 2Vt., 



7 — 2 and 77- — -;-. 

 Mo 3 



Energy of the Waves.^ — The formulas for the energies as given by Knott utilize 



the following abbreviations : 



C ■= cotangent of angle of incidence of longitudinal wave in layer 1. 



C = cotangent of angle of refraction of longitudinal wave in layer 2. 



7 = cotangent of angle of reflection of transverse wave in layer 1. 



7' = cotangent of angle of refraction of transverse wave in layer 2. 



n = modulus of rigidity of layer 1. 



n = modulus of rigidity of layer 2. 



A = energy factor of incident longitudinal wave. 



Ai ■= energy factor of reflected longitudinal wave. 



A' = energy factor of refracted longitudinal wave. f, 



Bi = energy factor of reflected transverse wave. 



B' = energy factor of refracted transverse wave. 



X = A + A^ Y = A-A, (5) 



Knott showed that the quantities just defined are related by the following 

 equations : 



Bi + CY = B' + C'A' 



yBi + X = — y'B' + A' 



-2yB,+ (y'-l) X= 2 ■^y'B' + —(y"-l) A' 

 n n 



(y'-\) Bi-2CY = — (Y'-l) B'-2 — C'A' 

 n n 



(6) 



The systems of equations (5) and (6) determine the values of Ai, A', Bi, and 

 B' as a function of A, C, C, 7, 7', n, and n. 



If, as before, the densities in the layers are denoted by di and rfa, the energy 

 equation may be written in the form : 



CdxA'= Cd^A^ + yd.B^+C'd^A'^ + y'd^B" 



t This section is taken largely from a paper by B. Gutenberg, H. O. ^Vood, and J. P. Buwalda, 

 "Experiments Testing Seismograph Methods," Bulletin of the Seismological Society of America, 

 Vol. 22, 1922. pp. 185-246. 



