Elastic Waves, with Seismologieal Applications. 93 



interface, or 



^- + ^- = ^— + -^— when .t' = 0. 

 d# dy ox dj/ 



(2) Equality of tangential displacements on each side of 

 the interface, or 



M_M = ^_M w hen»=0. 

 dy d-B oy o>» 



(3) Equality of normal stresses on each side of the inter- 

 face, or 



(m+ „)y*-HP - H) = k*-) w-K0 - g£) 



when .t' = 0. 



(4) Equality of tangential stresses on each side of the 

 interface, or 



/« W , B 2 ^ 3V\_ ,/ 9 W ,^ W 

 "lWy + BF" 5? J " l V 2 B^% + V W ) 



when # = 0. 



These lead to the equations 



(2) 



-7A1+ 



X = 7 'A' + 







B r 



A x - 



cY= A'- 







c'B' 



(c 2 -l)A 1 + 



2cY=-(V 2 -l)A' + 



71 







^ SB' 



n 



27A!+(c 2 - 



-1)X= -— y'A'4- 

 7 n ' 



ri 



n 



(0 



2 -l)B' 



whereX=B + B 1 



, andY=B-B 1 . 









Multiplying together the iirst and third of these and also 

 the second and fourth, and then taking the difference of the 

 two equations so obtained, we get 



(c 2 +l)7Ai 2 -(c 2 +l)cXY = -^(c' 2 + l)7'A' 2 -^(c' 2 + iyB^ 



which by (1) becomes 



r / pt±* + ry'p>A l2 + c'p'B' 2 =cpXY = cpB*-cpB 1 2 . ... (3) 



This is the energy equation showing how the original 

 energy (?pB 2 ) of the incident wave is distributed among the 

 four waves into which it breaks up at the interface. In the 



* These correspond to Kelvin's formulae Nos. 39-42 in his paper of 

 1888 (Phil. Mag. xxvi. p. 422). 



