448 



SEISMIC METHODS 



[Chap. 9 



directions are 0, = du/dx. Hence, X^ = (^ + 2}i) du/dx. Y^ follows 

 from eq. (9-13). Since the problem is limited to propagation in the 

 X direction and the shears in the y and z directions are zero, Y^ = y dv/dx 

 and Zx = V dw/dx. By differentiation of these components with respect 

 to X as required by (9-14), the following equations are obtained: 



> (9-15) 



The first equation indicates the gradient of the specific strain du/dx in 

 the X direction, that is, in the direction of propagation. It is thus the 



expression for a compressional 

 and longitudinal wave. The 

 second equation gives the gra- 

 dient or propagation of the 

 shear dv/dx in the x direction ; 

 and the third equation, the 

 gradient or propagation of the 

 shear dw/dx in the x direction. 

 This applies to a plane wave 

 traveling in one direction (see 

 Fig. 9-3); however, it can be 

 shown that the equations 

 here derived also hold for 

 space waves. 

 Eqs. (9-15) show that the acceleration in a longitudinal wave (or com- 

 pression d\/dx^) is proportional to (^ + 2|i) /5, while in the second case it 

 is proportional to yi/h. The two last waves are shear waves; they are not 

 propagated independently. Figure and equations merely indicate the two 

 (z and y) components of the wave whose relative amplitudes determine the 

 plane of polarization. 



The proportionality factors in the last three equations may be designated 

 by v', thus: vf = (3. + 2v)/6 and v' = y /6. Then d\/dt^ = v^^'^^/^a:'; 

 d\/dt^ = v\-d\/dx) d^w/d^ = v\-d^w/dx. The velocities of the longi- 



Fig. 



T7» 



9-3. Propagation of longitudinal and shear 

 deformations. 



tudinal and transverse waves are, therefore, 



and Vt = 



(9-16) 



