Chap. 9] 



SEISMIC METHODS 



447 



As dz<pxi = du and dx<ptx = dw, <pxt + ^«i = t- + ^- and 



a« ax 



2^" = a^ + a^ 



^^- = 91 + 32/ 



^^- = 9^ + 9i- 

 Combining eqs. (9-11) and (9-12): 



(9-12) 



X.(=Z,) = v('^ + '"' 



.92 



bx, 



-■<=-•> -(:-:+S 



x„(= 



=^-> = "&+!) 



(9-13) 



2. Proyagalion of deformations; longitudinal and transverse types of waves 

 {dynamic problem). For simplification of analysis, consider a plane wave 

 resulting from displacements u in the x direction. Then only the com- 

 ponents Xj; , Y;c , and Z^, shown in Fig. 9-1 have to be considered. These 

 forces are referred to unit area and therefore their action on the y-z side 

 of an elementary parallelopiped is given by expressions of the form 

 Xx-dydz, or dX^/dx-dV. Although the primary deformation is in the 

 X direction, deformations in the y and z directions result from contraction. 



The accelerations correspondmg to these deformations are d^u/df, 

 d%/dt^, and d^w/dt^. Since the force (dXx/dxdV) is mass (dV-h, with 5 = 

 density) times acceleration {d u/dt ), the equations for a (plane) wave in 

 the X direction are 





dXx 



dx 



aj; ^ 9Y. 

 ' 9^2 dx 



' (9-14) 



5. 



9<2 



dx 



In these equations, X: 

 e follows from (9-1). 



X, as given by formula (9-9), = 03t + 2^ du/Bx. 

 Since the initial specific strains in the y and z 



