Greens tensor for the Lame equation

Now let us consider the Green s tensor for the system of equations of dynamic elasticity theory, the vector form of which is called a Lame equation. We will call this tensor an elastic oscillation tensor G or Green s tensor for the Lame equation. As in the case of the vector wave equation, discussed above, the components of the elastic oscillation tensor describe the propagation of elastic waves generated by a point pulse force. In other words, it satisfies the following Lame equation (see equation (13.29))  

We have learned already that any elastic oscillation can be represented as a superposition of the compressional and shear waves, which correspond to the potential and solenoidal parts of the elastic displacement field. Therefore, it is clear that the elastic tensor G can also be represented as the sum of the potential and solenoidal components, described by tensor functions G W and G respectively  

Omitting the long derivation, which can be found in the relevant text books on the theory of elastic waves (see, for example, Morse and Feshbach, 1953 Aki and Richards, 2002), I present here the final expressions for functions G and G in a homogeneous medium  

Using Green s tensor G (r, t) for the Lame equation, we can express the solution of this equation for an arbitrary right-hand side F (r, t) as the convolution of the Green s tensor G (r, ) with the function F (r, t), i.e., 

Similar to the vector wavefield, we can obtain the integral representation of the frequency domain elastic wavefield 

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