Greens tensor for vector wave equation

Now let us analyze the fundamental solutions of the vector wave equation  

We found in the previous section that this equation describes separately the propagation of the compressional and shear waves in a homogeneous medium. 

Similarly to the scalar case (equation (13.65)), an arbitrary source field in the vector equation can be represented as the sum of point pulse sources  

Note that the last equation can be rewritten in the equivalent form 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G (r, t), which we call the Green s tensor of the vector wave equation. 

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