Dynamic equations of elasticity theory for a homogeneous isotropic medium

3 Dynamic equations of elasticity theory for a homogeneous isotropic medium Let us recall the general form of the equations of motion of a homogeneous isotropic elastic medium under the influence of its own elastic forces. These equations are of utmost importance for seismology and seismic exploration and lie at the foundation of the majority of their theoretical constructions. To obtain those equations according to Newton s law, let us represent the volume forces as a product of acceleration d U/dt and elastic medium density p  

This equation can be written for scalar components of the stress tensor and deformation vector as follows (Udias, 1999) 

Assuming all deformations to be small, we can employ Hooke s law (13.16) and the deformation tensor expressed through the displacement vector (13.8). Substitution of (13.16) into (13.21) yields the following form for the equation of motion of a homogeneous isotropic elastic medium  

Formulae (13.23) form a system of linear partial differential equations of motion of a homogeneous isotropic elastic medium. These equations can be presented in more compact form using vector notation. 

First of all, it is clear that the bracketed expression on the left-hand side of relation (13.23) is equal to the divergence of the displacement vector  

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