## Proof That the Equation of Motion For an Elastic Isotropic Material is a Wave Equation

A material is elastic if it returns to its original shape when a force is removed and isotropic if the material has no favoured direction for an applied force.

Let and be the position vectors of a point P before and after deformation. The displacement of the point P is then If this displacement is small then the force will be proportional to the displacement.

Let be the force density, then the motion of the material is where and are constants of the material.

Neglecting any body forces such as gravity, where is the density of the material.

Hence Any vector field can be written as the sum of solenoidal and irrotational fields and satisfying Hence Take the divergence of this equation to give Interchange the order of differentiation on the left and use  Move everything to the left and factor out  Since the curl of the expression in brackets is zero. The only way the curl and divergence can both be zero is if the expression is zero hence The wave equation takes this form.

Take the curl of to give     Hence This also has the form of a wave equation. 