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.
