Linear elasticity
The linear theory of elasticity studies  with mathematical methods 
the macroscopic mechanical properties of solids, assuming "small" deformations.
Basic equations
Linear elastodynamics is based on three tensor equations:
 dynamic equation
 constitutive equation (anisotropic Hooke's law)
 kinematic equation
where:
 is stress
 is body force
 is density
 is displacement
 is elasticity tensor
 is strain
Wave equation
From the basic equations one gets the wave equation

where

is the acoustic differential operator, and is Kronecker delta.
Plane waves
A plane wave has the form

with of unit length.
It is a solution of the wave equation with zero forcing, if and only if
and constitute an eigenvalue/eigenvector pair of the
acoustic algebraic operator

This propagation condition may be written as

where
denotes propagation direction
and is phase velocity.
In isotropic media, the elasticity tensor has the form

where
is incompressibility, and
is rigidity.
Hence the acoustic algebraic operator becomes

where
denotes the tensor product,
is the identity matrix, and
are the eigenvalues of
with eigenvectors parallel and orthogonal to the propagation direction
, respectively.
In the seismological literature, the corresponding plane waves are called Pwaves and Swaves (see Seismic wave).
 Gurtin M. E., Introduction to Continuum Mechanics, Academic Press 1981
 L. D. Landau & E. M. Lifschitz, Theory of Elasticity, Butterworth 1986