where is symmetric (i.e., ) and traceless (i.e., ), is isotropic, and only has three independent components.
Show that the coefficients in the previous expression transform under rotation of the coordinate axes like the components of a symmetric second-order tensor. Hence, demonstrate that the equation for the surface can be written in the form
where the are the components of the aforementioned tensor.
is an alternative, and entirely equivalent, definition.
then and are said to be eigenvalues and eigenvectors of the second-order tensor , respectively. The eigenvalues of are calculated by solving the related homogeneous matrix equation
Now, it is a standard result in linear algebra that an equation of the previous form only has a non-trivial solution when (Riley 1974)
Demonstrate that the eigenvalues of satisfy the cubic polynomial
where and . Hence, deduce that possesses three eigenvalues-- , , and (say). Moreover, show that
the isotropic stiffness tensor. Here, and are the bulk modulus and shear modulus of the medium, respectively. Show that the divergence and the curl of both satisfy wave equations. Furthermore, demonstrate that the characteristic wave velocities of the divergence and curl waves are and , respectively.