Solving for particle motions

Equation 26 is a wave equation that combines elastic strains, electric displacements, and entropy. It describes how, in a given medium, a traveling wavetype is typically neither purely elastic, nor purely electromagnetic, nor purely thermal. Instead, one wave includes all such attributes. However it is useful to term them quasi-elastic when most of the energy is transferred by strain changes, or quasi-electric when most of the energy propagates in form of electromagnetic waves.

Solving the eigenvalue equation 28, we obtain as eigenvalues the propagation velocities for the elastic waves, the electromagnetic and the thermal wave. Although ${\bf \epsilon_1}$ is a 9 component tensor there are only three non-zero eigenvalues, corresponding to three propagating elastic wavetypes. The electromagnetic part will typically have two non-zero eigenvalues, corresponding to two propagating wavetypes. The eigenvalue for the thermal wave will be complex. Consequently we can also calculate propagation velocities and particle motion for the electromagnetic wavetypes and the heat wave.

In order to calculate the elastic displacements we have to extract the three non-zero eigenvalues (v₁,v₂,v₃) and eigenvectors (${\bf V_1},...,{\bf W_3}$) from the coupled system. We can combine them as follows:  

$$\displaystyle \mathop{\mbox{${\bf A}$}}_{\mbox{$\sim$}} = \p... ...&0 \cr 0&0&v_3 \cr} \pmatrix{{\bf W_1}&{\bf W_2}&{\bf W_3}\cr}$$ (29)

to produce the purely elastic wave equation which uniquely describes the propagation of strains with the correct (coupled) velocity. This purely elastic equation can be turned into a generalized eigenvalue problem for the elastic displacements u, as follows:  

$$\displaystyle \mathop{\mbox{${\bf K}$}}_{\mbox{$\sim$}} ~ \d... ...laystyle \mathop{\mbox{${\bf K^T}$}}_{\mbox{$\sim$}} ~{\bf u} .$$ (30)

The eigenvalues and eigenvectors represent now propagation velocity and particle motion respectively and are physically meaningful.

APPENDIX B: SYMBOLS

The following definitions translate the symbols used in the notation into commonly used terms:

• ${\bf \sigma_1}$ = elastic stress (second rank tensor)
• ${\bf \sigma_2}$ = electric stress (``electric field'', first rank tensor, vector)
• ${\bf \sigma_3}$ = heat stress (``Temperature'', zeroth rank tensor, scalar)
• ${\bf \epsilon_1}$ = elastic strain (second rank tensor)
• ${\bf \epsilon_2}$ = electric strain (``electric displacement'', first rank tensor, vector)
• ${\bf \epsilon_3}$ = heat strain (``Entropy'', zeroth rank tensor, scalar)
• $$\mathop{\mbox{${\bf C}$}}_{\mbox{$\sim$}}$$ = generalized stiffness matrix
• $$\mathop{\mbox{${\bf K}$}}_{\mbox{$\sim$}}$$ = symmetric spatial derivative matrix

The matrix notation for $$\mathop{\mbox{${\bf K}$}}_{\mbox{$\sim$}}$$ is

$$\displaystyle \mathop{\mbox{${\bf K}$}}_{\mbox{$\sim$}} ~ = ... ...0 & k_y & 0 & k_z & 0 &k_x \cr 0 & 0 & k_z & k_y & k_x &0 \cr}$$ (31)

The matrix formulation for ${\rm \bf curl}$ is

$$\displaystyle \mathop{\mbox{${\bf R}$}}_{\mbox{$\sim$}} ~=~i... ...rix{0 & -k_z & k_y \cr k_z & 0 & -k_x \cr -k_y & k_x & 0 \cr}$$ (32)

${\rm \bf Grad}$ and ${\rm \bf Div}$ are denoted as

$${\rm \bf Grad}~=~i~\pmatrix{k_x \cr k_y \cr k_z \cr}$$ (33)

$${\rm \bf Div}~=~i~\pmatrix{k_x & k_y & k_z \cr}$$ (34)

APPENDIX C