FORMAL SPLITTING OF THE WAVE OPERATOR

The constitutive equation links basic wave propagation observables with the medium parameters. Combining a force balance equation with a constitutive equation results in a wave equation. In this exposition I use the elastodynamic wave equation. The constitutive equation and force balance without exterior forces in a perfectly elastic medium are

$$\begin{eqnarray} \epsilon & = &\nabla^t u \\ \sigma &=& { \displaystyle \mathop{... ..._{\mbox{$\sim$}} }\ \epsilon \\ \rho \ \ddot{u} &=& \nabla \sigma \end{eqnarray}$$ (1) (2) (3)

where u is the elastic displacement, $\epsilon$ the strain, $\sigma$ the stress and ${ \displaystyle \mathop{\mbox{${\bf C}$}}_{\mbox{$\sim$}} }$ the stiffness tensor. The resulting wave equation involves calculating tensor products and derivatives. This wave equation operator L can be defined as  

$$L u = \nabla ( { \displaystyle \mathop{\mbox{${\bf C}$}}_{\mbox{$\sim$}} } \nabla^t u )$$ (4)

$\nabla$ is a partial derivative operator matrix acting on its argument. ${ \displaystyle \mathop{\mbox{${\bf C}$}}_{\mbox{$\sim$}} }$ is the stiffness tensor and u is the displacement field. Using the chain rule for covariant derivatives and applying it to the tensor product, we can rewrite equation (4) to  

$$L\ u\ =\ ( \stackrel{\downarrow}{\nabla} \stackrel{\downarr... ...}$}}_{\mbox{$\sim$}} } ) \stackrel{\downarrow}{(\nabla^t u )}.$$ (5)

The $\downarrow$ pairs indicate the entity on which the matrix of derivative operator is acting upon. In the first term of equation (5) derivatives of the elastic constants are computed; in the second term only derivatives of the strain tensor are computed. The stiffness components in the second term act as constant coefficients for the derivate matrix elements. We can rewrite operator equation (5) in terms of its tensor elements by using (1) and get  

$$l_i = \sum_{jkl} [\ (\ {\partial_i\over{\partial_j}} + {\p... ...+ \ {\partial_j\over{\partial_i}}\ ) \ {\bf\epsilon_{kl}}\ ]$$ (6)

where [...] indicates the scope of the partial derivatives and where $\mbox{i,j,k,l~=~1,2,3}$.Equation (6) is written in its most general form and is valid for anisotropic media. The derivative operation in equation (6) appears in both terms. It acts, however, on two different entities. In the first term the derivative of medium properties is taken and in the second term the derivative of the wave field is computed. In the end summation of both terms produces the complete derivative without taking a derivative of the medium and wave field product, as equation (4) would suggest. For a medium with constant properties the first term vanishes, leaving the second term in equation (6), whose Fourier transform represents the Christoffel equation. From an economical point of view equation (6) doesn't seem attractive, but the opportunity lies in computing the solution to equation (4) more accurately and more realistically for certain types of media. In particular the properties of derivatives can be adjusted to match the properties of the observables.