Acoustic Medium

In the special case of an acoustic medium the stiffness matrix and the stress tensor reduces to a scalar value (impedance and pressure). Identifying u in (5) with the pressure gradient, we can write again the familiar set of first order equations.

$$\begin{eqnarray} \rho {{\partial}\over{\partial{t}}} u_i = -{{\partial}\over{\pa... ...\partial}\over{\partial{t}}}P = K {\partial\over{\partial x_k}}u_k\end{eqnarray}$$ (6) (7)

This corresponds to the general form (1), where here K is the bulk modulus and $\rho$ is the density.

It is obvious from the previous two paragraphs that the equation structure is identical for different type of media. One can either treat the acoustic case as a completely self contained problem or as an elastic problem where the material properties happen to be acoustic. Differences in spatial dimensionality or component dimensionality are reflected just in the bounds of the subscripts. From that we can conclude that if we have a given modeling structure we can use it for all our modeling purposes. It is just a matter of keeping track of the indices and to make sure those indices are efficiently mapped in each case. Thus all the building blocks can reuse the same pieces of code if they are written in a general form (i.e. using the full anisotropic formulation). Clearly for small spatial variations in density we could use a more compact Laplacian in case of the cascaded first order derivative operators.