How to model realistc (thermal and viscous) damping in fluid-gas scattering problem? I am looking for information on how damping can be incorporated into acoustic scattering problems.
Suppose we have some region $D$ filled with gas in a liquid in $\mathbb{R}^3$. The wave propagation in this case is governed by the Helmholtz equation both inside and outside $D$, with wavenumber $k_i$ inside $D$, and wavenumber $k_o$ outside $D$. In general, for a wavenumber $k$ we have the following relation:
$$k = \omega\frac{\rho}{\kappa},$$
where $\omega$ is the frequency, $\rho$ is the density, and $\kappa$ is the bulk modulus in the particular region under consideration.
I often read papers where the author will consider a wavenumber $k$ 'with a small imaginary part' to model damping. But I would like to have a much more accurate representation of damping than introducing an arbitrary small imaginary part to the wavenumber.
So how can a more accurate damping model that realistically accounts for thermal and viscous damping be developed. Does anyone have experience/knowledge on how to model realistic damping in a scattering problem?
 A: 
I often read papers where the author will consider a wavenumber kk 'with a small imaginary part' to model damping. But I would like to have a much more accurate representation of damping than introducing an arbitrary small imaginary part to the wavenumber.

I think you may be misunderstanding the motivation behind assuming an imaginary part to $k$ and/or $\omega$ in these problems.


*

*Physically, an oscillation that satisfies $\Im \left[ k \right] \neq 0$ corresponds to one whose amplitude changes spatially, i.e., it grows/damps as it propagates to different locations.

*An oscillation that satisfies $\Im \left[ \omega \right] \neq 0$ corresponds to one whose amplitude changes temporally, i.e., it grows/damps with increasing time.


The "assumption" here is largely dependent upon the system and initial conditions.  A more restrictive assumption would be that the oscillations satisfy $\Im \left[ k \right] = 0$ and $\Im \left[ \omega \right] = 0$, i.e., no damping or growth.

So how can a more accurate damping model that realistically accounts for thermal and viscous damping be developed. Does anyone have experience/knowledge on how to model realistic damping in a scattering problem?

I think you may be misunderstanding the motivation again.  A system that satisfies $\Im \left[ k \right] \neq 0$ and/or $\Im \left[ \omega \right] \neq 0$ can be modeled accurately and realistically.  I think you are assuming that the imaginary parts are constants but they can be functions of physical parameters like viscosity.  The functional form of the damping/growth terms depends upon the system.
For example, one can model a viscous fluid using Burgers' equation for a nonlinear acoustic problem like shock waves.
Of course, all of these things are largely applicable to linear and quasi-linear systems.  If the system in question is fully nonlinear (e.g., amplitude depends upon $k$ and/or $\omega$), then things become more complicated and the best (i.e., practical and feasible) approach is generally numerical simulations.
Side Note:  In some systems one can choose either $\Im \left[ k \right] \neq 0$ or $\Im \left[ \omega \right] \neq 0$ and end with effectively the same result.  Thus, the initial assumption is a matter of choice in these systems but as I stated before, there are physical interpretations to either choice.
