Consider the damped wave equation in 2 dimensions
$$u_{tt}+b(x,y)u_{t}=u_{xx}+u_{yy}$$
where $b(x)$ is not necessarily constant.
One way to try to understand it is to go to polar coordinates, assume that there is no angular dependence and write the solution as a sum of Bessel's functions.
In particular, we could also try to expand the damping $b(r)$ as a sum of Bessel's functions, and try to find some formula for the coefficients using the orthogonality property.
This leads us to the integral
$$\int_{0}^{1}J_{0}(\alpha_{0i}r)J_{0}(\alpha_{0j}r)J_{0}(\alpha_{0k}r)rdr$$
where $J_{0}$ is that Bessel function of first kind, and $\alpha_{0j}$ is its $j$th root.
If we could express it analitically as a function of $i,j,k$, we would have an expression realting the coefficients of the damping expansion with the coefficients of the position expansion.
After some search, I could not find any formula for it, I would appreciate if someone has a tip on how to proceed from here.