Webster equation is a popular generalization of 1D wave equation used for ducts of variable cross-section $S \equiv S(x)$. Assuming harmonicity in time, the spatial equation for propagation of variable $\Phi(x)$ is:$$\frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial (\ln S(x))}{\partial x}\frac{\partial \Phi}{\partial x} + k^2\Phi^2 = 0$$What FD scheme would be appropriate assuming that we have boundary conditions at $\Phi(0)$ and $\Phi(L)$, i.e. no $\partial_x\Phi(0)$ etc.?
Finite difference scheme for Webster equation
finite difference;boundary conditions;wave propagation