Case of an exact integrability of $SH$-wave equation

The equation for $SH$-wave is considered
$$ \frac\partial{\partial x}\left(\mu\frac{\partial w}{\partial x}\right)+\frac\partial{\partial y}\left(\mu\frac{\partial w}{\partial y}\right)=\rho\frac{\partial^2w}{\partial t^2}, $$
when $\mu=a(x)b(y)$, $\rho=a(x)b(y)(c(x)+d(y))$ ($a,b,c,d$ are known functions). The problem of interaction of a whispering gallery wave with a vertical interface is solved in explicit form.