Geodesic
Case of an exact integrability of $SH$-wave equation
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 22, Tome 203 (1992), pp. 12-16

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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. 
@article{ZNSL_1992_203_a1,
     author = {V. M. Babich},
     title = {Case of an exact integrability of $SH$-wave equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {12--16},
     publisher = {mathdoc},
     volume = {203},
     year = {1992},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1992_203_a1/}
}
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V. M. Babich. Case of an exact integrability of $SH$-wave equation. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 22, Tome 203 (1992), pp. 12-16. http://geodesic.mathdoc.fr/item/ZNSL_1992_203_a1/