Geodesic
The construction of Gaussian beams satisfying to the wave equation with exponential exactness
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 22, Tome 203 (1992), pp. 17-20

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A function $V$ having Gaussian beam asymptotic (the frequency $\omega\to+\infty$) is obtained. The expression $(\Delta+\frac{\omega^2}{c^2(x)})V$ is exponentialy small in the the neighbourhood of axial ray of Gaussian beam. The velocity $c(x)$ is analitical. 
@article{ZNSL_1992_203_a2,
     author = {V. M. Babich},
     title = {The construction of {Gaussian} beams satisfying to the wave equation with exponential exactness},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {17--20},
     publisher = {mathdoc},
     volume = {203},
     year = {1992},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1992_203_a2/}
}
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V. M. Babich. The construction of Gaussian beams satisfying to the wave equation with exponential exactness. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 22, Tome 203 (1992), pp. 17-20. http://geodesic.mathdoc.fr/item/ZNSL_1992_203_a2/