Geodesic
Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018), pp. 393-405

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We study the Cauchy problem for the inhomogeneous two-dimensional wave equation with variable coefficients and zero initial data. The right-hand side is assumed to be localized in space and time. The equation is considered in a domain with a boundary (shore). The velocity is assumed to vanish on the shore as a square root of the distance to the shore, that is, the wave equation has a singularity on the curve. This curve determines the boundary of the domain where the problem is studied. The main result of the paper is efficient asymptotic formulas for the solution of this problem, including the neighborhood of the shore. 
@article{JMAG_2018_14_a0,
     author = {Anatoly Anikin and Sergey Dobrokhotov and Vladimir Nazaikinskii},
     title = {Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {393--405},
     publisher = {mathdoc},
     volume = {14},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2018_14_a0/}
}
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Anatoly Anikin; Sergey Dobrokhotov; Vladimir Nazaikinskii. Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018), pp. 393-405. http://geodesic.mathdoc.fr/item/JMAG_2018_14_a0/