Abstract
In this paper, we study the wave equation on the simplest hybrid spaces of constant curvature, namely, on Euclidean space or a sphere with a glued ray. We obtain explicit formulas for solutions of the Cauchy problem, which are the simplest nontrivial analogs of Kirchhoff or Herglotz-Petrovsky formulas; especially simple formulas are obtained in the case of three-dimensional Euclidean space with a glued ray. The solutions depend on the boundary conditions at the point of gluing, and these conditions determine the choice of the domain of the Laplace operator; the conditions ensuring the full reflection or full passage of waves are described separately.
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The research was financially supported by the RFBR grants nos. 12-01-31235, 13-01-00664, and 14-01-00521.
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Shafarevich, A.I., Tsvetkova, A.V. Solutions of the wave equation on hybrid spaces of constant curvature. Russ. J. Math. Phys. 21, 509–520 (2014). https://doi.org/10.1134/S1061920814040098
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DOI: https://doi.org/10.1134/S1061920814040098