An asymptotic fundamental solution of the reduced wave equation on a surface

Author:
C. R. Steele

Journal:
Quart. Appl. Math. **29** (1972), 509-524

MSC:
Primary 73.35; Secondary 80.35

DOI:
https://doi.org/10.1090/qam/408397

MathSciNet review:
408397

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Abstract: The physical problem of steady-state heat conduction in a thin shell is described by the ``reduced wave equation'' in which the differential operator is the (generally noneuclidean) Laplacian for the surface. A similar equation gives the approximation for steady-state waves in a prestressed curved membrane. A modification of the ``geometric optics'' asymptotic expansion, involving a Bessel function, is given for the fundamental point source solution. This is proven to be uniformly valid in the large, until a ``caustic'' is reached. Various features of the solution for a surface, which do not occur for the plane, are discussed.

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DOI:
https://doi.org/10.1090/qam/408397

Article copyright:
© Copyright 1972
American Mathematical Society