Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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

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