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Asymptotic expansion of integrals occurring in linear wave theory

Published online by Cambridge University Press:  17 April 2009

P. van den Driessche  and
R.D. Braddock
P. van den Driessche
Affiliation:
Department of Mathematics, University of Victoria, Victoria, British Columbia, Canada
R.D. Braddock
Affiliation:
Department of Mathematics, University of Queensland, St Lucia, Queensland.
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The asymptotic expansion of an integral of the type , is derived in terms of the large parameter t. Functions Φ(k) and ψ(k) are assumed analytic, and ψ(k) may have zeros at a stationary phase point. The usual one dimensional stationary phase and Airy integral terms are found as special cases of a more general result. The result is used to find the leading term of the asymptotic expansion of the double integral. A particular two dimensional Φ(k) relevant to surface water wave problems is considered in detail, and the order of magnitude of the integral is shown to depend on the nature of ψ(k) at the stationary phase point.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 1 , August 1972 , pp. 121 - 130
Copyright
Copyright © Australian Mathematical Society 1972

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