High frequency asymptotic solutions of the reduced wave equation on infinite regions with non-convex boundaries
The asymptotic behavior as λ→∞ of the function U(x,λ) that satisfies the reduced wave equation Lλ[U]=∇⋅(E(x)∇U)+λ2N2(x)U=0 on an infinite 3-dimensional region, a Dirichlet condition on ∂V , and an outgoing radiation condition is investigated. A function UN(x,λ) is constructed that is a global appr...
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
1996-01-01
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| Series: | Mathematical Problems in Engineering |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S1024123X96000385 |
| Summary: | The asymptotic behavior as λ→∞ of the function U(x,λ) that satisfies the reduced wave equation Lλ[U]=∇⋅(E(x)∇U)+λ2N2(x)U=0 on an infinite 3-dimensional region, a Dirichlet condition on
∂V
, and an outgoing radiation condition is investigated. A function
UN(x,λ)
is constructed that is a global approximate solution as
λ→∞ of the problem satisfied by U(x,λ)
. An estimate for
WN(x,λ)=U(x,λ)−UN(x,λ)
on V
is obtained, which implies that
UN(x,λ)
is a uniform asymptotic approximation of
U(x,λ)
as
λ→∞, with an error that tends to zero as rapidly as
λ−N(N=1,2,3,...). This is done by applying a priori estimates of the function
WN(x,λ) in terms of its boundary values, and the L2 norm of rLλ[WN(x,λ)]
on V. It is assumed that
E(x), N(x), ∂V
and the boundary data are smooth, that
E(x)−I and N(x)−1 tend to zero algebraically fast as r→∞, and finally that
E(x) and
N(x) are slowly varying;
∂V may be finite or infinite. |
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| ISSN: | 1024-123X 1563-5147 |