The Reduced Wave Equation with a Different Radiation Condition

Ching, Chin-Hung

doi:10.1137/0505070

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1987

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“…In this connection we refer to the partly parallel result of the Russian school (see Eidus 1969 and the review therein). In Ching (1974) and Adams & Roach (1972), the radiation condition has a slightly different form, and the existence of such a radiating solution seems to remain an open question.…”

Section: Introductionmentioning

confidence: 99%

Weighted Sobolev spaces and exterior problems for the Helmholtz equation

Neittaanmäki

Roach

1987

Proc. R. Soc. Lond. A

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Weighted Sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the Helmholtz equation. Furthermore, it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity. In contrast to the Rellich–Sommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form ( ∂ / ∂ r − i k ) u ∈ L 2 ( Ω ) , r = | x | ∈ Ω ⊂ R n , it is shown here that radiating solutions satisfy a condition of the form ( 1 + r ) − 1 2 ( ln ( e + r ) ) − 1 2 δ u ∈ L 2 ( Ω ) , 0 < δ < 1 2 , and, moreover, such solutions satisfy the classical Sommerfeld condition u = O ( r − 1 2 ( n − 1 ) ) , r → ∞ . Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincaré inequality and the Sobolev embedding theorems.

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Section: Introductionmentioning

confidence: 99%