Energy Concentration and Sommerfeld Condition for Helmholtz Equation with Variable Index at Infinity |
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Authors: | Benoit Perthame Luis Vega |
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Institution: | (1) UMR 7598 LJLL, BC187, Université Pierre et Marie Curie-Paris 6, 4, place Jussieu, F-75252 Paris cedex 5, France;(2) Institut Universitaire de France, Paris cedex 5, France;(3) Universidad del Pais Vasco, Apdo. 644, 48080 Bilbao, Spain |
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Abstract: | We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like as . Under some appropriate assumptions on this convergence and on n
∞ we prove that the Sommerfeld condition at infinity still holds true under the explicit form It is a very striking and unexpected feature that the index n
∞ appears in this formula and not the gradient of the phase as established by Saito in S] and broadly used numerically. This
apparent contradiction is clarified by the existence of some extra estimates on the energy decay. In particular we prove that
In fact our main contribution is to show that this can be interpreted as a concentration of the energy along the critical
lines of n
∞. In other words, the Sommerfeld condition hides the main physical effect arising for a variable n at infinity; energy concentration on lines rather than dispersion in all directions.
Received: March 2006, Revision: July 2006, Accepted: July 2006 |
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Keywords: | and phrases:" target="_blank"> and phrases: Helmholtz equation Energy concentration Sommerfeld condition index of refraction |
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