High Frequency Wave Propagation

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High-frequency Behaviour of the Helmholtz Equation: Radiation Condition at Infinity, Bounds, and a Counter-example

François Castella

Mathematics at IMAR - Université de Rennes 1

Abstract:   We consider the high-frequency Helmholtz equation in the whole space, with a variable index of refraction. More precisely, we select the outgoing solution to the above-mentionned Helmholtz equation, i.e. the (unique) solution that satisfies the Sommerfeld radiation condition at infinity. Our main result establishes that this solution radiates energy in the outgoing direction only, uniformly along the high-frequency parameter. This result holds provided the associated rays of geometric optics nicely escape to infinity, and they satisfy an original non-focusing condition. The analysis relies on carefully following the dispersive properties of the Helmholtz equation, for small, moderate, and large values of the space variable. A counterexample shows the optimality of the non-focusing condition. Uniform boundedness of the solution is also considered.

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