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.  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 $$n(x)\rightarrow n_{\infty} (x/|x|)$$ as $$|x| \rightarrow \infty$$. 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     $$\int_{{\rm \mathbb{R}}^{d}}\left| \nabla u-in^{1/2}_{\infty}u\frac{x}{\mid x \mid}\right|^{2}\frac{dx}{\mid x \mid}     It is a very striking and unexpected feature that the index n ∞ appears in this formula and not the gradient of the phase as established by Saito in [S] and broadly used numerically. This apparent contradiction is clarified by the existence of some extra estimates on the energy decay. In particular we prove that     $$\int_{{\rm \mathbb{R}}^{d}}\left|\nabla_{\omega}n_{\infty}\left(\frac{x}{\mid x \mid}\right)\right|^{2}\frac{{\mid u \mid}^2}{\mid x \mid}dx     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.

Energy Concentration and Sommerfeld Condition for Helmholtz Equation with Variable Index at Infinity