On the shielding effect of the Helmholtz equation

Hadamard‐type instability has been known for over a century as a cause of ill‐posedness of the Cauchy problem for elliptic PDEs. This ill‐posedness manifests itself as evanescent modes growing exponentially when propagated in the reverse direction. Since every oscillating mode of the Laplace equation is evanescent, the ill‐posedness of its Cauchy problem is solely due to Hadamard‐type instability. The presence of the propagating modes and beams for the Helmholtz equation gives rise to an entirely different type of ill‐posedness, hitherto unknown to the practice, and untreated by the theory, of inverse scattering. We will present this fundamental phenomenon of ill‐posedness for the Helmholtz equation. © 2007 Wiley Periodicals, Inc.