# Solution of a Sommerfeld Diffraction Problem with a Real Wave Number

## Abstract

We consider a problem of wave diffraction by a half-plane with general boundary and transmission conditions of first and second kind. The problem is taken in the framework of Bessel potential spaces and several Wiener-Hopf operators are introduced in order to translate the conditions initially stated. Similar problems can be found in the work of E. Meister and F.-O. Speck (see, e.g., [1]). In the present work, the main difference is the possibility to consider a real wave number. The class in study contains, as a particular case, the Rawlins’ Problem [1] which was already considered by K. Rottbrand also in the limiting case of a wave number with a null imaginary part [2]. The study is carried out with the help of some factorization techniques, certain projectors and a representation due to Laplace-type integrals. As a consequence, the exact solution of the problem is obtained in a form that is still valid for the limiting case of a real wave number.

## Keywords

Helmholtz Equation Singular Integral Operator Representation Formula Transmission Condition Intermediate Space## Preview

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## References

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