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Friends in Partial Differential Equations
May 25, 2024 14:55, St. Petersburg, St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, online


High-frequency diffraction by a jump of curvature. Tangential incidence

E. A. Zlobina

Saint Petersburg State University



Abstract: We are concerned with the construction of asymptotic formulas within the framework of systematic boundary layer method [1]. We consider two-dimensional diffraction by the contour $C$ composed of half-line $C_-$ and part of smooth curve $C_+$ (see Fig. 1) with a jump in curvature at the conjugation point $O$. The plane wave $e^{ikx}$ with large wavenumber $k \to \infty$ grazes along $C_-$ to $O$, see Fig. 1. The outgoing wave $u^\mathrm{out}$ is governed by the Helmholtz equation and the Neumann boundary condition:
\begin{equation*} u^\mathrm{out}_{xx}+u^\mathrm{out}_{yy}+k^2u^\mathrm{out}=0, \quad \left. \partial_n \left( u^\mathrm{out} + e^{ikx} \right) \right|_C = 0. \end{equation*}
Here, $\partial_n$ denotes the derivative along the normal to the contour $C$.
fig1.jpg
Fig. 1. The geometry of the problem.

fig2.jpg
Fig. 2. The sketch of boundary layers.

The problem has been previously addressed by A.V. Popov [2] who, using a heuristic approach, derived an expression for the diffracted cylindrical wave diverging from the jump point $O$ (see Fig. 1).
Our aim is to develop a formal boundary layer approach and construct high-frequency approximations for the wavefield in the neighborhood of the limit ray (orange zone in Fig. 1). We base on the Leontovich—Fock parabolic equation method [1,3]. This allows an asymptotic description of the wavefield in boundary layers surrounding the limit ray (see Fig. 2), which involves novel special functions.
The talk is based on the joined work with A.P. Kiselev [4].

Language: English

References
  1. V.M. Babich, N.Ya. Kirpichnikova, The Boundary Layer Method in Diffraction Problems, Springer, Berlin, 1979  mathscinet  zmath
  2. A.V. Popov, “Backscattering from a line of jump of curvature”, Trudy V Vses. Sympos. Diffr. Raspr. Voln, 1970, Nauka, Leningrad, 1971, 171–175
  3. V.A. Fock, Electromagnetic Diffraction and Propagation Problems, Pergamon Press, Oxford, 1965  mathscinet
  4. E.A. Zlobina, A.P. Kiselev, “The Malyuzhinets—Popov diffraction problem revisited”, Wave Motion, 121 (2023), 103172  crossref  mathscinet  zmath


© Steklov Math. Inst. of RAS, 2024