University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Diffraction by wedges: higher order boundary conditions, integral transforms, vector Riemann-Hilbert problems, and Riemann surfaces

Diffraction by wedges: higher order boundary conditions, integral transforms, vector Riemann-Hilbert problems, and Riemann surfaces

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CATW02 - Complex analysis in mathematical physics and applications

Acoustic and electromagnetic diffraction by a wedge is modeled by one and two Helmholtz equations coupled by boundary conditions. When the wedge walls are membranes or elastic plates, the impedance boundary conditions have derivatives of the third or fifth order, respectively. A new method of integral transforms is proposed. It requires mapping the wedge domain into a right-angled structure and applying two Laplace transforms. The main feature of the method is that the second integral transform parameter is a specific root of the characteristic polynomial of the ordinary differential operator resulting from the transformed PDE . For convex domains (concave obstacles), the problems reduce to scalar and order-2 vector Riemann-Hilbert problems. When the wedge is concave (a convex obstacle), the acoustic problem is transformed into an order-3 Riemann-Hilbert problem. The order-2 and 3 vector Riemann-Hilbert problems are solved by recasting them as scalar Riemann-Hilbert problems on Riemann surfaces. Exact solutions of the problems are determined. Existence and uniqueness issues are discussed.

This talk is part of the Isaac Newton Institute Seminar Series series.

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