On Some Boundary Value Problems for the Helmholtz Equation in a Cone of 240º
In this paper we present the explicit solution in closed analytic form of Dirichlet and Neumann problems for the Helmholtz equation in the non-convex and non-rectangular cone Ω0,α with α = 4π/3. Actually, these problems are the only known cases of exterior (i.e., α > π) wedge diffraction problems explicitly solvable in closed analytic form with the present method. To accomplish that, we reduce the BVPs in Ω0,α each to a pair of BVPs with symmetry in the same cone and each BVP with symmetry to a pair of semi-homogeneous BVPs in the convex half-cones. Since α/2 is an (odd) integer part of 2π, we obtain the explicit solution of the semi-homogeneous BVPs for half-cones by so-called Sommerfeld potentials (resulting from special Sommerfeld problems which are explicitly solvable).
KeywordsWedge diffraction problem Helmholtz equation boundary value problem half-line potential pseudodifferential operator Sommerfeld potential
Unable to display preview. Download preview PDF.
- 7.T. Ehrhardt, A.P. Nolasco and F.-O. Speck, A Riemannn surface approach for diffraction from rational wedges, to appear.Google Scholar
- 8.G.I. Èskin, Boundary Value Problems for Elliptic Pseudodifferential Equations Translations of Mathematical Monographs 52, AMS, 1981.Google Scholar
- 9.P. Grisvard, Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics 24, Pitman, 1985.Google Scholar
- 11.G.C. Hsiao and W.L. Wendland, Boundary Integral Equations. Applied Mathematical Sciences Series 164, Springer-Verlag, 2008.Google Scholar
- 13.G.D. Malujinetz, Excitation, reflection and emission of the surface waves on a wedge with given impedances of the sides. Dokl. Acad. Nauk SSSR 121 (1958), 436-439.Google Scholar
- 15.E. Meister, Some multiple-part Wiener-Hopf problems in Mathematical Physics. Mathematical Models and Methods in Mechanics, Banach Center Publications 15,PWN–Polish Scientific Publishers, Warsaw (1985), 359-407.Google Scholar
- 27.P.Ya. Ufimtsev, Theory of Edge Diffraction in Electromagnetics. Tech Science Press, 2003.Google Scholar