On Some Boundary Value Problems for the Helmholtz Equation in a Cone of 240º

  • A. P. NolascoEmail author
  • F.-O. Speck
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)


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).


Wedge diffraction problem Helmholtz equation boundary value problem half-line potential pseudodifferential operator Sommerfeld potential 


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© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Department of MathematicsI.S.T. Technical University of LisbonLisbonPortugal

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