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Volume 12: Pages 492-498, 1999

The Many Faces of the Helmholtz Equation

Richard L. Liboff

Schools of Electrical Engineering and Applied Physics and Center for Applied Mathematics, Cornell University, Ithaca, NY 14843 U.S.A.

A number of equivalent configurations in the physical sciences are shown to be described by the Helmholtz equation applied to a simply connected domain in two and three dimensions bounded by a convex surface with either Dirichlet or Neumann boundary conditions. Such examples include the quantum billiard for a spinless particle of finite mass in the relativistic and nonrelativistic limits, the vibrating membrane, the quantum wire and quantum dot, sound waves propagating in a homogeneous cylinder, TM and TE modes in a metal‐walled homogeneous waveguide, the vibrating plate, and vector electric and magnetic free fields. Solutions of the Helmholtz equation for geometries which allow separation of variables in a given orthogonal coordinate frame, or which have indigenous symmetry, are also discussed.

Keywords: Helmholtz equation, Dirichlet and Neumann boundary conditions, quantum billiard, vibrating membrane, vibrating plate, wave‐guide modes, symmetric domains, electric and magnetic free fields, field chaos

Received: February 18, 1999; Published online: December 15, 2008