The many faces of the Helmholtz equation

A number of equivalent configurations in the physical sciences are shown to be described by the Helmholtz equation applied to a simply connected domai n in two and three dimensions bounded by a convex surface with either Diric hlet or Neumann boundary conditions. Such examples include the quantum bill iard for a spinless particle of finite mass in the relativistic and nonrela tivistic limits, the vibrating membrane, the quantum wire and quantum dot, sound waves propagating in a homogeneous cylinder, TM and TE modes in a met al-walled homogeneous waveguide, the vibrating plate, and vector electric a nd 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.