  We develop powerful numerical and analytical techniques for the solution of
the Helmholtz equation on general domains. We prove two theorems: the first
theorem provides an exact formula for the ground state of an arbirtrary
membrane, while the second theorem generalizes this result to any excited state
of the membrane. We also develop a systematic perturbative scheme which can be
used to study the small deformations of a membrane of circular or square
shapes. We discuss several applications, obtaining numerical and analytical
results.
