Abstract
The Helmholtz wave equation [∇2+ (ω/c)2]W − 0 is customarily solved by the familiar, college-taught method of separation of variables. Separable solutions so obtained can be considered a special—and very important—case in an infinite set of nonseparable solutions of this equation. [See Quart. Appl. Math. 22, 354 (1965).] Known nonseparables of this set can be derived from the separable by repeated application of a generative operator. In view of this, a separable is looked upon as a nonseparable of zero order. An example in two dimensions is w = W exp iωt, where W is amplitude, W(0) = sinax sinby, W(1) = bx cosax sinby − ay sinax cosby, the operator connecting them is b∂/∂a−a∂/∂b, and the frequency eqation for both is (ω/c)2=a2+b2. Some properties of nonseparable members of the set are presented, and application to the transverse displacement of a membrane is illustrated.
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