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Evanescence and Bessel functions in the vibrating circular membrane

The problem of the uniform circular membrane is reviewed in the light of whether or not evanescent-type behaviour can arise in the separated radial r and transverse theta parts of the solution. It is shown on phyiscal grounds that such behaviour inevitably must arise in the radial direction but cannot do so in the transverse. The consequences for the modal functions are considered and shown to be reflected in various well known, but otherwise physically unexplained, properties of the Bessel functions Jm(kr). It is hoped that these physical explanations will make Bessel functions seem more accessible to students.