Vibration of a circular membrane

Clearly, we are generating a series containing the binomial coefficients 

As our first example of a special function, we consider a two-dimensional problem the vibration of a circular membrane such as a drumhead. The amplitude of vibration is determined by solution of the Helmholtz equation in two dimensions, most appropriately chosen as the polar coordinates r, 9. Using the scale factors 2r = 1 and Q = r for the two-dimensional Laplacian, the Helmholtz equation can be written as 

This is subject to the boundary condition that the rim of the membrane at r = ro is fixed, so that P(ro, 9) = 0. Also, it is necessary that the amplitude be finite everywhere on the membrane, in particular, at r = 0. Assuming a separable solution T(r, 9) = R(r)Q(9), Helmholtz s equation reduces to 

We could complete a separation of variables by dividing by r R r). This would imply that the function of 9 in square brackets equals a constant, which 

Chapter 12 Partial Differential Equations and Special Functions 

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The solution of the differential equation for the vibrations of a circular membrane,... 

Some modes of vibration of a circular membrane, as labeled by the quantum numbers m and n, are sketched in Fig. 12.1. To simplify the figure, only... 

Fig. 20.—Some vibrational forms of a circular membrane fixed at the circumference the number of radial nodal lines is here (in disagreement with the custom in wave mechanics) called n, that of the aizimuthal nodal lines m n and 7n are the quantum numbers of the state of vibration.

FIGURE 12.1 Modes of vibration m, n) of a circular membrane. Wavefunctions are... 

A circular plate clamped at the edges has previously been considered in Section 13.10.6. In that example the deflection of a plate under uniform loading was considered. The example here is somewhat similar but is fliat of flie time dependent vibration of a clamped circular membrane such as a circular drum head. The describing PDE is of the form ... 

We next consider, as a two-dimensional example of a meckanical vibration, tbe vibrating circular membrane, the difEerential equation of which is... 

The VSEP (Vibratory Shear Enhanced Processing) system made by New Logic Research is an example of the vibration technique. It uses a stack of circular membrane discs, each with membrane media on both sides, enclosed in a relatively small housing (over lOOm of membrane area in less than 200 litres of housing volume). The stack is then vibrated torsionally, i.e. in a direction parallel to the discs. 

While vibrational spectroscopy is not capable of the structural resolution of X-ray diffraction, it nevertheless has some important advantageous features. First, it is not generally limited by physical state samples can be in the form of powders, crystals, films, solutions, membranous aggregates, etc. Second, a number of different experimental methods probe the structure-dependent vibrational modes of the system infrared (IR), Raman (both visible and UV-exeited resonance), vibrational circular dichroism, and Raman optical activity, many of these with time-resolution capabilities. Finally, in addition to providing structural information, vibrational spectra are sensitive to intra- and intennolecular interaction forces, and thus they also give information about these properties of the system. 

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