A method for improving the estimation of membrane frequencies

A method for obtaining approximations of the natural frequencies of membranes is developed. An approximate expression for the radius of the bounding curve is first written as a truncated Fourier series. The deflection, which is written as a superposition of the modes of the circular membrane, is forced to vanish (approximately) on the approximated boundary. This generates a system of linear homogeneous equations, the unknowns in which are the amplitudes of the modes of the circle. Equating the determinant of coefficients to zero yields an equation from which the approximate frequencies may be found. It is shown that the first-order approximation obtained through this procedure is identical to a method given by Rayleigh. Approximate frequencies of the first several modes of membranes in the shapes of a square, an ellipse, and the limacon of Pascal are then determined as demonstrations of the new second-order approximation. The approximations of the first three natural frequencies of the ellipse were found to be in error by less than 5% for eccentricities of 0·8 or less, and the approximations of the first four frequencies of the square were found to be in error by less than 3%.