PERRIN and SWALLOWE, 2003. Rayleigh's bell model revisited. IN: Proceedings of the Stockholm Music Acoustics Conference, Stockholm, Sweden, August 2003, pp 347-350.
It is now well over a century since Lord Rayleigh published his
model for western-style bells. He used a hyperboloid of
revolution plus a flat circular plate for the crown. By limiting
himself to inextensional modes of a very restricted type, and
exploiting the hyperbola’s parametric form, he produced an
equation whose roots give the locations of nodal circles.
Remarkably this equation involves neither the wall thickness nor
physical properties of the bell material and this approach
remains the only available analytical way of making such
predictions. Although he gave adequate accounts of the
derivation and method of solution of his equation, Rayleigh did
not present much in the way of comparison of its predictions
with experiment. Rather he focussed on using it to explain the
fact that the Hum note never has any nodal circles. In the present
paper we consider how well profiles of some modern church and
handbells can be fitted by hyperbolae. We compare the model’s
predictions for these bells with data for a range of inextensional
modes and report a new, surprisingly accurate, approximate
analytical solution of Rayleigh’s equation.