Rayleigh's bell model revisited

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.