Further observations on the anharmonic motion equation

Peters 1] discussed some theoretical aspects of the ‘ringing’ sound associated with barbershop quartet singing. These combination tones were first investigated theoretically by Helmholtz 2] who showed that the transient displacement of the ear membrane is governed by the anharmonic motion equation. Rayleigh 3] modelled the situation as an initial value problem and this is now referred to as the Rayleigh problem. Following 1] there has been a renewed interest in the problem (see 4‐8]). The intention of the present paper is to:

• illustrate the qualitative behaviour of the solutions of the Rayleigh problem by a phase plane analysis;

• describe how the method of multiple scales provides a framework within which previous analyses can be seen;

• describe an improvement to the method of multiple scales, which may also prove useful in other applications.