Solution to the Gliding Tone Problem

Abstract: The solution is given to the classical problem of an oscillator driven by a sinusoid of steadily-varying frequency. A closed analytical expression is obtained in the case where the Q-factor of the oscillator is high, equivalent to the rotating wave approximation of atomic physics. In this case all independent variables of the system combine into a single parameter. The results are compared with previous work: series and other approximations, numerical calculations, graphical solutions and analogue simulations. Attention is paid to the distortion of the resonance -- specifically frequency shift and amplitude attenuation -- consequent upon the finite sweep rate. The frequency shift is interpreted as a delay in the appearance of the resonance peak; to leading order this time delay is twice the oscillator's ring-down time. Measurements on a high-Q oscillator are consistent with the mathematical solution.