Solution to the Gliding Tone Problem

TL;DR

This paper provides an analytical solution to the classical gliding tone problem for high-Q oscillators, analyzing resonance distortion due to finite sweep rates and validating results with experiments.

Contribution

It derives a closed-form analytical expression for the oscillator's response under a steadily-varying frequency in the high-Q limit, linking it to the rotating wave approximation.

Findings

Frequency shift equals twice the ring-down time.

Analytical results agree with experimental measurements.

Resonance distortion depends on sweep rate and Q-factor.

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.