Comment on the paper “A mathematical analysis of a series circuit containing periodically varying resistance” by L. A. Pipes

Abstract

AT FIRST sight, the application of W.K.B. approximation to a time-dependent circuit seems perfectly straightforward. Unfortunately, there are two different and equally plausible ways to apply it to the circuit treated by Pipes, and the two results will generally not agree. The W.K.B. solution of the homogeneous equation (43) contains two arbitrary constants. These can be chosen so that at some particular time τ, q = 0 and dq/dt = 1. Call this solution q1(t, τ). Alternatively, the constants can be chosen so that q = 1 and dq/dt = 0 at time τ. Call this solution q2(t, τ). The response of the system at time t to a unit voltage impulse applied at some earlier time τ is q1(t, τ)/L, hence, by the superposition principle, we get a general solution of the inhomogeneous equation $q_1 (t) = {1 \over L} \int_{-\infty}^t q_1(t, \tau) E(\tau) d\tau. \eqno{\hbox{(1)}}$.