Geometric Reasoning about Damped and Forced Harmonic Motion in the Complex Plane
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Complex-valued functions are commonly used to solve differential equations for one-dimensional motion of a harmonic oscillator with linear damping, a sinusoidal driving force, or both. However, the usual approach treats complex functions as an algebraic shortcut, neglecting geometrical representations of those functions and discarding imaginary parts. This article emphasizes the benefit of using diagrams in the complex plane for such systems, in order to build intuition about harmonic motion and promote spatial reasoning and the use of varied representations. Examples include the analysis of exact time sequences of various kinematic events in damped harmonic motion, sense-making about the phase difference between a driving force and the resulting motion, and understanding the discrepancy between the resonant frequency and the natural undamped frequency for forced, damped harmonic motion. The approach is suitable for supporting instruction in undergraduate upper-division classical mechanics.