Complex Oscillations

Abstract

The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic oscillator. 4.1 The harmonic oscillator equation The damped harmonic oscillator describes a mechanical system consisting of a particle of mass m, subject to a spring force and a damping force: Spring force Damping force The particle can move along one dimension, and we let x(t) denote its displacement from the origin. The damping coefficient is 2mγ, and the spring constant is k = mω 2 0. The parameters m, γ, and ω 0 are all positive real numbers. (The quantity ω 0 is called the "natural frequency of oscillation", because in the absence of the damping force this system would act as a simple harmonic oscillator with frequency ω 0 .) The motion of the particle is described by Newton's second law: m d 2 x dt 2 = F (x, t) = −2mγ dx dt − mω 2 0 x(t). (1) Dividing by the common factor of m, and bringing everything to one side, gives d 2 x dt 2 + 2γ dx dt + ω 2 0 x(t) = 0. (2) We call this ordinary differential equation the damped harmonic oscillator equation. Since it's a second-order ordinary differential equation (ODE), the general solution must contain two independent parameters. If we state the initial displacement and velocity, x(0) and ˙ x(0), there is a unique specific solution. Note Sometimes, we write the damped harmonic oscillator equation a bit differently: d 2 dt 2 + 2γ d dt + ω 2 0 x(t) = 0. (3) The quantity in the square brackets is regarded as an operator acting on x(t). This operator consists of the sum of three terms: a second-derivative operator, a constant times a first derivative, and multiplication by a constant. We are interested in solving for x(t). For the simple (undamped) harmonic oscillator, which is the case where γ = 0, we know what the general solution looks like: x(t) = x 0 cos(ω 0 t + φ). (4) The particle oscillates around the equilibrium position, x = 0, because the spring force keeps pushing it towards the origin and its momentum causes it to "overshoot". For the damped 27