The harmonic oscillator plays a loftier role in physics than one might guess from its humble origin: a mass bouncing at the end of a spring. The harmonic oscillator underlies the creation of sound by musical instruments, the propagation of waves in media, the analysis and control of vibrations in machinery and airplanes, and the time-keeping crystals in digital watches. Furthermore, the harmonic oscillator arises in numerous atomic and optical quantum scenarios, in quantum systems such as lasers, and it is a recurrent motif in advanced quantum field theories. In short, if there were a competition for a logo for the universality of physics, the harmonic oscillator would make a pretty strong contender.
We encountered simple harmonic motion—the periodic motion of a mass attached to a spring—in Chapter 3. The treatment there was highly idealized because it neglected friction and the possibility of a time-dependent driving force. It turns out that friction is essential for the analysis to be physically meaningful and that the most interesting applications of the harmonic oscillator generally involve its response to a driving force. In this chapter we will look at the harmonic oscillator including friction, a system known as the damped harmonic oscillator, and then examine how the system behaves when driven by a periodic applied force, a system called the driven harmonic oscillator.