Resonance

Abstract

Up to now, we have discussed the general properties of waves, mainly using the principle of superposition. In this chapter, we turn to the interaction of a wave with the medium in which it is travelling, and we shall concentrate on the medium itself. We shall find that the driven damped harmonic oscillator serves as a model for the response of any medium to an incident wave. The line of argument is as follows. First we show that in general the small vibrations about equilibrium of any system are simple harmonic oscillations. This gives the simple harmonic oscillator great general significance. A wave of frequency w incident upon a system may then be represented by a force of frequency w driving the harmonic oscillators corresponding to the small vibrations. In order to make our model physically realistic, we must allow for some damping of the vibrations by dissipative forces analogous to resistance in electrical circuits. The most important quantity to calculate is the power absorbed from the driving force by the oscillator. As a function of frequency, the power absorbed is a very characteristic resonance curve, or lorentzian, with a maximum at the frequency of the undamped oscillator. We shall finally look at a few of the many examples of resonance curves in natural systems.