Thus far we have focused on the free oscillations of an isolated system. Some initial disturbance displaces the system from equilibrium, and it then oscillates freely until its energy is dissipated. These are very important situations, but they do not exhaust the possibilities. Another important situation is an oscillator that is subjected to a periodic external force. Its motion is called a driven oscillation.

A simple example of a driven oscillation is pushing a child on a swing, where your push is a periodic external force applied to the swing. A more complex example is a car driving over a series of equally spaced bumps. Each bump causes a periodic upward force on the car’s shock absorbers, which are big, heavily damped springs. The electromagnetic coil on the back of a loudspeaker cone provides a periodic magnetic force to drive the cone back and forth, causing it to send out sound waves. Air turbulence moving across the wings of an aircraft can exert periodic forces on the wings and other aerodynamic surfaces, causing them to vibrate if they are not properly designed.

As these examples suggest, driven oscillations have many important applications. However, driven oscillations are a mathematically complex subject. We will simply hint at some of the results, saving the details for more advanced classes.

Consider an oscillating system that, when left to itself, oscillates at a frequency $f_0$. We will call this the natural frequency of the oscillator. The natural frequency for a mass on a spring is $\sqrt{k\text{/}m}\text{/}2\pi$, but it might be given by some other expression for another type of oscillator. Regardless of the expression, $f_0$ is simply the frequency of the system if it is displaced from equilibrium and released.

Suppose that this system is subjected to a periodic external force of frequency $f_{\text{ext}}$. This frequency, which is called the driving frequency, is completely independent of the oscillator’s natural frequency $f_0$. Somebody or something in the environment selects the frequency $f_{\text{ext}}$ of the external force, causing the force to push on the system $f_{\text{ext}}$ times every second.

Although it is possible to solve Newton’s second law with an external driving force, we will be content to look at a graphical representation of the solution. The most important result is that the oscillation amplitude depends very sensitively on the frequency $f_{\text{ext}}$ of the driving force. The response to the driving frequency is shown in FIGURE 15.25 for a system with $m=1.0\text{kg,}$ a natural frequency $f_0=2.0\text{Hz,}$ and a damping constant $b=0.20\text{kg}\text{/}\text{s}$. This graph of amplitude versus driving frequency, called the response curve, occurs in many different applications.

FIGURE 15.25 The response curve of a driven oscillator at frequencies near its natural frequency of 2.0 Hz.

When the driving frequency is substantially different from the oscillator’s natural frequency, at the right and left edges of Figure 15.25, the system oscillates but the amplitude is very small. The system simply does not respond well to a driving frequency that differs much from $f_0$. As the driving frequency gets closer and closer to the natural frequency, the amplitude of the oscillation rises dramatically. After all, $f_0$ is the frequency at which the system “wants” to oscillate, so it is quite happy to respond to a driving frequency near $f_0$. Hence the amplitude reaches a maximum when the driving frequency exactly matches the system’s natural frequency: $f_{\text{ext}}=f_0$.

The amplitude can become exceedingly large when the frequencies match, especially if the damping constant is very small. FIGURE 15.26 shows the same oscillator with three different values of the damping constant. There’s very little response if the damping constant is increased to $0.80\text{kg}\text{/}\text{s}$, but the amplitude for $f_{\text{ext}}=f_0$ becomes very large when the damping constant is reduced to $0.08\text{kg}\text{/}\text{s}$. This large-amplitude response to a driving force whose frequency matches the natural frequency of the system is a phenomenon called resonance. The condition for resonance is

Within the context of driven oscillations, the natural frequency $f_0$ is often called the resonance frequency.

FIGURE 15.26 The resonance amplitude becomes higher and narrower as the damping constant decreases.

An important feature of Figure 15.26 is how the amplitude and width of the resonance depend on the damping constant. A heavily damped system responds fairly little, even at resonance, but it responds to a wide range of driving frequencies. Very lightly damped systems can reach exceptionally high amplitudes, but notice that the range of frequencies to which the system responds becomes narrower and narrower as b decreases.

This allows us to understand why a few singers can break crystal goblets but not inexpensive, everyday glasses. An inexpensive glass gives a “thud” when tapped, but a fine crystal goblet “rings” for several seconds. In physics terms, the goblet has a much longer time constant than the glass. That, in turn, implies that the goblet is very lightly damped while the ordinary glass is heavily damped (because the internal forces within the glass are not those of a high-quality crystal structure).

The singer causes a sound wave to impinge on the goblet, exerting a small driving force at the frequency of the note she is singing. If the singer’s frequency matches the natural frequency of the goblet—resonance! Only the lightly damped goblet, like the top curve in Figure 15.26, can reach amplitudes large enough to shatter. The restriction, though, is that its natural frequency has to be matched very precisely. And in practice, the sound also has to be extremely loud.