Chapter 4: Resonance

Resonance is a key concept in the production of sound in instruments and in acoustics. We will come across it many more times in this book. The natural frequency (f₀ measured in hertz) is the frequency at which an oscillating system naturally wants to vibrate. For a mass on a spring, this is determined by the size of the mass and the stiffness of the spring; a stiffer spring has a higher natural frequency as we saw in the previous chapter. To keep a system vibrating in the presence of friction we have to keep pushing it with a periodic force. The frequency of this periodic driving force is called the driving frequency, f which is totally independent of the natural frequency (we can push our mass on a spring at a frequency different from the frequency at which it wants to vibrate).

Key Terms:

Natural frequency, driving frequency, angular frequency, damped harmonic motion, driven harmonic motion, resonance, resonator, Helmholtz resonance, Quality or Q-factor.

4a: Resonance Examples

Resonance occurs in an oscillating system when the driving frequency happens to equal the natural frequency. For this special case the amplitude of the motion becomes a maximum. An example is trying to push someone on a swing so that the swing gets higher and higher. If the frequency of the push equals the natural frequency of the swing, the motion gets bigger and bigger.

For many systems we can make a graph of amplitude versus frequency and see where resonance occurs. Suppose we have a mass on a spring and attach a vibrator with a frequency we can choose. We start at low driving frequencies and measure the amplitude of the motion (how far it bounces) at each frequency. We might end up with a graph like the one below. Notice that the amplitude was a maximum at a driving frequency of 2.5 Hz. So the natural frequency of the system without the vibrator was also 2.5 Hz. In other words, we are driving the spring at resonance.

Video/audio examples:

Swing Resonance. What is the driving force in this case?
Air track resonance cart. Why is the amplitude of the cart larger at one particular frequency?
Resonance cart. Note there are three different natural frequencies in this example. Why is this so?
Tidal resonance at the Bay of Fundy. The high water mark is reached every 12 hours. Why is it 12 instead of every 24 hrs?

4b: A few other examples of resonance

Resonance can occur in any system that has a natural frequency. You probably have experienced a rattle or buzz in your car that only occurs at a certain speed. This is an example of resonance; the tires provide a periodic driving force which changes frequency as you change speed. Various parts of the car have different natural frequencies, particularly parts that have come loose. If the driving frequency of the rotating tires matches the natural frequency these parts will vibrate at a larger amplitude, making a buzz or rattle. Here are some other examples.

Video/audio examples:

Helicopter ground resonance. What is the driving force in this case?
Bridge resonances: bouncing, twisting, Tacoma Narrows, Russian bridge.
Breaking glass: 1, 2. What are the driving forces in these two cases? [NOTE: Don't do this at home! The breaking glass can go into your eyes or mouth.]
More Singing rod. What is the driving force that makes the rod resonate?
A web page with more examples of resonance.
A description of building resonance as a result of a dance class.

You probably have blown across the top of a bottle to get a note. Different sized bottles make different notes and partially filling the bottles with water also changes the pitch. These are examples of Helmholtz resonators; a container of gas with a single opening that will resonate at a specific pitch. Blowing across the top causes a driving force on the air inside which has a natural frequency due to the 'springiness' of the air and the size of the container. As we will see, acoustic stringed instruments such as the guitar consist of a hollow body which acts like a Helmholtz resonator. Here are more details on Helmholtz resonance.

If you have ever been annoyed by a low frequency sound while driving down the highway with one window partially open, you were experiencing a Helmholtz resonance where the air blowing past the open window caused the air inside the car to vibrate.