An analogy for resonance?

I learned that the phenomena of resonance occurs when the frequency of the applied force is equal to the natural frequency of an object. At this point, an object vibrates with maximum amplitude.

1. How can a force have a frequency? My teacher used the analogy of pushing a child on a swing. The person who pushes should apply a force of the same frequency of the swing (I didn't really get that).

2. Following the analogy, what if the person pushes harder, wouldn't the "amplitude" of the swing be larger... I mean is there any such thing a "maximum amplitude" for a vibrating object as I can always increase the magnitude of the force.

3. Do all objects always vibrate with some kind of frequency? I mean if I put a speaker in front of a goblet and produce a sound having a frequency equal to the "natural frequency", it will just start vibrating with maximum amplitude?

I know I asked a lot of questions at once but I am really confused about this phenomena and just cannot get the intuition behind it.

2 Answers

Re question 1: when you learn this stuff in school you usually simplify the system by modelling it as a simple harmonic oscillator so the amplitude of the system will be given by some equation like:

$$A(t) = A_0 e^{i\omega_0 t}$$

where $\omega_0$ is the natural frequency of oscillation. Typically you study what happens if you apply a force that also varies sinusoidally with time so:

$$F(t) = F_0 e^{i\omega t}$$

where the frequency of the applied force, $\omega$, is not necessarily the same as the natural frequency of the oscillator, $\omega_0$. This is what your teacher means by saying that the force has a frequency - they mean the frequency $\omega$. In your teacher's example of a swing the swing has some natural frequency. If you are applying a force periodically, i.e. pushing on the swing in a repetitive way, then the force you apply also varies with time (though it is more like a square wave than a sine wave). The amplitude of the swing is greatest when the frequency with which you push the swing matches the natural frequency of the swing.

Re question 2: when you start learning this stuff you typically start with an undamped simple harmonic oscillator, i.e. the oscillator doesn't lose any energy. If you solve the equations of motion you find that the amplitude goes to infinity when the frequency of the driving force $\omega$ is equal to the natural frequency $\omega_0$. This is because you're putting energy in but the oscillator doesn't lose any energy so the energy just keeps growing.

A real oscillator like a swing loses energy through friction, and we call it a damped harmonic oscillator. The rate at which the oscillator loses energy is related to its amplitude, so as you push your system (the swing in this case) the amplitude increases until the rate of energy loss matches the rate you're putting energy in. So the harder you push your system the more the swing will move. In principle there is no maximum amplitude, though in real life there obviously is since at some point the swing will go over the top and start revolving instead of swinging to and fro. A swing isn't a simple harmonic oscillator! It's only approximately simple harmonic for small swing amplitudes.

Re question 3: Most objects will have a range of resonant frequencies called normal modes. However there are usually many normal modes and the frequencies of these modes are related to the object's shape in a complicated way. The Wikipedia article gives some examples of normal modes, or do a YouTube search for "normal modes" to find loads of videos on the subject - some really impressive!

• ok, these Xmas "hats" are getting really silly :-) – Carl Witthoft Dec 17 '13 at 12:52
• @John Rennie: (Re 1) So if I follow the analogy... when the number of times the child makes a complete swing per second is equal to the number of times the person pushes the swing per second is when the swing "resonates". – Eliza Dec 17 '13 at 13:09
1. A force can be applied with a certain frequency. Most accurately, for your example, energy will be transferred with a certain frequency. This means that, you can interpret this as a wave.

2. The limit of amplitude of a vibrating object is related to the energy necessary to break or damage that object. The analogy of the swing may be misleading here.

3. Everything has a natural vibration frequency. It will depend on the dimensions, shape and materials.

To understand resonance you need to know that sound is mechanical vibrations (vibrations of matter) that propagate as a wave.

When these waves propagate through the object, some of them are reflected when they reach an interface. At the right frequency, you can create constructive interfering thus sum the amplitude of several propagating waves. This will make the overall amplitude of the waves to become higher. This is resonance.

• May someone please care to explain to me what is wrong with my answer? – cinico Dec 20 '13 at 11:07