This section will cover the following topics:

Waves on a String

All kinds of stringed instruments - guitars, pianos, violins - have stretched strings which oscillate when plucked or struck. This oscillation generates sound, which can be amplified acoustically by coupling the vibrations to a large sound-board (for a guitar, the top, and in an upright piano, at the back) or electrically by turning the string oscillations into an electrical signal (using a 'pickup') which can be sent to an electronic amplifier / loudspeaker. The 'note' depends on the frequency at which the string vibrates - and this depends on:

  • The tension in the string
  • The length of the string
  • The 'weight' (mass per metre) of the string - called in science the 'mass per unit length'.

This animation shows an impulse travelling along a string.

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The end of the string has been fixed - using a large metal block - to ensure a reflection of the wave when it meets this boundary. In the animation, there is a (crude!) attempt to show that when the wave is reflected, there is a phase change - if the displacement of the string is in an 'upwards' direction for the wave travelling left-right, then the reflected wave will be displaced in a downwards direction.

Compare the amplitude of the reflected wave and the incident wave - what does this tell you about the presence or absence of damping at the point of reflection? If damping was to take place (OK, we've given you the answer now!), what would happen to the amplitude of the reflected wave?

You can see this animation happening for real in this slow motion video of a transverse wave travelling along a bungee cord.

Downloadable High Quality Quicktime version (975 KB)

Downloadable High Quality Windows AVI version (1.44 MB)

The bungee cord is useful, because it's really stretchy (science-speak: compliant). This means that it's not hard to set up a wave with large amplitude that travels pretty slowly - it's easy to see. Here's another example of waves travelling along a compliant string which you could try - an elastic band:

Downloadable High Quality Quicktime version (1.56MB)

Downloadable High Quality Windows AVI version (8.17 MB)

It's just about possible to play a 'tune' on an elastic band, but for a real musical application we need a real musical instrument. Guitar strings are under quite high tension and are much less compliant, so wave amplitude is much smaller and the wave speed is much higher. Nonetheless, you can make out similar behaviour in this video:

Downloadable High Quality Quicktime version (3.31 MB)

Downloadable High Quality Windows AVI version (16.7 MB)

The examples above show what happens when a string is set into its free motion by an impulse - by being plucked. This is OK as far as it goes, but so far our explanation is a little 'simplistic' - it will help us to move on if we try to work our way towards thinking about what happens if we force a string to vibrate using some kind of sine-wave input.

Standing waves

Let's start by putting just one sine-wave into the left-hand end of the string, and allowing it to travel towards the reflecting boundary.

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Once again, note the phase-change on reflection. Now let's consider the case when we continuously excite the string at the left-hand end using a sine-wave input. This means that sine waves are continuously travelling from left to right, changing phase by 180 degrees on reflection, and travelling back from right to left. The incident and reflected waves interfere with each other by superposition and the following pattern results:

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The composite waveform is known as a standing wave. It differs from a travelling wave (transverse or longitudinal - in fact every type of wave we have considered so far has been 'travelling') since the pattern oscillates, but appears to be stationary - there is no apparent right-left or left-right movement of energy. Every standing wave has nodes (locations of minimum amplitude) and antinodes (locations of maximum amplitude).

A string (such as our guitar string we looked at earlier) will naturally 'want' to vibrate at a number of different standing wave frequencies. One way to find out at which frequencies these standing waves occur, is to connect a small shaker (like a loudspeaker coil and magnet, but with no cone) to one end of the string, and to vibrate it at various frequencies using an electronic sine-wave generator until we find those at which the amplitude of string vibration is greatest.

At the lowest-frequency standing wave we would find, the string would vibrate like this:

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This is actually rather similar to our kid-on-a-swing examples; the wave travels l-r, is reflected and travels r-l, arriving back at the shaker at just the right time to get another 'push'. The 'just-the-right-time' idea is another way of saying that the interference between the reflected wave and the 'next' wave driven by the shaker, is constructive.

We'll call this frequency which drives the lowest standing wave the 'fundamental frequency, F(Hz)'.

Here's an elastic band vibrating at its fundamental frequency - we have not used a shaker here, but have plucked it in the centre, which happens to exicite the fundamental standing wave very strongly.

Downloadable High Quality Quicktime version (2.24MB)

The 'just-in-time' idea can take us further. We know the period of vibration of the string driven at its fundamental frquency will be 1/F (s). Therefore, we know that it takes exactly 1/F seconds for a wave to travel l-r, be reflected and travel back r-l before meeting the next vibration of the shaker at exactly the right point in phase (a better way of thinking than 'point in time').

So - if we vibrate the string twice as fast and assume that the speed of the wave, and thus the time taken to travel 'there and back', do NOT change, then by the time the wave gets back to the start the shaker will have completed two complete oscillations...but exactly two. This means that the returning wave still arrives at just the right 'point-in-phase' - and so interferes constructively with the next wave sent out. Another standing wave results:

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The same rules apply if the string is vibrated 3, 4, 5 etc times as fast, as the fundamental case. Because the standing waves occur at integer (whole-number) multiples of the fundamental frequency, they are sometimes called harmonics.

Here the string is vibrating at three times the fundamental frequency (3F Hz)...

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...and here the string is vibrating at four times the fundamental frequency (4F Hz).

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If we drive the string with an oscillator and shaker, we can choose which standing wave we set up. Normally, if we pluck the string, we excite a bunch of harmonics, all with different amplitudes. The balance of these amplitudes depends heavily on where along the string, we pluck it.

The amination below shows the first four harmonics of a string simultaneously:

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Of course, the string can only be 'at one place at once' - and so these various standing wave patterns must add-up by superposition and create a 'total' waveform.

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The resulting combination of standing waves contains frequency components at F, 2F, 3F, 4F (Hz) etc - and this harmonic series determines the timbre of the instrument. The timbre can be changed by plucking the string at a different location - try a guitar string in the centre (lots of fundamental) and near the bridge (lots of high-order harmonics).

Earlier on this page there was a slow-motion video of someone plucking a number of guitar strings. If you replay this, you can actually see that they have different fundamental frequencies, which are set by the speed at which the wave travels along each string and their length. Remember - wave speed is fixed by tension and mass per unit length.

Harmonics are very good news musically, because the ear 'likes' to constuct pleasant-sounding tones from harmonic acoustic oscillations. One or two instruments do not produce harmonic standing wave vibrations - bells are a good example - and this makes them rather odd to listen to from a musical point of view.

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