Superposition

Superposition is a word used to describe what happens when one wave is superimposed - 'sat on top of' - another. To think about how two waves might add up, let's start by looking at the peaks and troughs which we see whenever we encounter waves in reality. Using water waves as an example, these troughs and peaks are literal - at an instant in time the 'height' of the water varies with distance from a vibration source, and at each point on the water's surface the height varies with time.

You can set up real waves like this by dropping pebbles into a still pond (or frozen peas into a still washing-up bowl!) - have a look at this video, where we allow a drop of water to fall into a still water tank, videoing it at high speed.

Note that in the animation, the water is driven continuously and we are looking at the forced response, whereas in the video the water is 'driven' impulsively by one droplet, so we are looking at the free response.

Now let's think about sound. For acoustic waves in air, the peaks represent high air pressure, and the troughs, low air pressure. At any point in space, the air pressure varies or oscillates from high to low with a frequency set by the pitch of the sound in question. As time moves on, if you stand in one place, peaks are followed by troughs which are followed by peaks again, with a repetition rate (or period) set by the frequency of the sound..

Oscilloscope showing a sine wave

When two waves overlap in space, their vertical displacements (water) or pressures (air) combine together; this is known as interference. Where both waves produce a peak, or a trough, simultaneously at a point in space, they combine together to create an even larger oscillation - this is constructive interference. On the other hand, if a trough and a peak combine together the opposing displacements cancel each other out and create a much smaller (or even zero) total displacement - this is destructive interference. This is shown below - amplitude is on the y-axis, and time on the x-axis.

Oscilloscope showing two sine waves combining together. When they are aligned the resultant wave is a large amplitude sine wave. When they are out of alignment the waves cancel each other out.

Phase

The alignment of two waves in space or time is a tricky thing to talk about, as we have seen. To save having to talk about 'troughs lining up with troughs' or 'peaks lining up with peaks' it is really useful to introduce the idea of phase.

When peaks line up with peaks and troughs line up with troughs, waves are said to be in-phase. Here, interference between two waves is constructive - in acoustics, the overall sum of the two pressures results in a bigger (louder) pressure.

When peaks line up with troughs, waves are out-of-phase - they cancel each other out (interfere destructively).

Two waves can have an infinite variety of relationships in phase. To describe them, we use the idea of phase angle.

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By placing two markers on the edge of the disc, we can generate two waves with the same amplitude and period. The really neat thing is that the physical angle between the two points on the disc will correspond to the phase angle difference between the waves which result!. Turn the animation on below, and try dragging the slider to change the angle between the two points. Note the phase relationship between the waves - try some significant angles (90°, 180°, 270°) and then play around for a while. What's the difference between about 20° and about 340°?

You can see that the addition (superposition) of the two waves is also shown. This means we can straight away see the interference which would result from the two waves combining. Which phase angles generate the largest total combined amplitudes, and which the smallest? Can you produce a combined pressure wave of zero amplitude - two waves which add up to silence?

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When the two markers are at the same location there is no angle (0° or 360°) between them and therefore no phase difference. They are in-phase. At each point in time the two waves combine together to create a wave with double the amplitude of the consituent parts. This is constructive interference. The combined wave (if a sound wave) will be the loudest it can be.

When the two points are on opposite sides of the disc there is a 180° (or π radians) angle between then. They are said to be out of phase. In the superposition, at each point in time a peak will line up with a trough and these cancel each other out. This is destructive interference. The combined wave has no amplitude: you get silence.

If the points are placed at any other angle, for example 90°, they are neither in- nor out- of phase. We say there is a phase angle of 90 degrees between them. As the animation has shown, the combined wave is always sinusoidal, with the same frequency as the two constituent parts, and the amplitude lies somewhere between 0 (silence) and 2 (loudest). What amplitude do you get when there is a phase difference of 90 degrees?

Radians

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The phase difference between two waves can be represented in degrees, or in radians. We know there are 360° in a circle - this corresponds to 2π radians. So - a phase of 90° corresponds to 2π x 90/360 = π/2 radians.

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