Superposition of Waves

We shall now look at what happens when two waves interact.  If two jets of water interact, they will mix and there are collisions between droplets causing a change in speed and direction.  This does not happen with waves.  If two waves interact, a new wave is temporarily formed, after which the two waves carry on with exactly the same properties as before, as if nothing had happened.  The waves are superposed.

Superposition can only be applied to waves of the same kind.  Light and sound waves cannot superpose; light and X-rays can.  Let us look at two waves of different wavelengths crossing and superposing:

The resultant wave can be worked out by the vector sum of the two waves.  The principle of superposition of waves can be used to explain the presence of beats in sound, interference effects and standing waves.

We can use the superposition of waves to explain interference.  When two waves meet, the amplitude of the resultant wave will not only depend on the amplitude of the two waves, but also their phase relationship.  Let us look at two waves of equal amplitude superposing:

In this case the waves are in phase.  The resultant wave is double the amplitude of the original waves.  This is called constructive interference or reinforcement.  If the waves are 180^o (p radians) out of phase, the waves cancel each other out.

This is called destructive interference or cancellation.  If the phases are different to these values, the resultant amplitude are between these two extremes.

Standing Waves

Sometimes waves appear to be standing still, i.e. the crests and the troughs appear to stay in the same place.  We can see them in water, especially water surrounded by walls.  We call them standing waves or stationary waves.  Musical instruments depend on standing waves:

• In a string, for example guitar, pianoforte, violoncello.

• In a column of air, e.g. clarinet, tuba, organ.

  Stationary waves are formed when two progressive waves are superposed:

• Equal frequency

• Nearly the same amplitude

• Same speed

• Travelling in opposite directions.

If we send an incident wave down a string, which is fixed at the end, the wave is reflected at the fixed end and undergoes a phase change of p radians or 180^o.  There is no phase change at the free end.

If we send a continuous stream of waves down the string, they are reflected and a standing wave gets set up.  The frequency will be the same, the amplitude very nearly the same and the speed will be the same.  The directions are opposite.  The phase change of p radians causes cancellation at the fixed end.  This region of zero displacement is called a node.

In a progressive wave, points X and Y would be in antiphase, p radians out of phase.  However, because the wave is reflected, the phase is changed by p radians.  So they are now 2p radians out of phase, which means that they are in phase.  Superposition is constructive.   The amplitude is now at a maximum, and this is called an antinode.

Notice:

• All particles between nodes are in phase.

• All particles either side of a node are in antiphase.

• Each “sausage” is half a wavelength.  

We can show standing waves with Melde’s Apparatus.

If we start the frequency of the vibration at a low level, increasing it slowly, we see little of significance until at a certain value, a single large vibration loop is seen.  This is due to resonance and is called the fundamental frequency or the first harmonic.  The second harmonic has two vibration loops. It is twice the fundamental frequency.

The frequency at which resonance happens depends on:

• The tension

• The length

• The mass per unit length (how thick the string is).

We can also have longitudinal standing waves, which we can show with a Kundst tube.  

We should note that:

• The wave is longitudinal so that all the particles vibrate parallel to the tube.

• The amplitude is at a maximum at the open end of the tube, so there must be an antinode.

• The amplitude at the closed end is zero; there is a node.

• All molecules between nodes vibrate in phase.

• All molecules either side of a node vibrate in antiphase.

• Adjacent nodes are half a wavelength apart.

Since this is ¼ of a wavelength, the organ pipe sounds a note whose wavelength is 4 times its length.  The antinode is formed by air passing a whistle arrangement.

A couple of rules when looking at air columns:

• At the open end, there is always an antinode.

• At the closed end, there is always a node.

This has important implications for wind instruments.  A brass instrument like a trombone has an antinode made by the player pursing his lips and vibrating them (an embouchure).  There is an antinode at the bell of the instrument, and a node half way down.

The fundamental frequency can be changed by altering the length of the tube.  In a trombone, the player moves part of the tube in and out.  A trumpeter changes the length by pressing in valves.

We can set up a standing wave using 3 cm wave apparatus.

We move the probe between the transmitter and the reflector and we detect maximum readings and minimum readings with the probe, which is connected to a microammeter.  The maximum readings coincide with the antinodes; the minimum readings with the nodes.