Waves Tutorial 3 - 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.

 

Question 1

Light and water waves are both waves.  Will they superpose?  Explain your answer.

Answer

 

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 180o (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.

 

Question 2

These two waves are p radians out of phase, but have different amplitudes.  Draw the output wave you would expect.

Answer

 

The diagram below shows the resultant of two waves that are 3/4 cycle (270 o or 1.5 p rad) out of phase.

 

 

The red and blue waves are superposing.  The green wave is the resultant.

 

Superposition of waves is the principle that explains:

 

Sound Pictures

Sound waves reflect.  We hear the reflections as echoes.  Bats use echoes to locate their prey, by building up a "sound picture" from the patterns produced by reflected sound pulses.  You can see the enlarged ears that pick up the reflected sound

 

Picture by Vehlo, Wikimedia Commons.

 

The waves produce the sound picture by superposing, as the reflected waves interact with each other.  The pattern of superposed waves will be altered by the presence of a moth (or similar juicy food item), and the animal will use that to aim at its target.  (In some places, bats have given up with this, and have reverse-evolved to their original form, a wingless, mouse-like animal that eats seeds.)

 

Stereophonic music reproduction works in a similar way.

 

 

If your stereo system is set up properly, i.e. the speakers are in the same plane and you sit at the apex of a triangle, you will be able locate the position of the instruments being played.  The same effect is heard when you wear headphones. 

 

 

The apparent placing of the instruments arises from the pattern of superposed waves.  As well as the quality and beauty of the music, a stable stereo image makes the pleasure of listening even greater.  The effect is somewhat diminished by placing one speaker on a bookshelf, and one behind the settee.  

 

Reflected sound waves and the way they superpose affects the acoustics of buildings.  In some halls and large rooms, the acoustics can be bad.  There may be spots where you cannot hear the musicians or actors properly.  There may be areas where the sound is fainter that it should be, while in other areas it is louder.  In some cathedrals, a sound may be accompanied by echoes that last three or four seconds; music or speech becomes inaudible or unpleasant to listen to.  In mediaeval times, when these churches were built, nobody knew a thing about acoustics.  Nowadays it is possible to alter acoustics so that certain echoes are absorbed by sound deadening material.  This can result in substantial improvements.

 

If the room has to much sound deadening material, the sound can be rather "dull" or "flat".  Scientific studies of audio equipment are carried out in anechoic chambers, but nobody would use one as a listening room.  An anechoic chamber is shown below:

 

Picture by Tlotoxl, Wikimedia Commons.