When Waves Collide

Demo using torsion generator

Principle of Superposition

1

2

Let ( , ) be the amplitude function for a certain wave

when no other waves are present on the wave medium.

Let ( , ) be the amplitude function for another wave

when no other waves are present on the

y x t

y x t

1 2

wave medium.

If both waves are present together on the medium,

the amplitude function for the combined wave is

( , ) ( , ) + ( , )y x t y x t y x t

Another way of saying this is that two waves don’t collide with each other. They simply pass right through each other.

Another way of saying this is that two waves don’t collide. They simply pass right through each other.

This phenomenon of wave amplitudes adding together is often called wave interference.

Another way of saying this is that two waves don’t collide. They simply pass right through each other.

This phenomenon of wave amplitudes adding together is often called wave interference.

John William Strutt (Lord Raleigh): “If this is interference, it is hard to know what one would mean by non-interference.”

That is not the end of the story, because a wave can collide with another object. In particular, suppose two different wave media are in contact. What happens when a wave from one medium enters the other?

Before answering that question, let’s ask a simpler question. What happens when a wave collides with a rigid boundary?

There is a brief time when the incident pulse and the reflected pulse overlap. What is happening at the fixed end at this time?

There is a brief time when the incident pulse and the reflected pulse overlap. What is happening at the fixed end at this time?

What is the wave amplitude at this point?

There is a brief time when the incident pulse and the reflected pulse overlap. What is happening at the fixed end at this time?

What is the wave amplitude at this point?

A point where the wave amplitude is always zero is called a node.

There is a brief time when the incident pulse and the reflected pulse overlap. What is happening at the fixed end at this time?

What is the wave amplitude at this point?

What is the net result when there are two waves present?

A point where the wave amplitude is always zero is called a node.

> with(plots):

> animate(exp(-(x-t+5)^2)-exp(-(x+t- 5)^2),x=-10..10,t=0..10, frames=50);

1

If the incident wave amplitude is given by some function

( , ) ( v )

what waveform has to be superposed with this one to give

zero at position x = 0 at all times?

y x t f x t

1

1

If the incident wave amplitude is given by some function

( , ) ( v )

what waveform has to be superposed with this one to give

zero at position x = 0 at all times?

Answer: y (0, ) ( v ). To get zero w

y x t f x t

t f t

2

2

1 2

e have to add another

waveform y (0, ) ( v ). We want the reflected wave to

be moving to the right, so we need y ( , ) ( v ).

The net wave is given by ( , ) y ( , )

( v ) ( v )

t f t

x t f x t

y x t x t

f x t f x t

Something different happens if a wave is incident on a moveable boundary.

Something different happens if a wave is incident on a moveable boundary.

The general rule is: Whenever a wave is reflected from a free boundary, it is reflected in phase. Whenever it is reflected from a fixed boundary, it is reflected 180 degrees out of phase.

What would happen if the string on the right had exactly the same properties as the one on the left?

What would happen if the string on the right was thicker and denser than the one on the left?

Consider two different wave media that meet at a boundary. The wave speed in medium #1 is v1 and the wave speed in medium #2 is v2. A wave that originates in medium #1 and strikes medium #2 will generally give rise to a reflected wave and a transmitted wave.

The reflected wave is in phase with the original wave if v2 > v1.

The reflected wave is 180 degrees out of phase with the original wave if v2 < v1.

Summary:

1. Two or more waves on the same medium produce a net wave that is determined by the principle of superposition: The amplitude of the resulting wave is the sum of the amplitudes of the individual waves.

2. The reflected wave from a fixed boundary is 180 degrees out of phase with the original wave. The reflected wave from a free boundary is in phase with the incident wave.

3. When two wave media share a common boundary, a wave incident on that boundary will generally produce both a reflected and a transmitted wave. The transmitted wave is always in phase with the incident wave while the reflected one is in phase if v2 > v1, out of phase if v2 < v1.