Interactive textbook: Travelling waves

What is a wave?

Gas in a container is not a room full of perfectly elastic spheres, but we can apply the knowledge we have of the motion of balls to derive equations that can be used to make predictions regarding the behaviour of a gas. It is important to remember this when we get onto quantum mechanics and consider the wave properties of electrons. Electrons aren't waves or balls but using models based on the behaviour of water waves and elastic balls enables us to predict their behaviour.

The original wave is the water wave. This is the only wave for which you can actually see wave properties in action. Other things that we call waves (e.g. sound) have the same properties but are totally different. It is the model that is the same, rather than the phenomena. What do waves have in common? Transfer of energy without transfer of matter.

Water waves

Water waves are actually rather complicated. The water doesn't simply move up and down; it moves in a circular motion. We won't go into all the details of how the individual particles move and will just look at the wave.

A water wave can be created by making a disturbance on the surface. In this section we will use a simulation of a ripple tank made by Paul Falstad to illustrate the way the wave moves and how they interact with boundaries and other waves.

This animation shows a point disturbance is creating a circular wavefront spreading out across the surface of the water. We can see that this is very different to the motion of a particle. To put it simply a wave spreads out but a particle doesn't. The water could have been disturbed by dropping a stone onto it. The energy from the stone was given to the water lifting up the surface. The energy then spreads out in a circle. A ball floating on the surface would receive energy from the wave but definitely not all of it. This is different to the interaction between two balls where all the energy from one ball could be given to the other:

  • A wave is the propagation of disturbance
  • Energy is transmitted by a wave

 

What forms of energy are transmitted by a water wave.

You could argue that there is some change in internal energy associated with the wave but a water wave isn't really the transfer of heat.


 

A single stone produces a simple wave - more of a pulse than a wave. To produce a continuous wave we could disturb the water with an oscillating body.

Here you can see the sinusoidal nature of the wave and how the height of the peaks decreases as the energy spreads out. If we are going to model the wave we will need to define some quantities, let's look at a vertical slice.

Wavelength (\(λ\)): The distance between two consecutive points that are in phase. This could be two peaks, two troughs or any points the same distance apart.

Amplitude (\(A\)): The maximum displacement from the undisturbed position.

Velocity (\(v\)): The distance moved by the wave profile per unit time.

Frequency (\(f\)): The number of complete cycles produced or passing a point per unit time.

The time taken for one complete cycle to pass is \(1\over f\). In this time the wave profile has moved a distance equal to \(\lambda\).

The velocity is therefore calculated using the following equation (as well as using distance and time):

\(v=f\lambda\)

 

The velocity of waves in the sea is related to depth so as waves approach a beach they slow down.

What happens to the wavelength of the waves?

f does not change so λ is prortional to v

 

Wavefronts: If we look at the wave from above we see a series of equally spaced concentric circles. These circles are lines joining points that are in phase. In this simulation the bright ones are peaks and the dark ones troughs.

A plane wavefront (look for the straight bright line) can be formed by adding a line of point sources:

This leads to Huygens' construction:

  • A wavefront propagates as if made on an infinite number of point wavelets.

We will use this construction to explain why waves behave as they do.

 

When using Huygens' construction we ignore the wave propagating backwards:

There aren't really lots of wavelets but it helps us to model the wave propagation.


 

Reflection

When a water wave hits a boundary such as a wall it reflects.

The boundary acts as a line of wavelets. These wavelets add up to make the new wavefront reflecting off the boundary at the same angle as the incident wave.

Refraction

When a water wave travels into a shallow region it slows down. This makes the wavelets propagate at a slower rate so the resulting wavefront changes direction. The effect of the changing speed and direction is called refraction.

Notice how the wave not only refracts at the boundary but also reflects.

The angle of refraction is given by Snell's law:

\({\sin(\text{angle of incidence})\over\sin(\text{angle of refraction})}={\text{velocity in depth or medium 1}\over\text{velocity in depth or medium 2}}=n\)

  • \(n\) is refractive index (dimensionless)

 

The refractive index of the boundary above is approximately:

The angle of incidence is about 60° and refraction 30°


 

The angles mentioned here are the angles between the wavefront and boundary. These are the same as the angles between the wave direction and normal to the boundary.

An increased difference in velocity (and, hence, refractive index) leads to an increased change in direction. You can try varying  angle of incidence and refractive index in this simulation.

Diffraction

Diffraction takes place when a wave passes through a narrow opening. If the opening is very small only one wavelet would be let through. This results in a circular wave front; the wave spreads out.

Gaps wider than a few wavelengths allow through more wavelets resulting in a pattern of areas of high and low amplitude.

 

If instead of using Huygens' construction we considered the wavefront to propagate as a straight line, we would predict that on passing through a slit the wave :

This is why we use Huygens'


 

Interference

When two waves meet they do not bounce off each other. They pass through each other.

Where waves overlap, the displacements superpose. Displacement is a vector so if the two displacements are in the same direction they add (constructive interference). If the displacements are in opposite directions they cancel (destructive interference). In the animation you can see two waves passing through each other, notice how the displacements add where the waves cross over.

If we have continuous waves we get areas of constructive and destructive interference depending on whether the waves are in phase or not when they arrive at that point. Let us consider two point sources.

This is difficult to look at so we will simplify in 1-dimension. If a ball is floating somewhere between the sources it will experience the combined effect of two waves - one from the left and one from the right.

The resultant displacement of the ball at any time is the vector sum of these displacements. We can see that the displacements above are always opposite so the waves cancel. They don't cancel everywhere though. Below, we have moved the ball to the left and the displacements add.

Whether the waves add or cancel depends on how far the ball is from each source: 

  1. If the ball is equally distant from each the waves add. The waves also add if the difference is a whole number (\(n\)) of wavelengths, \(nλ\).
  2. If the difference is half a wavelength the waves cancel. This is also true for \((n+{1\over2})λ\). In two dimensions this forms a pattern.

The lines show positions where the path difference is \(0\), \(λ\over 2\) and \(λ\).

 

How many wavelegths apart are the 2 sources above?

Count the regions of constructive interference. The region in line with the sources has path difference 4λ.

 

Why do the two sources have to be the same frequency?

if the sources had different frequency there would not be a constant phase difference

 

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