Student notes: 4.2-4.5 Transverse and longitudinal waves
Keeping it simple
The water wave demonstrates all the properties of a wave but isn't the easiest to understand. A wave in a string only propagates in 1-dimension so is a bit simpler.
In some ways a string wave is too simple (!) as it can't be used to to show how a wave changes direction when it enters a different medium and can't be passed through a narrow opening to show diffraction.
Transverse waves
Pulses
First we will consider a pulse moving along a stretched string. The pulse is formed by moving the end up and back down to the undisturbed position.
In this PhET simulation the string is actually made of a line of balls connected by springs. The wave motion is made up of the motion of the individual balls. This is a common trick - splitting a complex problem into a series of simple ones. Each ball simply moves up and down, the same as the first one. But they don't move at the same time. Each ball moves slightly later than the first. This is due to the inertia of the balls.
Make the balls heavier and the time delay will be greater, resulting in a slower pulse. Make the springs stiffer or stretch them and the pulse moves faster:
Notice how the pulse is broader, even though the movement of the end is of equal duration, since the disturbance spreads more quickly.
Note that this is an infinitely long string, which is not the same as a loose end. A loose end creates a change of medium, as if the string was connected to a very thin one. When there is a change of medium, the pulse reflects.
Reflection
Now we have a loose end for the string. When the wave reaches the end, the end moves up and down in exactly the same way as the other end moved when the pulse was created. This forms another pulse travelling back along the string.
If the end is fixed it is as if there is a really heavy string attached to the end.
This time the end doesn't move. The clamp exerts a downward force balancing the upward force of the pulse, which sends an inverted pulse back along the string. We didn't notice this when the water wave reflected off the boundary in the ripple tank but the same thing was happening there too.
- When a wave travels into a more dense medium the reflected wave is \(π\) radians out of phase with the incident wave.
By "more dense" we mean a medium where the velocity would be slower. This result will be used when we deal with stationary waves in sound and thin film interference of light.
Interference
We can't send two waves that meet at a point
but we can send two pulses along the same string. Here the pulses add during interference:
Here they cancel:
Polarisation
In a transverse wave the disturbance is perpendicular to the direction of propagation.
This means the disturbance could be in any direction provided it was perpendicular to the direction of propagation. In this example it is always vertical. When the disturbance is limited to one direction we say it is polarised.
Continuous waves
A continuous wave is formed by an oscillating source. In this case, the source is oscillating with SHM so the wave is sinusoidal. Each particle has the same motion but different phase.
If you focus on the points A and B marked below you will notice that B lags behind A by a quarter of a cycle. This is equivalent to a phase difference of \(π\over 2\).
A and C are in phase. You could also say that C lags behind A by \(2π\).
The wave speed is related to the mass per unit length (\(μ\)) and the tension (\(T\)):
\(v=\sqrt{T\over\mu}\)
In this example, the frequency is the same as in the previous but the tension is increased. This gives a higher velocity. Since \(v\propto λ\)the wavelength is increased.
Polarisation
The continuous waves above are polarised as the displacement is in one direction, vertical. We say the plane of polarisation is vertical.
This is a horizontally polarised wave:
A vertically polarised wave can pass through a vertical slit...
...but a horizontally polarised wave can't.
If the slit is at an angle to the plane of polarisation only the component of displacement in the direction of the slit passes.
Notice how the plane of polarisation of the transmitted wave is now in the direction of the slit and the amplitude is reduced.
An unpolarised wave has oscillations in ALL directions, if this is incident on a slit only components of oscillations in the direction of the slit pass, this causes a reduction in amplitude by a factor 1/√2.
Here we have the different directions of oscillation of a wave before and after polarisation. The slit only lets through the vertical components.
The unpolarised wave can be split into the sum of the vertical components, A and the sum of the horizontal components, also A. We can use Pythagoras to add these to give √2A. The polarised wave is just one of these components with amplitude A. The amplitude is therefore reduced by a factor 1/√2.
Note that later on you will deal with the polarisation of light. The principle is the same but light is polarised using a polarising filter such as Polaroid (and not by a slit). The intensity of light is proportional to amplitude2 so will be reduced by 1/2 when passed through a polarising filter.
Graphical representation
We can represent a wave using two different graphs:
- Displacement vs position
- Displacement vs time
Displacement-position
A displacement vs position graph is like a snapshot of the wave showing the displacement of each particle at one moment in time. Let's draw the graph at time \(0\text{ s}\).
Although this looks like a drawing of the wave, it's a graph. The shape of a graph depends on the scales of the axes. The wavelength of this wave is \(4\text{ cm}\) and the amplitude is \(1\text{ cm}\).
The graph will be different at different times. If the graph were drawn at time \(1\text{ s}\) the wave would have progressed \(1.3\text{ cm}\).
From the two graphs you can work out the velocity of the wave: \(1.3\text{ cm s}^{-1}\).
Displacement-time
A displacement vs time graph shows the displacement of one point at different times. The motion of each point is sinusoidal so the graph is a sine curve. Different points will have different graphs, so let's start with point A.
At time \(t = 0\text{ s}\) the particle has zero displacement. We can see from the displacement-time graph at \(t = 0\text{ s}\) that the next part of the wave is a trough - so the particle is about to move down. What happens next to this particle can be seen by looking to the right as time is progressing to the right.
The time period of the wave is \(3\text{ s}\) so the frequency is \({1\over 3}{ Hz}\).
The next graph is the displacement-time graph for point B.
This point starts at the top and is about to move down.
So we can see that v = fλ (1.3 = 0.33 x 4)
Superposition
It is difficult to send two waves along the same string in the same direction but we can send two waves in opposite directions, the easiest way to do that is to reflect the wave from a fixed end.
The result is a strange looking wave that doesn't progress. It is called a standing or stationary wave. Notice that the amplitude of each point is not the same: some points hardly move whereas others have double amplitude.
The formation of a standing wave can be understood by moving the waves past each other. Check out the following simulation:
Notice how, as the waves pass through each other, the peaks sometimes add and sometimes cancel. There are also some points that never move. These points are called nodes. The positions of highest amplitude are called antinodes. To represent a standing wave we often draw the displacement-position graph for the two positions of maximum displacement.
Standing waves
Stringed instruments such as the guitar and violin have strings stretched between two points. When plucked or bowed the string vibrates, which sends a wave moving along the string. The string is light and stretched tight so the wave moves very quickly, too fast to see. When the wave meets the ends it reflects and the reflected wave superposes with the incident wave to create a standing wave. Because the ends are clamped they can't move - so they must be nodes. This means that the wavelength of the standing wave can only be certain values. The diagram below shows three possible standing waves in a string of length \(L\).
The frequencies at which these standing waves form are called harmonics. We can show that the second and third have frequencies that are multiples of the first.
The 1st harmonic is half a wavelength. It's a bit confusing because (above) there is a peak and a trough, but these are the upper and lower positions of the string. If you just look at the top part you see it is half a wavelength:
\(\lambda=2L\)
\(v=f_1\lambda\)
\(f_1={v\over \lambda}={v\over 2L}\)
The 2nd harmonic is one wavelength:
\(\lambda=L\)
\(v=f_2\lambda\)
\(f_2={v\over\lambda}={v\over L}=2f_1\)
The 3rd harmonic is one and a half wavelengths:
\(\lambda={2L\over3}\)
\(v=f_3\lambda\)
\(f_3={v\over\lambda}={3v\over2 L}=3f_1\)
When the string is plucked all harmonics are produced - but the 1st harmonic has the biggest amplitude. This can be displayed on a frequency spectrum.
The vertical axis is scaled according to the fraction of the amplitude of the highest amplitude harmonic. So if the 1st harmonic has amplitude \(2\text{ mm}\) its value is \({2\over2}=1\).
To change the highest amplitude harmonic, the string can be restricted at one of the nodes of that harmonic.
Summary of travelling and standing waves
Travelling and standing waves have some crucial differences.
| Travelling | Standing |
Amplitude of all points is equal | Amplitude of all points between a node and antinode are different |
| All points within one wavelength are out of phase | All points between two nodes are in phase |
| Energy is transferred | No energy is transferred |
| Wave profile progresses | Wave profile is stationary |
Longitudinal waves
Waves in a slinky spring
A slinky is a toy that was very popular in the 1960s. They are still popular with physics teachers born around that time. The original purpose of the slinky was to go down stairs, but physics teachers use them to make waves.
This may not look like a wave but nevertheless energy is being transferred from one end of the spring to the other. The difference is that the disturbance is parallel to the direction of the propagation of energy not perpendicular to it - a longitudinal wave.
To model the wave we can use a series of balls connected with springs... but this time we will move the end ball in the same direction that the energy will progress.
This shows how the springs transfer energy from one ball to the next.
To show a continuous wave without it reflecting off the end we would need an infinitely long chain of springs. This would take rather a long time too set up (!) but we can fake it using a GeoGebra simulation. This is based on the same mathematics as the real wave but there is no physical connection between the balls.
Now you can see the wave travelling from left to right. Each ball is simply moving left and right with SHM iin exactly the same way as the first one. The only difference is the phase. The mathematics is the same as the above, transverse, waves except the oscillations are horizontal not vertical. We can show this by overlaying a second set of balls with equal vertical displacement.
The transverse wave is made of peaks and troughs and the longitudinal wave has compressions and rarefactions, but note that the peaks do not coincide with the compressions.
The centres of both compressions and rarefactions are at positions where the displacement of the balls is zero. The wavelength of the longitudinal wave is the distance between consecutive compressions or rarefactions; balls that are a whole number of wavelengths apart are in phase.
Graphical representation
As with transverse waves, we can represent a longitudinal wave by a displacement vs time graph for one point or a displacement vs position graph for all the points.
Displacement-time
Displacement-time graphs are easy. Each point simply moves with SHM. If this is the graph for the first ball...
...then the graph for a ball \({1\over4}\lambda\) to the right will be like this:
Displacement-position
Displacement-position graphs for longitudinal waves are challenging but can be made easy if you have the equivalent transverse wave overlaid. The first thing to do is to mark the positions where the displacement is zero, this is the middle of the compressions and rarefactions.
Now estimate the amplitude by looking at how far the most displaced ball has moved, about \(0.5\text{ cm}\).
There are now two possibilities:
To decide which one to choose you need to look at the ball to the right of the first one. Is it displaced to the left (negative) or the right (positive). It's displaced to the right so the correct graph is the first one.
The other graph would be for this situation:
Standing waves
If a longitudinal wave reflects off the end it will superpose with the incident wave to produce a standing wave.
Notice how the nodes are in the centre of both compressions and rarefactions, a separation of half a wavelength.