### Waves Tutorial 2 - Transverse and Longitudinal Waves

We will assume that all waves are sinusoidal.  A sinusoidal wave-form (sine wave) is the simplest kind of wave.  Sound waves of sinusoidal form are rather boring to listen to.  The waves made by musical instruments are more interesting, but more complex.  However it can be shown that even the most complex wave-form can be broken down into sine waves.  Wave motion can be analysed in terms of circular motion and simple harmonic motion (SHM).  Since these topics are in the A2 unit 4, we will not attempt to do that in these tutorials.  (Originally the topic on wave properties was in the A2 syllabus, so waves could be discussed in terms of SHM.)

A transverse wave is one in which the displacement of the particles is at 90o to the direction of travel.  In a water wave, the particles move up and down while the wave travels horizontally.  All electromagnetic waves are transverse.

We can show the features of a transverse wave in the diagram below:

The equilibrium position in the diagram is the position that the material would take if the wave motion stopped.  We could also call it the rest position.  Both terms are used in SHM.

Strictly speaking, water waves are not transverse.  They are a type of wave called a roller and are only transverse when the amplitude is much less than the wavelength. When the amplitude is large, the wave is no longer transverse; it takes on the characteristic shown below.

A cork on the surface does not go up and down; it takes on a circular motion.

As the amplitude gets bigger compared with the wavelength, the crest then breaks.  This often happens near the shore because the bottom part of the wave is travelling more slowly than the top.  Surfers use the rolling nature of water waves.

In longitudinal waves, the displacement is parallel to the direction of travel of the wave.  There are regions of high pressure, compression, and regions of low pressure, rarefaction.  In a sound wave the air molecules move forwards and backwards; where they are squashed together, a compression results, where they are forced further apart, there is a rarefaction.  Like all mechanical waves, a medium or material is required.  The speed of sound in air is 336 m s-1, in water 1400 m s-1, in steel it is 6000 m s-1.  Other examples of longitudinal waves include some kinds of earthquake waves (the pressure or P-wave).  We can see the features of a longitudinal wave in the diagram below.

 Write down two similarities and two differences between transverse and longitudinal waves.  Give one example of a transverse wave and one example of a longitudinal wave.

Electromagnetic Waves

Electromagnetic waves have the following properties:

• They are transverse;

• They consist of an electrical wave component and a magnetic component;

• They travel in straight lines at 3.0 × 108 m s-1;

• They can travel in a vacuum;

• The magnetic and electrical waves are at 90o to each other and are in phase.

The diagram shows the idea:

Electromagnetic waves form a large family of waves.  The EM spectrum is shown in the diagram:

Image courtesy of Inductiveload, NASA.  Wikipedia

The energy in the wave is inversely proportional to the wavelength:

This becomes:

We have seen this in the context of photon energy:

• E - energy (J)

• h - Planck's Constant = 6.63 × 10-34 J s;

• c - speed of light = 3.0 × 108 m s-1;

• l - wavelength (m).

In this topic, we are counting EM radiation as a wave, although we know that they travel in trains of waves called photons.

 This equation applies to electromagnetic waves, NOT to mechanical waves.

 For a wave of wavelength 95 nm, calculate: (a) The frequency; (b) The photon energy in J and eV

Visible Light
Visible light is a very small part of the electromagnetic spectrum. We can see wavelength from about 300 nm to 600 nm. Below 300 nm we have ultra-violet.  Many animals, e.g. insects and fishes can see in ultra-violet.  We cannot see UV.  No animals can see infra-red, which has a wavelength longer than 700 nm, and this allows biologists to observe animals at night without disturbing them.

The picture shows relative wavelengths of the primary colours: red, blue, and green.

If the light rays are super-imposed on each other, we get:

• Red + blue = magenta;

• Red + green = yellow;

• Green + blue = cyan;

• Red + green + blue = white.

This process is called colour addition.

Image by courtesy of SharkD.  Wikipedia

 The artist's primary colours are red, blue, and yellow.  This is because mixing of artists' colours works by colour subtraction.

Speed of Electromagnetic Waves (Extension)

This is on the Scottish Advanced Higher Syllabus

The speed of light is given to a large number of significant figures as:

2.99792458 × 108 m s-1

It is a fundamental constant on which other constants rely.

We use c = 3.00 × 108 m s-1 to 3 significant figures.  But how is it worked out?  Initially it was carried out by astronomical observations by Olaus Roemer, a Danish astronomer, in 1676.  He observed the eclipses of the moons of Jupiter.  In the 19th Century, several physicists used rotating mirrors to determine the speed.  The details of such experiments are not needed here, but you can see some video clips of how it's done.  See HERE.

The Scottish theoretical physicist James Clerk Maxwell used the complex Maxwell's equations to derive this simple relationship:

The terms are:

• Speed of light - c = 3.00 × 108 m s-1;

• Permittivity of free space - e0 ("epsilon-nought") = 8.85 × 10-12 F m-1;

• Permeability of free space - m0 ("mu-nought") = 4p × 10-7 H m-1 » 1.257 × 10-6 H m-1.

(Epsilon is a Greek letter 'e'.  Mu is a Greek letter 'm'.)

 Worked Example Use the equation: to show that the speed of light is approximately 3.0 × 108 m s-1. Answer c = 1 ÷ (8.85 × 10-12 F m-1 × 4p × 10-7 H m-1)0.5 = 2.999 × 108 m s-1 (which is pretty close to 3.0 × 108 m s-1)

We can work out the value for e0 using a parallel plate capacitor.  We can use a current balance to work out m0.

### Polarisation of Transverse Waves

Polarisation is a feature of transverse waves only.  Longitudinal waves are never polarised. We say that a wave is plane polarised if all the vibrations in the wave are in a single plane, which contains the direction of propagation of the wave.  Suppose we have a rope and make waves down it.  We could make waves in any direction we liked.  But if we made waves through a narrow vertical slit, we would find that the waves would only pass through if they were vertical. This would be a vertically polarised wave.

Light waves are easily polarised using polaroid filters.  Light waves, like all electromagnetic waves, consist of an electric field component perpendicular to a magnetic field component, which are always in phase.  We normally consider only the electric field component in polarisation, because the electrical effects are those that dominate.  The unpolarised waves are normally oriented in any direction.

If two polaroid filters are mounted such that they are parallel, the light will pass through both the first at which point it is vertically polarised, and then through the second.

If the two filters are crossed, so that the transmission planes are at 90o to each other, the vertically polarised light gets blocked, because it cannot pass the horizontal transmission plane. No light passes.

Crossed polaroids are found in liquid crystal displays on calculators and petrol pumps.

 Radio aerial rods must be in the correct plane, vertical or horizontal in order to work properly, otherwise the signal is weak.  Use the information above to explain why this is the case.

The diagrams in the answer to Question 3 show the ideas.

Malus' Law (Extension)

The quantitative nature of polarisation is summed up by Malus' Law.

The intensity (power per square metre) of light passing through crossed polaroids is related to the angle of rotation by the following relationship:

The experiment is carried out by measuring the angle between the two polaroids.  The intensity is represented by the output voltage of a solar cell.  The experiment uses the electric field vector of the EM wave.  The measured intensity is the result of the transmitted electric field.  Some of the electric field vector is absorbed.  The idea is shown in the diagram below.

The apparatus is set up like this:

Source: AQA Specimen practical paper

In this experiment the intensity of the light is reflected in the output of the solar cell.  As you change the angle of the second polaroid, the output of the solar cell changes.  You take readings every 20 degrees.

Data modelling shows what the graph should look like, if Malus' Law is followed:

It is quite challenging to get decent data from this experiment.  At least two repeat readings should be made and an average taken.  Also the model above assumes that the crossed polaroid filters block out all the light.  In reality they do not.

Graphical Representation of Waves

We can show that both longitudinal and transverse waves can be represented by a displacement-distance graph.  If we take a snapshot of a wave at any instant, we see:

For a transverse wave we see that the graph looks very similar to the actual wave.

For a longitudinal wave the graph is not so easy to see.

Let us look at the air molecules in their undisturbed positions and compare them as a sound wave passes by.

If we plot displacement on the y-axis and distance on the x-axis, we get the same graph to what we had before.   The shape is a sine wave.

If we plot a displacement-time graph for a single particle we see:

This is true whether we have a longitudinal or transverse wave.  If we connect a microphone to a CRO, the CRO displays a displacement-time trace.  It is important that we do not confuse the displacement-distance with the displacement-time graph. The latter tells us nothing of the wavelength, only the period (hence the frequency) of the wave.

The simplest shape of graph we see is the sine wave.  The sine wave equation links wave motion with simple harmonic motion.   Sine waves in sound are very boring to listen to.  The quality of the sound gives us important clues as to the source.  Sounds made by different musical instruments have very different wave patterns, even if the notes sounded and the volumes are the same.

 The diagrams show the variation with time t of the displacement x of the two identical cones of loudspeakers A and B in air. Calculate: (a)    the frequency of the vibration of the speaker cones. (b)   the phase difference between the speaker signals. (c)    What kind of wave is being produced in the air by each speaker? (d)   Which speaker produces the loudest sound?  Explain your answer. (e)    The speed of sound in air is 340 m s-1.  What is the wavelength of the sound waves?

Gravitational Waves

The notion of gravitational waves has been around since 1893 when first proposed by Oliver Heaviside (1850 - 1925).  They were also proposed in 1905 by Henri Poincaré (1854 - 1912).  The theory was developed by, and is credited to, Albert Enstein (1879 - 1955) in 1916, using his Theory of General Relativity.  They can only be observed when a disturbance arises from the interaction of two or more very heavy objects, like stars.  A single star on its own will not set off a gravitational wave.

The general features of gravitational waves are:

• Speed is 3.0 × 108 m s-1;

• They are transverse;

• They follow the normal wave equation (c = fl);

• The period of such wave varies from a few milliseconds to millions of years.

Image Credit: NASA Goddard Space Flight Center

Gravitational waves from two merging black holes were detected in 2015 and announced in 2016.