Waves

Two Examples of Waves

Mechanical Waves

A wave is a method of transferring energy from one place to another without transferring matter. Mechanical waves are those that require a medium for their transfer and include water waves, sound waves and waves in stretched strings. A disturbance at A causes a disturbance of a particle, that drags its neighbour's particles along with it until the disturbance reaches B. If the disturbance at the source continues, the wave is maintained, and if it is simple harmonic, then a plot of the displacement of the particles at a single point in time is a sine curve. This is the basic nature of a mechanical wave that is considered when looking at mechanical waves.

The graph below shows the displacement of a single particle plotted as a function of time.

These two situations both apply to transverse waves, where the displacement is perpendicular to the direction of travel. For longitudinal waves, where the displacement is parallel to the direction of travel (e.g. sound waves) it is necessary to adopt an arbitrary convention as to which direction is plotted above the line and which below.

The amplitude of a single particle is also the amplitude of the wave.

One thing to point out is that the only thing that actually moves in a wave is the energy. The actual particles, after the wave has gone by, have no overall displacement from their original position.

Electromagnetic Waves

Electromagnetic waves consist of varying electric and magnetic fields. The two fields are perpendicular to each other and to the direction of travel of the wave. Each vibrates at the same frequency - the frequency of the wave. The waves all travel at the same speed in a vacuum - 2.998x108 ms-1. They are unaffected by electric and magnetic fields, and in general travel in straight lines. They are transverse, and therefore can be polarised. They can be diffracted, and can interfere with one another.

The spectrum of electromagnetic waves is shown below:

The following table summarises all of the important information about electromagnetic radiation:

Type

Examples of Generation

Main Properties and uses

Detection

g-rays

Radioactive decay

Nuclear fusion and fission

Interactions between elementary particles

 

Gieger-Müller tubes

Ionisation chambers

Solid state

X-rays

Rapid deceleration of fast electrons

Atomic transitions involving innermost electrons

 

detectors

Scintillation counters

Photographic film

Ultra-violet

Atomic transitions (e.g. mercury vapour lamp)

Produces ionisation and florescence

Promotes chemical reactions

Affects photographic film

Produces photoelectric effect

Absorbed by glass

Photoelectric cell

Florescent materials

Photographic film

Visible

Atomic transitions (e.g. in incandescent lamps)

Stimulates the retina

Affects photographic film

Initiates photosynthesis

Photoelectric cell

The eye

Photographic film

Infrared

Atomic transitions

Molecular vibrations

Produces heating

used in "night sights"

Thermopile

Special photographic film

Micro-wave

Klystrons and Magnetrons

Masers

Radar

Telemetry

Electron spin resonance

Crystal detectors

Radio

Electrical oscillations

Radio communication

Radio receivers

General Wave Properties

Definitions

The wavelength, l, is the distance between any particle and the nearest one which is at the same stage of its motion (same displacement and velocity). In particular, it is the separation of two adjacent peaks or troughs.

The amplitude, a, of the wave is the greatest displacement of any particle from its equilibrium position.

The period, T, is the time taken for any particle to undergo a complete oscillation. It is also the time taken for any wave to travel one wavelength.

The frequency, f, is the number of cycles that any particle undergoes in one second. It is also the number of wavelengths that pass a fixed point in one second.

In one second, a wave goes through f cycles. In each cycle it moves on by the wavelength (l). Therefore, in one second, the wave moves on by fl metres. Therefore, c=fl, where c is the speed of light.

Wavefronts

A wavefront is a line or surface, in the path of a wave motion, where all the displacements at any point have the same phase. A point source leads to circular wavefronts, at large distances from the source they are straight lines. A ray is a line that shows the direction of the wave and is perpendicular to the wavefronts.

Types of waves

Waves are generally divided up into two types of wave - longitudinal and transverse. Longitudinal waves are waves where the displacement of the particles is parallel to the direction of travel of the wave (the vibrations are along the direction of the travel of the wave). The example of this is sound waves, where there are compressions and rarefactions in the air that cause the sound to be transmitted. Transverse waves are waves where the displacement of the particles is perpendicular to the direction of travel of the waves (the vibrations are perpendicular to the direction of travel). One obvious example of this is light.

The critical difference between transverse and longitudinal waves is that transverse waves can be polarised whereas longitudinal ones cannot. Polarised waves are ones where the vibrations of the wave are in a single plane. Most light sources produce light that is unpolarised - the polarity of the light changes constantly from one polarity to another. However, there are certain substances where light that hits them becomes polarised. One example of this is Polaroid. The intensity of light that passes through a sheet of Polaroid is approximately half of the intensity that went in, since half of the light has been cut out by the polarisation. However, the light that comes out s all vibrating in one field. If another piece of Polaroid is placed in front of the first piece in the same optical orientation, then no difference is observed. However, if the second piece is rotated, then the intensity of the light slowly decreases, until it reaches 90, where no light penetrates at all.

Light reflected off a relatively even surface (e.g. a wet road) is plane polarised. Therefore, sunglasses with a suitably oriented piece of Polaroid will filter out the glare from such a surface, making it easier to drive / see fish under water.

Microwaves can also be polarised, using a grille of metal rods, separated by approximately the same distance as the wavelength of the waves.

Diffraction and Interference

Superposition

The principle of superposition states that when two waves are travelling in the same region the total displacement at any point is equal to the vector sum of the individual displacements at that point. If two sources of light are coherent (i.e. they are the same frequency, and therefore have a constant phase difference) and they have the same amplitude, interference takes place. For interference there are two important possibilities:

1. The two waves arrive exactly in phase (i.e. the phase difference=0) When this happens, the two waves interfere constructively, and form a similar wave with twice the amplitude.

2. The two waves arrive exactly out of phase (i.e. the phase difference = 180 = p radians = l/2) In this case the waves interfere destructively, and the output has 0 amplitude.

In between these two posibilities, light is produced of amplitude that varies between the two extremes.

This is shown best by an experiment using two slits, through which comes light that is prescisely in phase. This leads to a pattern as is shown below.

The circles represent the wavefronts of two waves that are in the same phase and are being emmitted from the points marked (possibly two slits). As the two sets of waves are coherent, they interfere with each other, leading to the pattern shown here. The dashed lines represent points where there is constructive interference only (where two peaks coincide), This leads to the maximum possible amplitude. The dotted lines represent sets of points where a trough meets a peak (where there is total destructive interference). This leads to a totally dark point here. If a screen were placed where the dark black line is now, then the classic interference pattern of dark and light stripes would be seen. This provides very strong evidence that light exists as waves, as only waves can interfere in this way.

Stationary Waves.

A stationary or standing wave results when two waves which are travelling in opposite directions and which have the same speed and frequency and approximately equal amplitudes are superposed.

This superposition results in the formatiuon of nodes and antinodes. Nodes are points at which the amplitude of the standing wave is zero, and the antinodes are points at which it is 2a, where a is the amplitude of the original wave. The nodes are seperated from each other by a distance of l/2. This wave is stationary - it does not move, but stays in the same position.

Fraunhoffer Diffraction at a Single Slit

For this type of diffraction, a distant source can be used or a source at the focal point of a converging lens. Eiother way, there need to be a set of non-diverging wave-fronts moving towards a slit. The equipment is shown below. It is possible to omit the lens L2 and place the screen a long distance from the slit.

O is a light source at the focal point of L1. P is the point of maximum intensity.

When light is brought to a focal point by the means of a lens then the number of wavelengths in each of the wave paths is the same, and therefore the optical path length of each ray is the same. For example, on the diagram shown below, there are the same number of wavelengths on the path OAAB and the path OBBC. This is because the wavelength of light is shorter in glass than in aitr, and the ray X travells farther in glass than Y.

When diffraction takes place at a single slit, a pattern is obtained with a single maximum that is very bright, surrounded by two minima that are very dark, next to two maxima that are slightly bright, surrounded by two minima, and so on. This can be shown on a graph of distance, intensity as follows:

The reasons for this are (not) shown by the diagram below:

In this diagram, a is the width of the slit. Thius is split into 2n sections, where n is one of the natural numbers (1,2,3...). q is an angle to the normal, and l is the wavelength of the light.

If q is used such that , then the wave from A is completely out of phase with the wave from C. As this is the case, the waves froim A and C completely destroiy each other (complete destructive interference). Also, the waves from all the points between A and C are destroyed by the waves from corresponding points between C and D. This is true for all the other pairs of sections (there are always an even number of sections, as there are 2n sections). Therefore, in sections where . Looking at a close up of the diagram:

Looking at the angles on the diagram,

Looking at the triangle ACN, we can see that , and therefore:

This allows us, from the width of the slit and the wavelength of the light used to find out the angular placement of any of the minima. This also shows that the minima are not equally spaced, however, when , and therefore the first few minima are equally spaced. The formula also applies for the minima to both sides of the central maximum.

The Central Maximum

This diagram explains why there is a central maximum. All the rays arriving at O are in phase, as only the rays focused on this point have not been diffracted at all. This means that they all interfere constructively, and there is a brightness.

Geometrical Optics

When talking about reflection and refraction concerning plane mirrors, there are several important terms that need to be defined

1. Incident ray - the ray that hits the surface / interface / substance in question

2. Reflected ray - the ray that is reflected off a surface / interface

3. Normal - line drawn from the point where the incident ray hits the interface perpendicular to the interface.

4. Angle of incidence - angle between the incident ray and the normal.

5. Angle of reflection - angle between the reflected ray and the normal.

There are two laws of reflection:

1. The reflected ray is in the same plane as the incident ray and the normal to the reflecting surface at the point of incidence.

2. The angle of reflection is equal to the angle of incidence

Refraction

When light passes between two substances, it changes direction (it is bent). Any material has a refractive index that quantifies how much light will refract by when it moves from one material to another.

The absolute refractive index of a material is called n:

Therefore, the refractive index of air 1, as air has very little effect on light.

There are two laws of refraction:

1. The incident ray, the refracted ray and the normal at the point of incidence are all in the same plane.

2. At any boundary between two similar substances:

Where is the angle of incidence, is the angle of refraction, l is the wavelength of the light in the material, c is the speed of light in the material, and n is the refractive index of the material.

The frequency of the light going through stays constant, but the speed and the wavelength change. These are connected by the equation . This shows that the speed of waves in a material may be dependant on the frequency of the light.

The higher the absolute refractive index is for a material, the higher the optical density of that substance is.

Total internal reflection

When light travels into an optically less dense material, then it is possible for the angle of incidence to be such that the angle of refraction is 90. The angle of incidence at which this happens is called the critical angle for a particular material. At the angle of refraction, the light is refracted so that it travels parallel to the side of the material in question. At any angle of incidence greater than the critical angle, the light is all reflected within the substance, and none comes out of the other side. It can be shown, therefore, that , and therefore, when the light is leaving into air, .

This is useful in optical instruments, where prisms can be made so that they reflect the light that enters them, this being better than using mirrors for this purpose. It is also useful in optical fibres (see later).

Optical fibres

Information can be transmitted by varying the intensity of light or infra-red radiation travelling through a transparent glass or plastic fibre. An analogue signal would be sent as a continuously variable light intensity but pulsed signals are more common. Serial rather than parallel data transfer keeps the number of fibres to a minimum.

Because a negative light intensity is not possible a sine wave must either modulate an otherwise constant light intensity (the carrier) or be converted to a digital signal and be transmitted as pulses of some sort.

Fibre optic waveguide action.

Light within an optical fibre striking the boundary between the fibre and the cladding of lower refractive index will be refracted only if the angle of incidence is less than the critical angle (see earlier). When the angle of incidence is greater than the critical angle for that boundary the light is totally internally reflected, and can be kept within the fibre by repeated total internal reflections. A fibre with a sharp change between the refractive index of the core fibre and the refractive index of the cladding is called a step index fibre.

Due to interference light cannot follow all the conceivable zigzag paths with the necessary internal reflection. The paths that are allowed are called the transmission modes of the fibre. In a thick fibre there will be a large number of modes.

Some light will travel directly parallel to the walls along the centre of the fibre (axial mode). Light of the highest order mode will be making the largest possible number of zigzags as it travels along the fibre and will be most steeply inclined to the fibre wall (hitting it at the critical angle).

Light travelling in a higher mode travels a greater distance than that in a low-order mode, and therefore will take a longer time to pass through a long fibre. Therefore, a pulse entering the fibre will come out as a pulse of longer duration and a different shape. Successive pulses may overlap at the end of the fibre. This can also cause distortion of analogue signals. Modal dispersion can be eliminated by using very thin fibres where only the axial mode exists (monomode fibres) or graded index fibres, where the light in the higher modes travels faster and therefore reaches the end of the fibre at the same time as the axial mode light.

The speed of light in the fibres is also affected by the frequency of the light, and therefore similar problems to those of modal dispersion can occur. The solution is to use filtered monochromatic light.