Waves: Part 6 - Wave Fronts

Refraction is a topic that is at the heart of waves of all kinds. It affects the broadcaster in many ways, in lenses, optical fibers and in the way transmissions propagate.

Whilst the actual process of wave propagation is complicated, two simplifications have been made that give good enough results for a lot of purposes. It should be understood that in the case of light, the world we perceive is macroscopic: most things are a lot bigger than light waves/photons and we can get away with not treating them individually.

The simplifications are rays and wave fronts. Rays are conceptual; they don’t exist, whereas wave fronts do. Over short distances, light rays seem to travel in straight lines and assuming the existence of rays allows certain calculations to be made that will tally with reality. For example in artificially rendered imaging, ray tracing is in extensive use to establish what a virtual camera would see from a given viewpoint in a virtual model.

Digital video effects units used in broadcasting work almost entirely on ray tracing.

On the other hand, direct light from air into glass or water and it will bend at the interface. The concept of the ray can’t explain why it bends or by how much. Wave fronts can explain both.

Sound waves and surface waves in water are very much bigger than light waves, even though a lot of the principles are the same. We don’t talk about sound rays or water rays because the concept is not useful. But wave fronts work in nearly all cases and provide more answers. In other words they are a better model of reality. Whereas rays have zero width, beams have finite width and it is usual to talk about beams from lighthouses and headlights. Loudspeakers that only direct good sound in one direction are said to suffer from beaming.

Fig.1 shows a number of points radiating in-phase with one another. The radiation will interfere constructively straight ahead. If the radiating points are evenly spaced, constructive interference will also occur at certain angles, because we have made a diffraction grating. However, if the number of radiating points tends to infinity, most of the energy will be concentrated in the wave front going straight ahead. That is why light travels in straight lines, under certain circumstances. This explanation was first put forward by Christiaan Huygens before 1700.

Fig.1 - A wavefront propagates forwards because in that direction an infinite number of spherical waves can add constructively.

Fig.1 - A wavefront propagates forwards because in that direction an infinite number of spherical waves can add constructively.

At first sight Fig.1 would appear to be bad news for the broadcaster trying to radiate a signal on a spherical planet, because the radiation would go off at a tangent to the Earth’s surface where there are no listeners. Marconi was told that his efforts to communicate around the globe by radio would not work because the signals would go into space instead of following the surface. Obviously there must be more to it than that, because Marconi succeeded and radio communication has become extremely popular.

There is an element of contradiction here, because a lot of broadcasting relies on short range, or line of sight reception so that there can be re-use of the same channels when the signals from one reception area cannot reach another. In that case atmospheric conditions that extend the range of radio signals are a nuisance because they cause interference. On the other hand, if it were not for such conditions long distance broadcasting would be impossible.

Fig.1 shows that a small amount of energy does not go straight ahead. The longer the wavelength, the less directional the wave front becomes. That was the argument for using so-called long wave radio signals, having wavelengths of the order of a kilometre and frequencies of a few hundred kilohertz. The exact frequency used by Marconi is unknown, but has been estimated to be around 850kHz, which would be considered a medium wave transmission by later standards.

Very low frequency radio signals can also be received under water and are useful for communication with submarines.

Radio communication around the Earth was shown to be possible, but the losses suffered by the signal were enormous. Marconi was working before the invention of the vacuum tube and he had no access to electronic amplification. Marconi’s transmitters were essentially gigantic vehicle ignition systems. A resonant circuit formed by a capacitor and an inductor was excited by passing current through the inductor and interrupting it with a contact breaker or spark gap.

Such a transmitter could not produce a continuous carrier, and instead produced a burst of RF energy each time the contacts opened. It was not possible to modulate such a carrier, so the transmission of speech was not possible. The best that could be done was to send Morse Code by keying the power supply. The transmitted spectrum was extremely wide and such transmitters cannot meet modern regulations as they are essentially sources of RF noise.

Subsequently high frequency alternators were developed that could generate low radio frequencies directly by having hundreds of poles on a large rotor. These made continuous carrier waves possible, but were soon rendered obsolete by the vacuum tube.

Marconi researched radio propagation by placing receivers in ships that would undertake long voyages away from his transmitter. Amongst other things, he found that the signals received were stronger at night. This was not understood, and would not be for some time.

Clearly Fig.1 will hold for all kinds of waves, sound, light water and so on, but only if the speed of the waves in the medium is constant. The speed change can be sudden, as in light reaching the surface of a lens, or gradual. A large number of effects result from gradual changes in a medium.

Fig.2a) - In open water the bow wave of a ship makes a V-shape.  Fig.2b) - In an estuary the bow wave travels faster where the water is shallow.

Fig.2a) - In open water the bow wave of a ship makes a V-shape. Fig.2b) - In an estuary the bow wave travels faster where the water is shallow.

Fig.2 shows a version of Fig.1 in which the propagation speed changes in the vertical axis. The plane wave fronts no longer go straight ahead, but follow a curved path because the wavelength is no longer constant. Fig.2a) shows that the bow wave of a ship in open water forms a constant V-shape. That can be explained in due course, but Fig.2b) shows the same ship in an estuary where the water gets shallower towards the banks. The propagation speed rises in shallow water and the bow wave takes on a curve.

In air, sound will take a curved path if there is a transverse temperature gradient. The usual lapse rate means that temperature falls with altitude and the speed of sound reduces. Sound will be refracted upwards. However, some meteorological conditions result in what is known as a temperature inversion, where a layer of air fails to follow the usual lapse rate. Under these conditions the inverted layer can act like a wave guide, bending radiation down from the top and up from the bottom. The inverse square law no longer applies and radiation can travel extreme distances. Since temperature changes affect the speed of sound and the speed of electromagnetic radiation, both can be guided in this way.

Radio signals in the VHF and UHF bands are normally intended for small service areas, but tropospheric ducting can allow such signals to propagate over thousands of miles in extreme cases. Such propagation cannot be relied upon.

Something similar to ducting happens in optical fibres. The refractive index of the fibre is graduated so that light travels a little slower in the center. As light approaches the surface of the fiber, it will be curved back by the mechanism of Fig.3. If light can be persuaded to follow a single path down the center of an optical fiber, the light pulses at the receiver will suffer less time smear and the bandwidth of the fiber will be increased.

Fig.3a) - Normal fall of temperature with height directs sound away from the Earth.  b) - Temperature inversion bends sound the other way.  c) - A temperature layer can act as a waveguide. A similar concept is used in optical fibers.

Fig.3a) - Normal fall of temperature with height directs sound away from the Earth. b) - Temperature inversion bends sound the other way. c) - A temperature layer can act as a waveguide. A similar concept is used in optical fibers.

The atmosphere is also capable of bending light as its density is not uniform. The sun will often appear to be flattened just before sunset whereas at midday it is circular. One of the most spectacular atmospheric effects is the mirage, in which light leaving the Earth is turned down by a vertical temperature gradient so that it may be visible at a considerable distance.

Mirages are popularly associated with desert regions, but the correct conditions can be present in many places, including over the sea, where ships and islands become visible even if they are over the horizon. It is possible for miraging to turn an atmospheric layer into a mirror, with the result that a mirage is seen which is upside down.

Whatever the atmosphere may do, the observed result is also affected by the imagination of the observer. A great deal of naval folklore from sea monsters to mythical lands stems from unsophisticated interpretation of mirages.

On land, temperature gradients can turn hot surfaces into reflectors. A common example is where a hot road surface reflects the sky and so appears to have pools of water. Such conditions play havoc with laser-based distance and speed measuring equipment which may give erroneous results because the assumption that distance is proportional to the transit time of light pulses fails when the propagation is not in a straight line. 

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