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There is no escape. Almost everywhere we look in broadcasting, there will be something that is dependent on waves of some kind. Happily, many different kinds of wave have a lot in common, so it is possible to get quite a long way with one's understanding by exploring that commonality, whilst watching out for the differences.

With the exception of light, waves need a medium in which to propagate. That is because waves carry energy, and perhaps information. Media have in common the ability to store energy locally and temporarily. Transmission lines use inductance and capacitance for that. Other types of wave store energy in some combination of compression, shearing, kinetic or potential energy. The interplay of the stiffness and the density determines the propagation speed. High stiffness speeds things up; high density slows things down.

But what is a wave? Basically some form of energy has caused an excursion away from equilibrium, a disturbance to the calm, if you will, having some implied periodicity. Many types of wave are invisible, but we have all seen waves on water. It really looks as if the water is moving at the speed of the waves. It's an illusion: only the shape of the wave moves and in the absence of a current the long-term position of individual water molecules doesn't change. Instead they move around their average position. This is also true of most other types of wave. The propagation velocity of the wave and the velocity of the particles involved are completely different. Signals pass down a transmission line almost at the speed of light, but the electrons involved move much more slowly.

Sound waves propagate within a medium, whereas surface acoustic waves (SAW) propagate across a boundary between a solid and some other substance. Gravity waves, the kind that we see on the surface of water, propagate at the boundary between two fluids having grossly different densities. We should not be surprised that in deep water they behave differently to sound waves yet in shallow water there are remarkable similarities.

Light is another example of wave propagation. Light is an example of an electromagnetic wave that has the remarkable property that it appears to require no medium in which to propagate and can travel indefinitely through the vacuum of space. One might almost say that light makes its own medium. Having done so, wherever the light has come from, it always appears to the onlooker to have the same speed in vacuo. The theory of relativity tells us that nothing having rest mass can move at the speed of light, because its mass becomes infinite. Light has energy on account of its velocity. Having no rest mass, it is the only thing that can move at the speed of light. It can't stop, of course.

Light can also travel through certain media, and when it does it slows down. The extent to which it slows is denoted by the refractive index and this is not constant, but varies with wavelength. When wave velocity varies with wavelength, the propagation is said to be dispersive. Gravity waves in deep water are strongly dispersive. Light waves in a medium are dispersive, which is why we see rainbows, why prisms and diffraction gratings can split white light into its constituents and why lenses suffer chromatic aberration.

In non-dispersive wave propagation, such as most cases of sound waves in air and their electrical analogs in an XLR cable, we can say that the velocity is constant and is the product of the frequency and the wavelength. We know from Fourier analysis that waveforms are made up of harmonics in a defined amplitude and phase relationship. For example the spectrum of a square wave has a sinx/x envelope with no even harmonics. A non-dispersive medium preserves the phase relationships between the harmonics so the waveform can propagate unchanged. The system is said to display linear phase. The common use of square wave testing in audio is checking for phase linearity. 

Once we have dispersion none of that works. Once the propagation velocity is a function of wavelength, it is no longer possible to preserve a waveform because the phase relationships of the harmonics must change. If we attempt to propagate a pulse, containing energy spread over a range of frequencies, the waveform changes constantly as the different components adopt different phases. The pulse appears where the components are in phase and reinforce. This means that the pulse does not travel at the speed of the components. The pulse travels at the group velocity, where a group is a set of components that are reinforcing one another.

Thus in wave propagation we have three different velocities. There is the particle velocity of the medium due to the disturbance of the equilibrium, there is the propagation velocity of a single frequency, which is known as the phase velocity or the celerity. The use of c to denote the speed of light comes from celerity. Then there is the group velocity, which is the velocity with which energy and/or information travels. Group velocity is invariably slower than phase velocity. In gravity waves it is half as fast. In a standing wave the group velocity is zero. It is important to realize that the phase and group velocities are the velocities of the disturbance, whereas the physical particle velocities may much lower.

Fig.1 shows the electromagnetic wave, which consists of varying electric fields at right angles to varying magnetic fields at the same frequency. Maxwell discovered that the energy constantly changes in the two fields. Magnetic and electric fields can both store energy in a vacuum. The varying magnetic field generates an EMF and this causes what Maxwell called a displacement current to flow. The current then generates a magnetic field and so on. The magnetic and electric fields carry equal energy and can have various phase relationships that will be considered in due course. Einstein's equivalence of energy and mass allows the electromagnetic wave to have momentum without rest mass.

Waves that depend on media vary as a function of the way the medium tries to restore equilibrium after a disturbance. All such waves are subject to energy loss or damping over time, which may be viscous, frictional or ohmic. If the energy source ceases, eventually calm returns.

Fig.2 shows a sound wave propagating in a compressible medium. Only the wave propagates, the individual particles of the medium maintain the same average position, but oscillate to-and-fro in the direction of propagation. The wavefront is at 90 degrees to the direction of propagation, which is described as longitudinal. The particle movement must be longitudinal, because only that movement can cause pressure changes that allow restoration.

Fig.3 shows a transverse wave propagating along a flexible medium that is under tension, such as a guitar string, a helicopter blade or the cables of a suspension bridge. Again it is the disturbance that travels, not the medium. In this case the oscillations of the medium are at right angles to the direction of propagation. The restoring mechanism is the tension. A transverse wave in air or water is not possible because there is no resistance to shear.

Fig.4 shows a gravity wave propagating across the surface of water. The restoration mechanism is gravity, which tries to pull down the peaks of the waves and to fill in the troughs. The water must flow locally as the wave propagates. In deep water, particles move in a circle as the wave passes. The rotation is with the wave propagation, in other words a wave propagating from left to right has clockwise rotation. The diameter of the circle is equal to the peak-to-peak amplitude of the wave. As depth increases, the diameter of the circle decays exponentially.

Waves can also propagate across the surface of a solid in what are known as Rayleigh waves. Rayleigh waves are found in SAW (surface acoustic wave) devices used in electronics and in some touch screens. Some of the energy of an earthquake propagates as Rayleigh waves. As the medium is a compressible solid rather than an incompressible liquid, the propagation is somewhat different. The restoration method is a combination of the pressure produced as the solid is compressed and the energy stored in shearing. The Rayleigh wave is a hybrid wave. The surface is moving up and down as a transverse wave and from side to side as a longitudinal wave. The two motions are 90 degrees out of phase, so the surface motion is circular, but the rotation is in the opposite direction to that of gravity waves. It is the horizontal component of the motion that does a lot of the damage in earthquakes as traditional buildings oppose gravity and may be weak against horizontal forces.