EC-2.3 Plane Waves, and How They Propagate

What this is and why it exists

Combine the two equations that couple the fields, and a wave equation falls out. Its speed is built from two constants measured in entirely static experiments.

That is the moment the subject stops being about circuits and becomes about light. What follows is the behaviour of the simplest solution, and every quantity in it reappears later as a specification on a radio link.

The vocabulary

• Wave equation — the equation whose solutions travel at a fixed speed.
• Uniform plane wave — the simplest solution, with the fields constant across any plane at right angles to the travel direction.
• Intrinsic impedance — the ratio of the electric field to the magnetic field in a medium.
• Attenuation constant — how fast a wave decays as it advances through a lossy medium.
• Phase constant — how fast its phase advances with distance.
• Skin depth — the depth at which the amplitude has fallen to about a third.
• Polarization — the path the electric field traces as the wave passes.
• Phase velocity — the speed of a point of constant phase.
• Group velocity — the speed at which a packet of energy travels.
• Poynting vector — the cross product of the two fields, giving the direction and density of energy flow.

The mental model

The derivation is short and worth doing once. Take the curl of one of the two coupled equations, then substitute the other into it. The magnetic field disappears and you are left with an equation in the electric field alone. It is a wave equation, and its speed is one divided by the square root of the product of the two constants of the medium. Put in the values measured for empty space and out comes the speed of light. Those constants came from experiments with static charges and steady currents, which is why the result astonished the people who found it.

The simplest solution is the uniform plane wave. Its two fields are at right angles to each other and both at right angles to the direction of travel. Their ratio is the intrinsic impedance of the medium, which is about three hundred and seventy seven ohms in free space. That single ratio carries most of what you need to know about a material.

In a conducting medium the wave decays as it goes. Two constants describe it: one for how fast the amplitude falls and one for how fast the phase advances. The skin depth is the distance over which the amplitude falls to about a third of its value. It shrinks as frequency rises. That is why a high-frequency current rides on the outside of a conductor, and why plating a cheap conductor thinly with a better one works.

Polarization is the shape the electric field traces as the wave goes past. It is decided by the relative amplitude and phase of two perpendicular components. Equal amplitudes in phase give a straight line. Equal amplitudes a quarter cycle apart give a circle. Anything else gives an ellipse. A receiving antenna aligned to the wrong polarization loses most of the signal, so this is a practical matter and not a curiosity.

Phase velocity and group velocity part company in a dispersive medium, where the speed depends on frequency. The phase velocity is the speed of a point of constant phase, and it can exceed the speed of light without anything being wrong. Nothing is carried by a point of constant phase. The group velocity is the speed of the packet, and it is the one that carries information, and it never exceeds the speed of light.

The Poynting vector closes the topic. The cross product of the electric and magnetic fields gives both the direction of energy flow and its density. Its time average is the power a receiver can actually collect, and that is the quantity a link budget is written in.

At a boundary the wave splits. Part reflects and part passes through, and the fractions are set by the intrinsic impedances on either side. At normal incidence that gives a simple ratio, identical in form to the transmission line result in a later topic. At oblique incidence the angle enters, and the answer depends on the polarization as well.

What you should now be able to explain or do

Derive the wave equation from the two coupled equations, and say where the speed comes from. Describe a uniform plane wave and use the intrinsic impedance of a medium. Compute a skin depth and explain why high-frequency currents stay near the surface. Identify linear, circular and elliptical polarization from two components. Explain why phase velocity may exceed the speed of light while group velocity may not, and compute reflected power at an interface.

Check yourself