EC-2.3 Plane Waves, and How They Propagate
You can derive the wave equation from Maxwell's equations, describe how a uniform plane wave behaves in a lossless and a lossy medium, and compute what happens when one meets an interface.
Combining the two curl equations produces a wave equation whose speed is built from two constants that were measured in entirely static experiments, which is the moment the subject stops being about circuits and starts being about light. What follows is the behaviour of the simplest solution: how fast it goes, how deep it gets into a conductor, which way it is oriented, how much power it carries, and what fraction of it comes back at a boundary. Every one of those quantities reappears later as a specification on a radio link.
Work through these
Deriving the wave equation, and where the speed of light comes from
Taking the curl of one of the two coupled equations and substituting the other leaves a wave equation for each field alone. Its speed is set by two constants from static experiments, which is why the result was so surprising.
NPTEL: Transmission Lines and EM Waves · CourseThe uniform plane wave: the two fields, their ratio, and the direction of travel
The electric and magnetic fields are at right angles to each other and to the direction of travel, and their ratio is the intrinsic impedance of the medium. That one ratio carries most of what you need about a material.
NPTEL: Electromagnetic Theory · CoursePropagation in a lossy medium: attenuation, phase constant and skin depth
In a conducting medium the wave decays as it advances, and the depth at which it falls to about a third is the skin depth. It shrinks with frequency, which is why high-frequency currents ride on the surface of a conductor.
NPTEL: Electromagnetic Fields · CoursePolarization: linear, circular and elliptical, and how it is produced
Polarization is the path the electric field traces as the wave goes past, decided by the relative amplitude and phase of two perpendicular components. A receiving antenna aligned to the wrong one loses most of the signal.
Phase velocity and group velocity, and why they can differ
One is the speed of a point of constant phase and the other the speed a packet of energy travels, and in a dispersive medium they part company. Only the second one carries information, which is the answer to the usual puzzle about the first exceeding the speed of light.
The Poynting vector, and power carried by a wave
The cross product of the two fields gives both the direction of energy flow and its density, and its time average is the power a receiver can collect. This is the quantity a link budget is written in.
MIT OpenCourseWare 6.013: Electromagnetics and Applications · CourseReflection and refraction at an interface, at normal and oblique incidence
At a boundary the wave splits, and the fractions are fixed by the impedances on either side and by the angle. Normal incidence gives the simple ratio; oblique incidence adds an angle-dependent term and a polarization dependence.
NPTEL: Electromagnetic Theory · Course
Sign in to keep your progress.
Free resources
We haven't checked most of these for screen reader use yet.
Links last checked 3 Sept 2026.
Stuck here?
Ask a mentor. A real person answers, and they can see exactly which topic you're on. Usually within a couple of working days.
Checking your session…
Topics shown in module order.