Wave motion
A signal traveling along a cable is a wave, with its speed, its wavelength and its reflections, and it obeys the same equation as a shaken string. All of transmission-line theory comes from there.
01What travels in a wave
A wave carries energy and momentum, not matter. The particles of the medium oscillate around their position and stay where they were; what moves forward is the disturbance. This holds for sound, for a string and also for the signals in a cable.
| Wave | What oscillates | Speed |
|---|---|---|
| On a string | Transverse position | \( \sqrt{T/\mu} \): tension and linear density |
| Sound in air | Pressure | 343 m/s at 20 °C |
| In a coaxial cable | Voltage and current | \( 1/\sqrt{LC} \) per unit length |
| Electromagnetic | Fields \( E \) and \( B \) | \( c/n \) |
The third row is the one that connects this unit with electronics: a signal traveling along a cable is a wave, with its speed, its wavelength and its reflections, and it obeys the same equation as a plucked string.
02Superposition
When two waves coincide at a point, the displacements add. It is a principle of deceptive simplicity: from it come interference, beats, standing waves and, at bottom, the whole of Fourier analysis.
Two waves and their sum. Change the frequency of the second, its amplitude and its phase, and try reversing its direction of travel: that is where the standing wave appears.
With equal frequency and the same direction, the sum is another wave of the same frequency whose amplitude depends on the phase shift: maximum in phase, zero in antiphase. With slightly different frequencies a beat appears, a slow envelope of frequency \( |f_1 - f_2| \), which is what you hear when tuning a guitar and what makes a superheterodyne receiver work. And with opposite directions, the wave stops advancing: it becomes standing, with fixed nodes where the amplitude is always zero.
03Standing waves and reflection
A wave that reaches the end of its medium is reflected. If the end is fixed, it comes back inverted; if it is free, it comes back upright. The superposition of the incident wave with the reflected one gives the standing wave.
The same phenomenon, in a cable carrying a signal, is a problem. If the end is not terminated with the correct impedance, part of the signal comes back and superimposes on the one arriving: steps appear, overshoots and —in radio frequency— a standing wave along the line. The measure of how much comes back is the standing wave ratio:
When the source and the observer approach each other, the perceived frequency rises; as they move apart, it drops. It is the phenomenon of the sound of a passing siren, and also the principle of the speed radar, which measures the shift of the wave bouncing off a car. In electronics it also appears as a problem in mobile communications: the Doppler shift of a moving device has to be compensated in the receiver.
04In the lab
With a generator and an electromagnetic driver on a taut string, find the frequencies that produce one, two and three antinodes. Verify the proportionality with \( n \) and obtain the wave speed; compare it with \( \sqrt{T/\mu} \).
With two audio generators near 440 Hz and a speaker, separate them little by little and count the beats per second. Verify that they match the difference in frequencies, and observe the envelope on the oscilloscope.
Send a narrow pulse down a long coaxial cable and observe the end with the oscilloscope, first open, then short-circuited and finally terminated with 50 Ω. Identify the sign of each reflection and verify that with the correct termination it disappears.
05Common mistakes
- Believing that the wave carries matter. It carries energy; the medium oscillates and stays.
- Thinking that the frequency changes when the medium changes. It is set by the source; what changes is \( \lambda \).
- Confusing the speed of the wave with the speed of the particles of the medium. They are different things and very different values.
- Believing that there is no energy in a standing wave. There is: what there is not is net transport.
- Leaving a signal cable unterminated. What is not absorbed comes back, and superimposes on what arrives.
- Adding intensities instead of amplitudes. What superimposes is the amplitudes; the intensity goes with the square.
06Self-assessment
What does a wave carry?
Energy and momentum. The medium oscillates around its position and does not move along.
When passing to another medium, what changes?
The speed and the wavelength. The frequency is set by the source and does not change.
What is the frequency of a beat?
The difference between the two frequencies that are superimposed.
When does a standing wave appear?
When two equal waves traveling in opposite directions are superimposed, almost always by reflection at an end.
What is the SWR of a properly terminated line?
One: with \( Z_L = Z_0 \) the reflection coefficient is zero and nothing comes back.
What happens on reflection at a fixed end?
The wave comes back inverted. At a free end it comes back upright.
07Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage, 2019. Wave motion, superposition, standing waves and the Doppler effect.
- Frank S. Crawford. Waves: Berkeley Physics Course, Vol. 3. McGraw-Hill, 1968 (Spanish edition, “Ondas. Berkeley Physics Course, vol. 3”, Reverté). An entire volume devoted to waves, with the modes approach that later reappears throughout engineering.
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. The natural continuation: transmission lines, reflection coefficient and SWR treated seriously.