Finding a Cable Fault from Its Echo

Send a short voltage pulse down a long cable. It travels at a fixed speed, and wherever the cable is damaged part of it reflects back. Timing the echo locates the fault: the distance is half the round-trip time times the wave speed.

How it works

D'Alembert's solution: waves traveling both ways, and reflections

In one sentence

On an infinite line, every solution of $u_{tt} = c^2u_{xx}$ has the form $F(x - ct) + G(x + ct)$ , and d'Alembert's formula builds $F$ and $G$ from the initial displacement and velocity. At an impedance change, continuity splits an incident wave into a reflected wave with coefficient $\Gamma$ and a transmitted wave with coefficient $1 + \Gamma$ .

General formula

u(x, t) = {\htmlData{sym=0}{F}}(x - {\htmlData{sym=2}{c}}t) + {\htmlData{sym=1}{G}}(x + {\htmlData{sym=2}{c}}t), \qquad {\htmlData{sym=3}{x_{\text{fault}}}} = \frac{{\htmlData{sym=2}{c}}\,{\htmlData{sym=4}{t_{\text{echo}}}}}{2}

Waves on a line just travel, keeping their shape. Everything you learn about a fault comes back as a delayed copy of the pulse.

What the symbols mean

$F$
Right-moving wave
A shape traveling toward the far end at speed c.
$G$
Left-moving wave
A shape traveling back toward the tester.
$c$
Signal speed
About 0.2 meters per nanosecond in a typical cable.
$x_{\text{fault}}$
Fault position
Distance from the tester to the damage.
$t_{\text{echo}}$
Echo time
How long the reflected pulse takes to come back.

On a lossless cable the voltage obeys the wave equation $u_{tt} = c^2u_{xx}$ , with $c$ about two thirds of the speed of light: 0.2 meters per nanosecond.
D'Alembert: every solution is a shape $F$ moving right plus a shape $G$ moving left, $u = F(x - ct) + G(x + ct)$ , each keeping its form.
At a fault the cable's impedance changes, so a fraction $\Gamma$ of the pulse reflects into a left-moving wave (inverted for a short, $\Gamma < 0$ ; upright for a near-break, $\Gamma > 0$ ) and the rest, $1 + \Gamma$ , carries on.
The echo returns after a round trip, $t_{\text{echo}} = 2x_{\text{fault}}/c$ , so $x_{\text{fault}} = c\,t_{\text{echo}}/2$ .

Learn the math

The Wave EquationPDE · Lesson 5, Foundations D'Alembert's SolutionPDE · Lesson 11, Separation of Variables Electromagnetic WavesE&M · Lesson 28, Maxwell's Equations and Light