Locating a Lightning Strike by Its Thunder

Two microphones a few kilometers apart hear the same thunderclap at slightly different times. The time difference says how much farther the strike was from one mic than the other, and every point with that difference lies on a hyperbola. A third mic adds a second hyperbola, and they cross at the strike.

How it works

Hyperbolas: a constant difference of distances

In one sentence

A hyperbola with foci $F_1, F_2$ is the set of points $P$ with $\left||PF_1| - |PF_2|\right| = 2a$ . With foci $(\pm c, 0)$ and $0 < a < c$ it's $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with $b^2 = c^2 - a^2$ ; its asymptotes are $y = \pm\frac{b}{a}x$ , and each branch is the set where one particular distance is the larger.

General formula

|{\htmlData{sym=0}{d_1}} - {\htmlData{sym=1}{d_2}}| = {\htmlData{sym=2}{v}}\,{\htmlData{sym=3}{\Delta t}} = 2{\htmlData{sym=4}{a}}, \qquad \frac{x^2}{{\htmlData{sym=4}{a}}^2} - \frac{y^2}{{\htmlData{sym=5}{b}}^2} = 1, \qquad {\htmlData{sym=5}{b}}^2 = {\htmlData{sym=6}{c}}^2 - {\htmlData{sym=4}{a}}^2

An ellipse keeps the sum of the two distances fixed; a hyperbola keeps their difference fixed.

What the symbols mean

$d_1$
Distance to mic 1
How far the strike is from the first microphone.
$d_2$
Distance to mic 2
How far the strike is from the second microphone.
$v$
Speed of sound
About 343 meters per second.
$\Delta t$
Time difference
How much later one microphone hears the thunder than the other.
$a$
Half the distance difference
Half of v·Δt: the hyperbola's vertex distance from the center.
$b$
Second axis
Sets the hyperbola's asymptote slopes, b/a.
$c$
Half the mic spacing
Each mic (focus) sits c = 2 km from the center.

Sound travels at about 343 m/s, so an arrival-time difference $\Delta t$ means the strike was $v\,\Delta t$ farther from one microphone than from the other.
All points whose distances to two fixed points differ by the same amount $2a$ form a hyperbola with those points as foci. With the mics at $(\pm c, 0)$ it's $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ , where $b^2 = c^2 - a^2$ .
Which mic heard it first picks the branch: the strike lies on the branch nearer that mic.
One time difference leaves a whole curve of possibilities. A third microphone gives a second hyperbola, and the strike is where they cross.

Learn the math

HyperbolasAlgebra · Lesson 30, Common Graphs Nonlinear SystemsAlgebra · Lesson 47, Systems of Equations

Locating a Lightning Strike by Its Thunder