Fog Horns and
Fourier Transforms

For most of two centuries the fix looked obvious. Build a bigger horn. It failed, and the way it failed taught the coast something that signal processing still lives on.

In 1874 John Tyndall, running fog-signal trials for Trinity House, reported that 273 vessels had been lost on British coasts in fog or thick weather over ten years, in spite of the most powerful coast lights then built.¹ Tyndall in England and Joseph Henry in the United States both set out to prove that a warning made of sound could be relied on, and both kept running into the same wall. A signal could be deafening at the source and gone a mile out, then plain again far past that.¹

The sharpest case came off a battlefield. Tyndall's paper carried a letter from R. G. H. Kean, who watched the Battle of Gaines's Mill: at least fifty thousand men and a hundred field guns in action for two hours, and from a rise a few miles off Kean saw the muzzles flash and heard nothing, while that same cannonade was heard clearly at Amherst Court House, a hundred miles west of Richmond.¹ Loud sound, clear air, dead silence up close. No theory of absorption can lose a sound at short range and return it a hundred miles away.

The old answer blamed the fog. The fog turned out to be nearly innocent.

Reverend William Derham had written in 1708 that thick vapors and particles deaden sound, and that stood as expert opinion well into the 1800s.¹ The numbers say otherwise. A fog droplet runs about ten microns across; the wave from a deep horn runs more than a meter. A wave that long does not catch on a droplet that small, and the scattering it does cause is far too weak to account for the silences keepers kept logging.¹

What bent the sound was the air itself. Wind speed climbs with height, and air temperature falls with altitude by day and inverts at night, so the speed of sound changes from one layer to the next. Either gradient curves the sound ray upward, lifts it over a near listener into an acoustic shadow, then sets it down again far downrange. George Stokes proposed the wind-gradient idea in 1857; Osborne Reynolds tested it in 1874 and found that lifting a ringing bell onto a post four feet high doubled the range of the sound.¹ The channel between horn and boat does as it likes with a sound, so the engineering had to move off raw power and onto the sound itself: choose one the channel treats kindly, and give it a shape a listener can name.

To choose the sound, you first take sound apart. The tool for that came out of heat, not water.

In 1822 Joseph Fourier published Théorie analytique de la chaleur, working out how heat spreads through a solid; to solve it he claimed that any reasonable curve can be written as a sum of sines and cosines.² Lagrange and Laplace doubted him. He was right enough that the claim outgrew the heat equation entirely.

A horn's blast, drawn as pressure against time, is one jagged line. Fourier says that line is really a stack of pure tones added together, each a smooth sine wave at its own frequency and strength. Pull the stack apart and you can read exactly which tones the blast is made of. Then keep the ones that carry, and let the coast have the rest.

Three choices hide inside a fog signal. Read them the way Fourier reads a wave, one component at a time.

Fourier said you can take a sound apart into tones. He also said you can add the tones up and get the sound back.

Run the transform forward on short, overlapping slices of a recording and you get a spectrogram, the blast drawn as frequency against time; the fast Fourier transform makes that cheap enough to watch as it happens.¹¹ The same decomposition runs inside the files that carry voices now. An audio codec cuts each slice into frequency components, spends bits on the parts an ear will catch, and drops the rest.¹¹ It is the diaphone's logic at a finer grain: choose the components that matter, send those, release the others.

A coast pays for one kind of sound: a low note the mist cannot take, marked in a rhythm the boat already knows.

What the keepers settled by ear, signal processing now settles by arithmetic, and the rule held. The work is never to be the loudest thing on the water. It is to build the one sound made to survive the crossing and to be known when it lands. On a falling tide off the Alsea, with the bar gone to milk and the channel somewhere under the fog, that is still the only sound worth making.