Rocky Has Sonar
Passive echolocation and Fourier transform
Spoiler warning: this page discusses the novel's plot up to this chapter. It doesn't quote passages from the book.
Grace discovers Rocky doesn't see with light but with sound: passive echolocation. Aluminum is "transparent" to Rocky. Using Fourier analysis, Grace begins decoding the Eridians' musical language.
Sonar instead of vision: how Rocky "sees"
Rocky can't distinguish ink marks but can "see" through an aluminum plate. Grace deduces he uses passive sonar.
The science behind it
Humans perceive the environment with photons (light). Rocky uses sound waves. On his planet, with 29 atm of ammonia and possibly no surface light, optical vision never evolved.
Unlike bats and dolphins (active sonar: emit pulses and listen for echoes), Rocky uses passive sonar: his brain processes ambient sound bouncing off objects.
A thin aluminum sheet blocks nearly all light but lets enough sound through. To Rocky, the Hail Mary's metal hull is like a slightly fogged window: it dims what's on the other side but doesn't hide it.
Ink marks on a measuring tape produce no relief, so sonar can't detect them. Only physical relief is "visible" to sonar. That's why the Eridian clock has raised numbers (3 mm high).
Key terms
Fourier transform: decoding the language
Grace installs Fourier analysis on his laptop. When Rocky speaks, the software shows each chord's frequencies.
The science behind it
The Fourier transform is an algorithm that decomposes any complex signal into its component frequencies. Record a piano chord, and the algorithm tells you exactly which notes compose it.
Human speech is chaotic: constantly changing formants, noisy consonants, blending vowels. It's hard to cleanly decompose.
Eridian chords have discrete, stable frequencies: each "phoneme" is a precise combination of notes. The Fourier algorithm identifies them unambiguously.
5 simultaneous notes in a chord implies 5 independent "vocal cords." Grace can transcribe each word as a list of numerical frequencies and build a dictionary.
Key terms
Try it yourself
How to use the simulator
When several notes sound at once, the air vibrates with a single complicated wave. The Fourier transform does the reverse: it takes that wave and splits it into the pure frequencies it's made of. That's what a spectrum analyzer would do to make sense of a language built from chords, like Rocky's. The visualizer shows the combined wave and the active frequencies side by side.
Controls
- Note buttons: eight notes spanning one octave, from C4 (261.6 Hz) to C5 (523.3 Hz). Each click turns one on or off.
Things to try
- Turn on a single note: the wave is a clean sine and the spectrum shows one bar.
- Turn on C4 and C5 together. The second frequency is exactly double the first, and the wave repeats in a simple pattern.
- Turn on four or five notes. The wave looks chaotic, but the spectrum still picks out every component clearly. That's the power of Fourier analysis.
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Open “Fourier transform visualizer” on its own page, with an explanation and a worked example →
Real science vs fiction: how much sound gets through a metal hull?
When sound crosses from one medium into another, part of its energy bounces back at the boundary. How much depends on each medium's acoustic impedance, which is its density multiplied by the speed of sound in it. Air has an impedance of about 400 rayls; aluminium, about 17 million. With a gap that large, only around one ten-thousandth of the energy gets across a single air-to-aluminium boundary.
A thin sheet does better than that figure suggests, because the whole panel vibrates like a drumskin. Acoustic engineers estimate it with the "mass law": the loss is about 20·log₁₀(f·m) − 47 decibels, where f is the frequency in hertz and m the panel's mass per square metre. A 1 cm aluminium sheet weighs about 27 kg/m²; at 500 Hz it cuts the sound by about 36 dB.
That's roughly what an interior wall in a flat blocks: you can hear the neighbours talking, just muffled. So the hull isn't as transparent to sound as a window is to light, but it lets enough through for a very sharp ear like Rocky's to work out what's on the other side.
The Eridian atmosphere helps. At 29 atmospheres and around 210°C, ammonia has a density of about 12 kg/m³, ten times that of air, and sound travels through it at about 550 m/s. Its impedance is some 17 times that of air, so it couples better to solid walls and carries more sound energy. A dense world is a world where hearing makes a lot of sense.
Key terms
Worked example: pulling apart the notes of a chord
To analyse a sound, Grace's computer records it as samples: a typical microphone takes 44,100 samples a second. The Nyquist theorem says that captures frequencies up to half that rate, 22,050 Hz, which comfortably covers human hearing.
Frequency resolution depends on how long you listen. With a window of length T, the Fourier transform can separate frequencies at least 1/T hertz apart. A 0.1-second window (4,410 samples) tells apart notes that are 10 Hz or more apart.
Try an A major chord: A (220 Hz), C sharp (277 Hz) and E (330 Hz). The notes are about 50 Hz apart, so a 0.1 s window separates them easily. But two notes a semitone apart near 220 Hz differ by only 13 Hz, which needs almost 0.08 s. At 110 Hz the semitone shrinks to 6.5 Hz and the window grows to 0.15 s. The lower and closer the notes, the longer you have to listen.
This trade-off between precision in time and in frequency can't be avoided: it's the same mathematics that underlies the uncertainty principle in quantum mechanics. The algorithm matters too. Computing the transform directly on 4,096 samples takes about 16.8 million operations; the fast Fourier transform (FFT) does it in about 49,000, roughly 340 times fewer. That's why a laptop can translate chords in real time.