How Bats See With Sound
A bat shouts into the dark and builds a picture out of the echo. Every part of that picture - how far, how big, how fast - is a number squeezed out of timing.
Shouting at the world to see it
Junior level — plain language, no maths
A bat hunting on a moonless night catches a moth the size of your thumbnail, in mid-air, without ever seeing it. It does it by shouting - loudly. Bat calls are among the loudest sounds any animal makes, well over 100 decibels at the mouth, which is jackhammer territory. You do not hear them only because they are pitched far too high for human ears.
The trick is timing. Sound travels at about 343 metres per second, which is fast but not instant. Shout at something five metres away and the sound takes about 15 milliseconds to get there and another 15 to come back - about 29 milliseconds in total. The bat's brain measures that gap and turns it straight into a distance. Twice as far away, twice the wait. That is all echolocation fundamentally is: a stopwatch, run very precisely.
Pitch controls the detail. Sound travels in waves, and a wave cannot show you anything much smaller than itself - like trying to feel the texture of a page with a boxing glove on. A 55 kHz call has waves about 6 mm long, so 6 mm is roughly the smallest thing that call can pick out. Shout higher and the waves get shorter and the detail gets finer - but high notes are absorbed by the air much faster, so they do not travel as far. Every bat has to choose between seeing detail and seeing far, and different species have chosen differently.
Then there is the problem that makes the whole thing dramatic. A bat cannot shout and listen at the same time - its own voice would deafen it. So it has to shout, wait for the echo to come home, and only then shout again. Far away, the wait is long and the bat calls slowly, maybe ten times a second. As it closes in, the wait shrinks and it can call faster and faster, until in the last fraction of a second before the strike it is firing off nearly two hundred calls a second. That machine-gun rattle is called the terminal buzz, and if you ever hear a bat detector click into a frenzy, something just got caught.
Things worth knowing
- Bat calls exceed 100 decibels at the mouth — as loud as a jackhammer. You cannot hear them only because they are pitched far above human hearing.
- Echolocation is a stopwatch. Sound covers about 343 metres a second, so an echo from 5 m away returns in 29 milliseconds, and the bat reads distance straight off that delay.
- Some moths hear bats coming and drop out of the air. Others answer back with ultrasonic clicks that jam the bat's sonar — an arms race running for fifty million years.
Delay, wavelength and the timing constraint
Student level — the core equations
Range comes from time of flight. An echo from a target at distance \(d\) returns after \(t = 2d/c\), with \(c \approx 343\ \text{m/s}\) in air, giving about 5.8 ms per metre of range. Bats resolve these delays remarkably finely - behavioural experiments put jitter discrimination at well under a microsecond in some species, corresponding to sub-millimetre range resolution, which is at the edge of what the auditory system should be able to do and is still argued about.
Resolution comes from wavelength. \(\lambda = c/f\), so a 55 kHz call has \(\lambda \approx 6.2\ \text{mm}\) and a 120 kHz call about 2.9 mm. A target much smaller than a wavelength scatters in the Rayleigh regime, with returned power falling as \(f^4\) - so detecting small insects strongly favours high frequencies. Working against that is atmospheric absorption, which rises steeply with frequency: roughly 1 dB/m at 40 kHz but 3 dB/m or more above 100 kHz. With a finite echo budget, range and resolution trade directly against each other, and the observed spread of call frequencies across species maps onto exactly this trade-off.
The third constraint is temporal. Because a bat cannot emit and receive simultaneously without self-deafening, the pulse interval must exceed the round-trip delay plus the call duration - the overlap constraint. This is not a curiosity; it dictates the whole structure of an attack. In search phase a bat emits long, narrowband, low-frequency calls at 5–15 per second. In approach it shortens and sweeps them. In the terminal buzz it fires 150–200 calls per second of under a millisecond each, which is the maximum the returning echoes permit - the call rate is set by the physics of the closing distance.
Velocity comes from the Doppler shift, \(\Delta f/f \approx 2v/c\) for a reflector moving along the line of sight. A moth closing at 5 m/s shifts a 60 kHz call by about 1.75 kHz. Horseshoe bats exploit this with constant-frequency calls and Doppler-shift compensation: they lower their emitted frequency in flight to hold the returning echo at a fixed 'acoustic fovea' frequency where their cochlea is hugely over-represented, then read the residual flutter of insect wingbeats as a modulation on that carrier.
Key Formulas
| Echo delay | \(t = \dfrac{2d}{c}\) | ≈ 5.8 ms per metre |
|---|---|---|
| Speed of sound | \(c \approx 343\ \text{m/s}\) | |
| Wavelength | \(\lambda = \dfrac{c}{f}\) | 55 kHz → 6.2 mm |
| Rayleigh scattering | \(P_{\text{echo}} \propto f^4\) | small targets need high f |
| Atmospheric absorption | \(\alpha \approx 1\text{–}3\ \text{dB/m}\) | rises steeply with f |
| Overlap constraint | \(T_{\text{pulse}} > \dfrac{2d}{c} + \tau\) | |
| Doppler shift | \(\dfrac{\Delta f}{f} \approx \dfrac{2v}{c}\) | |
Things worth knowing
- Some bats discriminate echo delay jitter of well under a microsecond — sub-millimetre range resolution. How the auditory system achieves this is still not fully explained.
- High frequency means fine detail but short range: absorption in air runs about 1 dB/m at 40 kHz and over 3 dB/m above 100 kHz. Every species sits somewhere on that trade-off.
- Horseshoe bats lower their call frequency in flight so the returning echo lands on a hyper-sensitive 'acoustic fovea' in the cochlea — active Doppler compensation, in a living animal.
Time-domain imaging, the jitter problem and an arms race in ultrasound
Scholar level — full mathematical depth
01Sonar, biologically
Echolocation is active time-domain imaging. The information available is entirely contained in the echo waveform: delay gives range, interaural level and time differences plus spectral notches from the pinnae give bearing and elevation, spectral colouration gives target structure, and Doppler gives radial velocity. Bats operate with a very poor antenna - a head a few centimetres across - and compensate with signal design. Frequency-modulated sweeps give excellent range resolution through their large time-bandwidth product; constant-frequency calls give excellent velocity resolution. The two families of call design are the two ends of the classical radar ambiguity trade-off, arrived at independently by evolution.
02The jitter problem
Simmons reported that Eptesicus fuscus discriminates echo delay jitter of order 10–50 ns, implying range resolution well below a millimetre. The difficulty is that the auditory nerve's temporal precision is of order tens of microseconds, so no straightforward neural stopwatch suffices. Proposed resolutions include coherent cross-correlation receivers operating on the carrier phase, spectral-cue readout in which the interference between overlapping echo components is read in the frequency domain rather than the time domain, and methodological artefacts in the original paradigm. The debate is not settled and is one of the sharper open questions in sensory neuroscience.
03The acoustic fovea
Rhinolophid bats emit long constant-frequency calls near 83 kHz and lower the emitted frequency in flight so that the Doppler-shifted echo returns at a species-specific reference frequency. The cochlea is grossly over-represented at that frequency - an expanded region of the basilar membrane with sharply tuned filters, the auditory analogue of a visual fovea. Its function is not range-finding but recognition: the wingbeat of a fluttering insect imposes amplitude and frequency modulations on the returning carrier, and the acoustic fovea resolves those glints. The bat is not measuring where the moth is so much as reading its wingbeat signature.
04Solving self-deafening
Emission at over 120 dB SPL beside an ear that must detect echoes tens of decibels above threshold demands isolation. Middle-ear muscles contract a few milliseconds before each emission, attenuating by 20 dB or more, and relax within milliseconds - a neuromuscular cycle repeated up to 200 times a second during the terminal buzz, among the fastest sustained muscle contractions in mammals. Neural gain in the auditory pathway is additionally modulated in time with emission. The overlap constraint that sets pulse rate is thus enforced both acoustically and neurally.
05The moths answer
Tympanate hearing has evolved independently many times in Lepidoptera, tuned to bat call frequencies, and drives evasive dives at high stimulus intensity. Tiger moths go further: some produce ultrasonic clicks that startle, some advertise unpalatability - the first proven acoustic aposematism - and Bertholdia trigona demonstrably jams bat sonar by emitting clicks timed to the returning echoes, degrading ranging rather than merely warning. Some moths carry scales that absorb ultrasound like acoustic metamaterial, cutting echo strength by up to 85%. A few bats have countered by shifting call frequency outside moth hearing, or by hunting in near-silence and listening for prey-generated sound.
06Convergence, twice over
Laryngeal echolocation and the odontocete biosonar of dolphins converged on the same solution from unrelated hardware, and both converged on the same molecular detail: the motor protein Prestin, responsible for outer hair cell electromotility and cochlear amplification, shows striking sequence convergence between echolocating bats and toothed whales - the same amino acid substitutions arrived at independently. Within bats, the phylogenetic position of the non-echolocating pteropodids implies either that laryngeal echolocation arose once and was lost, or arose twice; the question is still contested, which means a sense this elaborate may have been invented more than once.
Key Formulas
| Range from delay | \(d = \dfrac{c\,t}{2}\) | |
|---|---|---|
| Range resolution | \(\Delta d = \dfrac{c}{2B}\) | B = sweep bandwidth |
| Time-bandwidth product | \(TB \gg 1\) | why FM sweeps beat pure tones |
| Rayleigh regime | \(\sigma \propto \dfrac{a^6}{\lambda^4}\) | targets smaller than λ |
| Two-way spreading | \(P_{\text{echo}} \propto d^{-4}\) | |
| Absorption | \(P \propto 10^{-2\alpha d/10}\) | |
| Doppler | \(f_{\text{echo}} = f_0\,\dfrac{c+v}{c-v}\) | |
| Overlap constraint | \(\text{PRI} > \dfrac{2d_{\max}}{c} + \tau\) | |
Things worth knowing
- Middle-ear muscles contract before every call to protect the ear, then relax in time to hear the echo — up to 200 times a second in the terminal buzz, among the fastest sustained muscle contractions in mammals.
- The moth Bertholdia trigona genuinely jams bat sonar, timing ultrasonic clicks to the returning echoes to corrupt ranging. Other moths carry scales that absorb ultrasound, cutting echo strength by up to 85%.
- The hearing protein Prestin shows the same convergent amino acid substitutions in echolocating bats and toothed whales — two lineages that invented biosonar separately and landed on the same molecule.