Nature’s physics, explained.

Acoustics · Doppler effect

Hearing speed

A bat flying towards a moth hears the echo at a higher pitch than the call it sent. The size of this Doppler shift tells it how fast it is closing in. Horseshoe bats even lower their call in flight, so that the echo always lands in their most sensitive band of hearing.

At v = 5 m/s and f₀ = 50 kHz, the echo returns about 1.5 kHz higher.

Geometry · Trigonometry

Measuring distance with time

Sound travels at about 343 m/s in air. Halve the round-trip time of the echo and you have the distance. The tiny difference in arrival time between the left and the right ear adds the direction – two measurements, one position.

An echo after 17.5 ms puts the moth 3.0 m away. Δτ: time difference between the ears, b: distance between the ears.

Signal processing · Sonar

From chirp to code

Autonomous drones and parking sensors use the same trick. A matched filter slides the known chirp along the recording; the peak of the cross-correlation marks the echo – robust even in noise.

import numpy as np

SPEED_OF_SOUND = 343.0  # m/s, air at 20 °C

def echo_distance(chirp, recording, sample_rate):
    """Find the echo with a matched filter; return metres."""
    corr = np.correlate(recording, chirp, mode="valid")
    delay = np.argmax(np.abs(corr)) / sample_rate  # round trip, s
    return SPEED_OF_SOUND * delay / 2