Sound speed 0 m/s Range 0 m Echo delay 0 ms Clicks 0 /s Prey - FPS 0
Drag to orbit, wheel or pinch to zoom. Gold cap: the outgoing click beam; cyan cap: the echo from the salmon school (it spreads in all directions; only the part heading back to the whale is drawn). Sound is slowed by the Slow-motion setting (real sound crosses 100 m in about 67 ms). Scene depth is schematic: the depth slider only changes the sound speed.
Orca Echolocation: Click Trains, the Sonar Equation and Seawater Sound Speed
An orca sends a broadband click, waits for the echo from a salmon school and converts the delay to range. Whether the echo is detectable is a bookkeeping of decibels: source level, spreading, absorption, target strength and noise.
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View: pod View: click path View: salmon school
Water
Click and target
Receiver and noise
Readouts
Echo excess SE vs range SE = SL − 2·TL + TS − NL + DI − DT. The animal detects the school while SE ≥ 0 (dashed zero line). Dotted: the same budget without absorption.
Sound speed vs depth Mackenzie's equation with a mixed layer to 30 m and a cold deep layer (4 °C): speed falls with temperature down to the thermocline, then rises with pressure.
Experiment: range error from the wrong sound speed
The animal times the echo and converts with an assumed speed of 1500 m/s. Each row simulates the click wavefront at the current range in water of a different surface temperature (current salinity and depth); measured = discrete-step simulation of the wavefront and echo timing, predicted = r·cassumed /c.
Run temperature sweep
Independent numerical checks
Controls Water: surface temperature, salinity and the depth of the whale set the sound speed through Mackenzie's equation. Click and target: click frequency, source level, the range to the school, the number of fish and the single-fish target strength. Receiver and noise: ambient noise spectral level, receiving directivity index and the detection threshold. Slow motion scales simulated time so you can see the click and echo travel; the readouts always use real time. The science
Sound speed. Mackenzie (1981): c = 1448.96 + 4.591T − 0.05304T² + 2.374·10⁻⁴T³ + 1.340(S−35) + 0.0163D + 1.675·10⁻⁷D² − 0.01025T(S−35) − 7.139·10⁻¹³TD³ (T in °C, S in ‰, D in m), valid for T 2–30 °C, S 25–40 ‰, D up to 8000 m. The published check value is 1550.744 m/s at 25 °C, 35 ‰, 1000 m.
Echo ranging. A click leaves at t = 0, reaches the target at r/c and returns at t = 2r/c, so the animal infers r = c·t/2. If it assumes a speed cA different from the true c, its range is off by the factor cA /c. The click interval must exceed the two-way travel time, so the click rate falls as the target gets farther and rises into a rapid "buzz" at close range.
Transmission loss. One-way TL = 20 log₁₀ r + α r (spherical spreading plus absorption). α is from Thorp's empirical formula, α(dB/km) = 0.11 f²/(1+f²) + 44 f²/(4100+f²) + 2.75·10⁻⁴f² + 0.003 with f in kHz, so high frequencies are absorbed faster.
Sonar equation. Echo level EL = SL − 2TL + TS; the receiver must beat noise: NL − DI + DT, where NL = spectral level + 10 log₁₀(bandwidth). The detection range is where EL − (NL − DI + DT) = 0; a school of N fish adds 10 log₁₀ N to the target strength if their echoes add incoherently.
Simplifications and honest limits. Source level (typically quoted around 190–225 dB re 1 µPa peak-to-peak for killer whales), single-fish target strength (about −35 dB is illustrative), the 30 kHz click bandwidth, noise level, directivity index and detection threshold are typical or made-up round numbers, not measurements of a particular animal. The model ignores refraction, surface and seabed reflections, reverberation, the real transmit-beam shape, click-to-click variation and the animal's receiving physiology. Thorp's formula is a fit for the open ocean. Slow motion and the schematic scene depth are display choices.