Sound speed in seawater
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 with T in °C, S in ppt, D in m. About 1500 m/s, rising with temperature, salinity and depth (pressure).
Echo delay and the click train
t = 2R/c. A dolphin waits for the echo before the next click: ICI ≈ 2R/c + lag (lag roughly 20–45 ms in published tank work; here adjustable). A fixed ICI behaves like a radar PRF: a target beyond Ramb = c·ICI/2 returns after the next click and appears at the wrong range (range ambiguity).
Sonar equation
SL − 2TL + TS − (NL − DI) = DT. Spreading + absorption: TL = 20 log10R + αR, with α from Thorp’s formula (dB/km, f in kHz): 0.11f²/(1+f²) + 44f²/(4100+f²) + 2.75×10⁻⁴f² + 0.003 (extrapolated above its 50 kHz fit; real absorption at 100 kHz is of the same order, about 35 dB/km). Noise: NL = NSL + 10 log10B. Receiving gain from the same aperture as the transmitter: DI = 20 log10(ka). Levels are rms-equivalent: peak-to-peak source level minus 9 dB.
Detection threshold
The detector sums echo-plus-noise power over the N receiving animals and fires above a threshold set by the false-alarm probability: for unit noise power the sum is Gamma(N,1), so Pfa = e−tΣk<Ntk/k!. Requiring the mean of echo+noise to reach the threshold gives DT = t/N − 1 (as a power ratio). The simulation draws Rician samples, so the measured 50% range differs slightly from this closed form.
Target strength
Fish with swim bladder: TS = 20 log10Lcm − 67.4; without bladder: 20 log10Lcm − 84.9 (textbook regressions measured near 38 kHz; values at 100 kHz differ). Rigid sphere with ka>>1: 10 log10(a²/4). Flat plate at normal incidence: 20 log10(Aeff/λ) with Aeff limited by the Fresnel zone λR in the near field, and a sinc² fall-off with tilt. Cylinder broadside: 10 log10(aL²/2λ) with the same tilt factor.
Beam pattern
The melon and skull act like a circular piston of effective radius a: D(θ) = 2J1(ka sinθ)/(ka sinθ), −3 dB at ka sinθ = 1.616. Transmit and receive both use it, so a target off-axis loses 2·20 log10|D|.
Range resolution
Δr = c/(2B): with B = 40 kHz two highlights closer than about 1.9 cm merge into one. The zoom inset shows this for each target’s highlights.
Pod and buried objects
Other dolphins hear the echo of a neighbour’s click (bistatic path Rtx+Rrx) and their power is added before the threshold, which lowers the needed SNR per animal. A buried canister adds interface and sediment loss, and below the critical grazing angle (about 25° for sand, from cwater/csand) sound hardly enters the sand – the penalty used here is qualitative.
Simplified
Isotropic homogeneous water (no refraction or sound-speed profile), no surface/bottom multipath or reverberation, no Doppler, a single frequency per click instead of the real broadband spectrum, peak-to-rms conversion for a tone, Gaussian noise. Species parameters are approximate. Wave speed on screen is slowed by the chosen time ratio.