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The science
Sound speed and refraction. Sound in seawater travels at c ≈ 1450-1550 m/s, rising with temperature, pressure (depth) and salinity. A ray with angle θ to the horizontal obeys Snell's law, cosθ / c(z) = const, in differential form dθ/ds = -(cosθ/c) dc/dz: rays bend toward lower speed. Where c has a minimum (the SOFAR channel axis, near 1000 m in the deep ocean) rays are trapped and oscillate around it without touching the surface or seafloor. The page integrates the ray equations with a 4th-order Runge-Kutta scheme and the Munk canonical profile c(z) = 1500 [1 + ε(η - 1 + e-η)], η = 2(z - 1300 m)/1300 m, ε = 0.00737.
Spreading. In open water intensity falls as 1/R² (spherical): TL = 20 log10 R dB re 1 m. Once the sound meets the surface and seafloor (or a channel) it spreads only horizontally: TL = 10 log10 R (cylindrical). The ray model gets this from geometry: the power carried by each ray (a fan of rays covers the launch angles) is counted where it crosses range gates, I = ΣPray / (2π r Δz), then TL = -10 log10(I/I1m). This also reproduces shadow zones and convergence zones.
Absorption. Thorp's empirical formula gives α(f) = 0.11 f²/(1+f²) + 44 f²/(4100+f²) + 2.75×10-4 f² + 0.003 dB/km, f in kHz (0.069 dB/km at 1 kHz, 1.19 dB/km at 10 kHz). That is why low-frequency whale calls travel across ocean basins while high-frequency clicks fade within kilometres.
Detection (passive sonar equation). A call is heard when SL - TL - NL ≥ DT: source level in dB re 1 µPa at 1 m, transmission loss, band noise level and a detection threshold. The ambient spectrum level is modelled illustratively as NS(f) = A - 18 log10(f/100 Hz) dB re 1 µPa²/Hz with A set by shipping, band level NL = NS + 10 log10(bandwidth). More shipping noise raises NL and shrinks the radius within which whales can hear each other; with cylindrical spreading the radius scales as 10-ΔNL/10 so +10 dB cuts the radius tenfold and the communication area a hundredfold.
Honest simplifications. Range-independent profile, flat seafloor with a fixed reflection loss, flat sea surface (no scattering), no bottom penetration, no sound-speed gradient in the water column besides the chosen profile. Ray (geometrical) acoustics is only an approximation when the wavelength is comparable with the channel features: at 20 Hz the wavelength is about 75 m, so convergence-zone details are smoothed in reality (the page warns when λ > depth/10). Source levels, bandwidths and call frequencies of the whale presets are typical literature-order values, not measurements of any individual; the noise model is an illustrative fit, not the full Wenz curves; detection thresholds depend on the listener. The whale, ship and buoy are drawn far larger than to scale, and the vertical axis is exaggerated (shown on screen). Whales are shown only as sound sources; nothing here predicts how an animal is affected.