Shallow-Water Wave Celerity Calculator

At a glance:When a water surface is disturbed (a stone, a gate, wind), a wave train propagates outward. If the wavelength is far greater than the depth — a shallow-water or long wave — the celerity (the wave's speed relative to the water) depends only on depth, very simply: c = √(g·y), where c is the wave speed (m/s), g the gravity (9.81 m/s²) and y the depth (m). This comes from shallow-water theory: the whole water column moves nearly together, the restoring force is gravity and the inertia is set by depth, so the result involves only gy and is independent of wavelength and (for small amplitudes) wave height — a key non-dispersive property. Intuition: deeper water → faster waves. Example: y=1 m → c=√(9.81×1)≈3.132 m/s; y=4 m → c≈6.264 m/s (4× depth → 2× speed, because of the square root). This celerity is crucial in open-channel hydraulics: it is the speed at which a small disturbance can travel upstream. If the flow velocity v<c, disturbances travel upstream and affect conditions above (subcritical flow, Fr<1); if v>c, disturbances are washed downstream and cannot go upstream (supercritical flow, Fr>1); v=c is critical (Fr=1). The Froude number Fr=v/c=v/√(gy) is exactly this velocity-to-wave-speed ratio. Uses: (1) basis of open-channel regime classification and Froude number; (2) estimating the speed of tidal bores, flood waves propagating upstream or into an estuary; (3) tsunami speed (wavelength ≫ depth, c=√(gy) — in 4000 m deep ocean c≈198 m/s≈715 km/h); (4) determining the influence range of gates and weirs.