Shallow water wave speed calculator

It evaluates the standard shallow-water relation c = sqrt(g h), reports the result in metres per second, and shows the substituted values used in the calculation. This approximation is useful for tides, tsunami estimates, coastal modelling, and introductory fluid mechanics when the wavelength is much greater than the water depth. Enter a positive depth and optionally replace standard Earth gravity with a local value.

Understand the shallow-water approximation

A water wave is called shallow-water or long not because the water must look shallow to a person, but because its wavelength is large compared with the water depth. In that regime, pressure is approximately hydrostatic through the water column and horizontal motion extends nearly from the surface to the bed. Those assumptions reduce the dispersive surface-wave relation to the compact expression c = sqrt(g h). Here c is phase speed in metres per second, g is gravitational acceleration in metres per second squared, and h is still-water depth in metres. The result does not depend directly on wave amplitude, period, or wavelength under this approximation. A common practical guideline is that shallow-water behaviour is strong when depth is less than roughly one twentieth of wavelength, although the precise tolerance depends on the accuracy required. The calculator therefore reports the mathematical result but cannot decide whether the regime is valid without wavelength information. Treat the output as a long-wave estimate, not as a universal speed for every ripple or swell at the entered depth.

Enter depth and gravity consistently

Provide water_depth as a positive depth measured vertically in metres. Use a representative still-water or mean depth for the reach being studied rather than instantaneous wave height. Gravity defaults to 9.80665 metres per second squared, the conventional standard value for Earth, but you may supply another positive value for a location-specific approximation, a laboratory model, or another planetary body. The units matter because the returned phase speed is in metres per second: entering centimetres as though they were metres increases the implied depth by a factor of one hundred and the speed by a factor of ten. The calculation multiplies gravity by depth and takes the positive square root. For example, increasing depth by a factor of four doubles wave speed, while doubling depth increases speed only by the square root of two. Zero and negative values are rejected because this physical model requires positive gravity and a positive fluid depth. Non-finite values and values outside the declared safety limits are also rejected instead of producing misleading JSON such as Infinity or NaN.

Interpret the result in real settings

The returned phase_speed_m_s describes how a constant phase of an ideal linear long wave travels through uniform water. In the non-dispersive shallow-water limit, group speed is also approximately equal to this phase speed, so the value is often used as a first estimate for the propagation of tidal signals and tsunami wave packets. Real coastlines require more care. Depth varies along a route, currents add or subtract from the ground-relative propagation speed, and friction, refraction, shoaling, breaking, finite amplitude, and complex bathymetry can all affect observed arrival and waveform. For a varying-depth path, calculate a local speed from each depth segment and sum each segment length divided by its local speed to estimate travel time; taking one average depth can hide important slow regions. For shorter waves, use the full linear dispersion relation instead of this formula. The response includes the inputs, a substituted formula string, and a validity reminder so results remain auditable when copied into a worksheet, report, teaching example, or automated coastal-analysis pipeline.