How this instrument works
The Froude number compares how fast something moves through a fluid with a free surface to the speed at which a shallow gravity wave can travel there. Formally Fr = v ⁄ √(gL): velocity divided by √(gL), the speed of a long surface wave over a depth or length scale L. A dimensionless ratio, it carries no units — it says nothing about how fast anything actually is, only how that speed stands against the fluid's own capacity to carry disturbances away.
The square root of gL is not an arbitrary pairing — it falls straight out of shallow-water wave physics, where gravity is the restoring force pulling a disturbed surface back level, and that restoring force sets how quickly a ripple can outrun the flow that created it. When v is smaller than √(gL), the wave wins the race and can push upstream; the water ahead of an obstacle effectively gets advance notice and adjusts smoothly. When v exceeds √(gL), the flow outruns its own waves, nothing ahead is warned, and disturbances pile up into a steep front instead of spreading out — the same logic that gives compressible gas flow its Mach number.
L is the part people get wrong, because it is not a fixed property of the fluid but a choice that depends on what is being modelled: a ship's waterline length, a channel's hydraulic depth, a culvert's diameter, a pier's width. Swap in the wrong length scale and the number still comes out but no longer means what you think it means. The formula also assumes one dominant length and leaves viscosity out entirely, so it says nothing about friction or turbulence — matching Froude number alone reproduces wave patterns at model scale, not the drag a full-size hull or channel would actually feel.
- Enter the flow speed in the Velocity field (v) — the speed of the current, hull, or object relative to the surrounding fluid.
- Enter the Characteristic length field (L) — pick the length that actually sets the local wave speed: hull waterline length, channel hydraulic depth, or obstacle width.
- Read the result in the Froude number field (fr).
- Compare fr to 1: below is subcritical, where waves can travel upstream; above is supercritical, where they can't; exactly 1 is critical flow.
Worked example — a 2 m/s current past a 1 m obstacle
Take a current of 2 m/s flowing past a submerged piling 1 m across, so v = 2 and L = 1. The formula gives Fr = 2 ⁄ √(9.80665 × 1) = 2 ⁄ 3.132 = 0.638659913562, which the instrument displays as about 0.639. Because that sits comfortably below 1, the flow is subcritical: the piling's wake spreads out as ordinary ripples in every direction, and a disturbance dropped just downstream can still work its way back upstream against the current.
A naval architect reads that same 0.639 as a design number, not a curiosity: it predicts how much of a hull's resistance at that speed-to-length ratio comes from making waves rather than friction. Push the same 1 m length past roughly 3.13 m/s and Fr crosses 1 — the flow turns supercritical, the piling's wake steepens into a standing wave instead of ripples, and wave-making drag rises sharply, which is exactly the wall ship designers push against when choosing a hull's length for a target speed.
Questions
What counts as the characteristic length L?
Whatever length actually sets the local wave speed for the situation at hand: a ship's waterline length, a channel's hydraulic depth (cross-sectional area divided by surface width), a culvert's diameter, or an obstacle's width. Pick the wrong one and the ratio still computes but no longer compares the speed to the wave speed it's supposed to.
Why does the formula use the square root of g times L?
Because √(gL) is the speed at which a long gravity wave travels over a depth or length scale L — a standard result from shallow-water wave theory. Comparing velocity to that wave speed is literally what the ratio measures, in the same way Mach number compares speed to the local speed of sound.
What happens physically at Fr = 1?
That's critical flow: the object or current moves exactly as fast as the waves it generates. Disturbances can no longer outrun the flow or fall behind it, so they pile up at the source instead of radiating away, and small changes in depth or geometry produce disproportionately large changes downstream. Engineers generally try to avoid holding a channel right at this point.
Does Froude number stay constant along a river or channel?
No. Because v and L both change with local depth, width, and slope, Fr is a local quantity rather than a fixed property of the whole channel. A river can sit subcritical in a wide, slow pool and cross into supercritical flow a few metres downstream, over a steep drop or a weir crest, before returning to subcritical in a hydraulic jump.
How is this different from Reynolds number?
Reynolds number weighs inertial force against viscous force and governs whether flow is laminar or turbulent. Froude number weighs inertial force against gravitational force and governs free-surface wave behaviour. A scale-model ship test that matches Froude number gets the wave pattern right but not the friction, since matching both at once is essentially impossible at reduced scale.