The Froude number calculator above takes a velocity and a length scale and returns the dimensionless ratio that decides how a free surface behaves. It is the single most important number in open-channel hydraulics and in ship resistance, and although it is the same quantity in both fields, the length that goes into it is different and the interpretation is different.
Arb Digital builds free physics calculators that state their boundary. The open channel flow calculator derives the Froude number as one output of Manning's equation, starting from channel shape, slope and roughness. This page starts from the velocity and depth you already have, so it works for a measured flow, a laboratory result or a hull, none of which need a Manning calculation at all.
What This Froude Number Calculator Does
The Froude number is the flow velocity divided by the speed of a shallow-water gravity wave at the relevant length scale. It compares inertial forces against gravitational ones, and its square is that force ratio directly. When it equals one, the flow is moving at exactly the speed at which its own surface disturbances travel, and that coincidence has dramatic consequences.
In channel mode the tool uses hydraulic depth — flow area divided by top width — which is the correct length scale for a channel of any cross-section, and it classifies the result as subcritical, critical or supercritical. In hull mode it uses waterline length and reports the speed-length interpretation instead, including where the classical hull-speed limit falls.
You can enter the hydraulic depth directly or let the tool compute it from a flow area and a top width. For a rectangular channel the two are the same thing, but for a trapezoidal or part-full circular section they are not, and using the maximum depth instead of the hydraulic depth is one of the most common errors in this calculation.
How to Use It
- Choose the mode first. It changes which length scale the page expects and how the result is described, though the arithmetic is identical.
- Enter the velocity in whatever unit you have. Metres per second, feet per second, knots and kilometres per hour are all converted internally.
- Enter the length scale. Hydraulic depth for a channel, waterline length for a hull. For a hull, length overall will give an optimistic answer.
- Use the area and top width boxes for an irregular channel. Enter both and the hydraulic depth is derived from them.
- Read the regime, not just the number. A result near one is the interesting case, because that is where the surface becomes unstable and small changes in energy produce large changes in depth.
The Formula: How the Froude Number Is Calculated
The Froude number is Fr = v ÷ √(gL), where v is the mean velocity, g is gravitational acceleration and L is the characteristic length. For open-channel flow L is the hydraulic depth, defined as the flow area divided by the width of the free surface. The US Bureau of Reclamation's Water Measurement Manual, in its section on hydraulic mean depth and hydraulic radius, defines that depth as the value which, multiplied by the top water surface width, equals the irregular section area, and notes it is the length used in Froude number and energy relationships.
The quantity √(gL) is the celerity of a long gravity wave in water of that depth — the speed at which a shallow disturbance propagates. So the Froude number is literally a Mach number for surface waves, and the analogy runs deep: critical flow behaves like sonic flow, a hydraulic jump behaves like a shock wave, and supercritical flow cannot transmit information upstream.
The reason a value near one matters so much comes out of specific energy, the energy of the flow measured above the channel bed. The same manual's section on energy balance in open channel flow defines specific energy as depth plus velocity head. That curve has a minimum, and the minimum occurs exactly at critical flow. Near the minimum the curve is almost flat, so a tiny change in energy produces a large change in depth, which is why near-critical flow is visibly unstable.
Work the defaults by hand. A channel with a mean velocity of 2 m/s and a hydraulic depth of 1 m: √(9.80665 × 1) = 3.1315 m/s, so Fr = 2 ÷ 3.1315 = 0.6387. That is subcritical, and Fr² = 0.4079 says gravitational forces are dominating inertial ones by about two and a half to one. For the hull preset, 7 knots is 3.6011 m/s and a 10 m waterline gives √(98.0665) = 9.9029 m/s, so Fr = 0.3636 — just under the classical hull-speed value.
Subcritical, Critical and Supercritical
Below a Froude number of one the flow is subcritical: deep, slow and tranquil. Surface waves travel faster than the water, so a disturbance downstream propagates upstream and the flow is controlled from downstream. Most rivers and nearly all drainage channels are in this state, and it is stable and forgiving.
Above one the flow is supercritical: shallow, fast, and unable to carry information upstream. Control comes from upstream instead. A steep spillway or a lined chute runs in this state, and any obstacle produces standing waves rather than a backwater. If the flow has to return to subcritical, it does so through a hydraulic jump — a short, violent, turbulent transition that sheds a great deal of energy as heat and noise.
Right at one, the specific energy curve is at its minimum and the surface becomes wavy and unstable. Channels are deliberately designed away from that condition. Where a jump is wanted, it is engineered into a stilling basin so that it happens where the structure can take it. Our hydraulic jump calculator handles the conjugate depths and energy loss across that transition.
Why a Hull Has a Speed Limit
A vessel moving through water makes a wave system whose wavelength grows with the square of speed. At a Froude number of about 0.4, the wavelength of the bow wave equals the waterline length, and the ship settles into the trough of its own wave. Pushing further means climbing that hill, and the resistance rises steeply.
That is the origin of the traditional hull speed rule, usually written as 1.34 times the square root of the waterline length in feet, giving knots. It is not a hard barrier — a hull with enough power can push through it, and a light planing hull escapes it entirely by lifting out of the water — but it is a very real efficiency cliff for a displacement hull.
It is also why the Froude number is the governing similarity parameter in towing-tank testing. A model tested at the same Froude number as the full-size ship makes a geometrically similar wave pattern, so wave-making resistance scales properly. Viscous drag does not scale the same way, which is exactly why the Reynolds number has to be handled separately and why model results are corrected rather than simply multiplied up. Our Reynolds number calculator covers that second similarity requirement.
Choosing the Right Length Scale
Getting the length wrong is the commonest way to produce a Froude number that means nothing. For a channel, use hydraulic depth: area over top width. For a rectangular channel that equals the water depth, which is why textbooks often present them as the same thing, but for a trapezoidal ditch the top width exceeds the bed width and the hydraulic depth is smaller than the maximum depth. For a part-full pipe the difference is larger still.
Note that hydraulic depth is not the same as hydraulic radius. Hydraulic radius is area divided by wetted perimeter, and it belongs in Manning's equation and in friction calculations, not in the Froude number. The two are easy to confuse because both are an area divided by a length. Our hydraulic radius calculator deals with that second quantity.
For a vessel, use the waterline length. Length overall includes overhangs that make no wave, and using it flatters the Froude number and understates the speed the hull is really working at. For a partly submerged structure such as a bridge pier, the appropriate length depends on what phenomenon you are studying, and the choice needs to be stated explicitly whenever a Froude number is quoted.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using maximum depth instead of hydraulic depth — in any channel that is not rectangular the two differ, and the Froude number is defined with area over top width.
- Confusing hydraulic depth with hydraulic radius — the radius is area over wetted perimeter and belongs in friction calculations, not in this one.
- Using length overall for a vessel — overhangs make no waves. The waterline length is what sets the wave system and therefore the Froude number.
- Treating a value near one as just another number — near-critical flow is genuinely unstable, and a design that lands there will behave unpredictably in service.
- Scaling a model on Froude number alone — wave-making scales with Froude, viscous drag with Reynolds, and no single model speed satisfies both at once.
Related Free Tools From Arb Digital
To get the velocity and depth from a channel's shape, slope and roughness, use the open channel flow calculator. For the transition from supercritical back to subcritical, use the hydraulic jump calculator, and for the geometry term in friction calculations the hydraulic radius calculator. The second similarity parameter is covered by the Reynolds number calculator, discharge arithmetic by the flow rate calculator, and speed conversions by the velocity calculator. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is the ratio of flow velocity to the speed of a gravity wave at the relevant length scale, calculated as velocity divided by the square root of gravity times length. Its square is the ratio of inertial to gravitational forces.
The flow is subcritical: deep, slow and tranquil. Surface waves travel faster than the water, so disturbances propagate upstream and the flow is controlled from downstream. Most rivers and drainage channels are in this state.
The flow is supercritical: shallow, fast, and unable to send information upstream. Control comes from upstream, obstacles create standing waves, and returning to subcritical requires a hydraulic jump.
The hydraulic depth, which is the flow area divided by the width of the free surface. For a rectangular channel that equals the water depth, but for trapezoidal or part-full circular sections it is smaller than the maximum depth.
No. Hydraulic depth is area divided by top width and belongs in the Froude number. Hydraulic radius is area divided by wetted perimeter and belongs in Manning's equation and friction calculations.
Using waterline length as the length scale, it describes the wave system a hull makes. At about 0.4 the bow wavelength matches the hull length and resistance climbs steeply, which is the classical hull speed limit.
Because matching it between model and full scale reproduces the wave pattern correctly, so wave-making resistance scales. Viscous drag follows the Reynolds number instead, which is why model results must be corrected rather than scaled directly.
Because the specific energy curve is nearly flat at its minimum, which is exactly where critical flow sits. Tiny changes in energy then cause large changes in depth, and the water surface becomes unstable and wavy.
This tool is provided for educational and study use. It gives an ideal one-dimensional Froude number from the velocity and length you supply, and does not account for velocity distribution, channel curvature, air entrainment or hull form, so treat its output as a physics result rather than a design calculation.
