River waves · building, running, surfing

How a standing wave forms in a river

An ocean wave travels. A river wave stays where it is – the water rushes through it, the shape stands still. Whoever surfs it does not move from the spot and can ride the wave for as long as their legs hold out. This text explains why the water does that, which number decides it and what that means for a small river like the Rems.

Fast water meets slow water

When water flows over a step or a ramp, it becomes fast and shallow. Below, the river water backs up, slow and deep. Where the fast, shallow water meets the slow, deep water, it cannot simply carry on flowing: it has to brake abruptly and gain height. It does so in a wall of water that breaks on the spot. Hydraulic engineers call this a hydraulic jump. Every standing wave, from the Eisbach to the Fuchslochwelle, is such a jump – more or less cleanly shaped.

The Froude number

Whether a jump forms and what it looks like depends on a single parameter: the Froude number. It relates the flow velocity to the speed at which a small disturbance can spread across the water surface – and that depends on the water depth.

Fr = v / √(g · h) (v flow velocity, h water depth, g gravitational acceleration)

If Fr is less than 1, the flow is "subcritical": calm, deep, like a lowland river. If Fr is greater than 1, the flow is "supercritical": fast, shallow, like at the foot of a weir. A hydraulic jump is the transition from supercritical to subcritical. The hydraulic engineering literature distinguishes by the Froude number upstream of the jump:

Froude number	Name	Appearance
1.0–1.7	undular jump	smooth standing waves without breaking
1.7–2.5	weak jump	small roller at the surface, water behind it smooth
2.5–4.5	oscillating jump	pulsating jet, irregular waves
4.5–9	steady jump	strong roller fixed in place
> 9	strong jump	rough, lots of air, large energy dissipation

(Classification after Ven Te Chow, Open-Channel Hydraulics, 1959, and the U.S. Bureau of Reclamation.)

Only a narrow window is surfable. Froude numbers that are too low give smooth waves without push; too high, a roller that holds a surfer down instead of carrying them. Which range suits a surf wave depends on the design, the depth and the desired wave height; as a rule of thumb, a Froude number of about 2.3 and above is quoted for a powerful, breaking surf wave.

Four quantities, one balance

Research at the University of Innsbruck (Unit of Hydraulic Engineering, Prof. Markus Aufleger, in cooperation with the wave builder Dreamwave) describes what makes a good surf wave: a height between roughly 50 and 150 centimetres, not too flat and not too steep, an evenly smooth surface and enough width. And it describes the difficulty: the discharge has to match the height difference between headwater and tailwater very precisely. If either of the two quantities changes, the wave tips over – it goes flat, or it turns into a dangerous recirculation.

So the four adjusting screws are:

Discharge – how much water arrives per second (in a natural river this fluctuates daily),
Head – the height difference the water loses at the weir or the ramp,
Width – the narrower the outlet, the faster the water,
Tailwater – how deep the water backs up below, because that determines where the jump stands.

Natural waves like the one on the Eisbach or in Thun work because these quantities happen to fit together – and they disappear when the water level rises or falls too far. Built waves replace chance with technology.

How to help it along

Impounding. A weir raises the headwater. The greater the height difference to the tailwater, the faster the water at the foot – and the higher the Froude number. A constriction impounds too: if the river is made narrower than its discharge needs, the water rises upstream – the acceleration always comes from a difference in height.

Constricting. If the outlet becomes narrower, the same amount of water has to pass through faster. Canal waves like those in Munich and Salzburg live on this.

Ramps and adjustability. A ramp shapes the jet, and an adjustable ramp adapts the shape to the discharge. The Fuchslochwelle in Nuremberg works with an adjustable ramp technology in modules (up to 8 metres wide), the Leinewelle in Hanover with a hydraulically controlled system. Both run without pumps, on the energy of the river alone – and both can be taken out of the water completely during floods.

And on the Rems?

At Winterbach, the Rems flows subcritically on the vast majority of days, with a Froude number well below 1 – roughly estimated at around 0.2. So a wave needs help – with impoundment, a constriction or a ramp. The outlet structure of the flood retention basin (locally the "Remswehr") narrows the river and can impound it; at normal water levels, however, the Rems flows straight through. To what extent head for a wave can be gained from it is for the feasibility study to establish. Depending on the variant, the wave would run directly at the structure or at the stepping stones. Which combination of discharge, head and width produces a surfable wave on how many days a year is the core question of the ongoing feasibility assessment. → Site

Frequently asked questions

Why does a river wave stay in one place?

Because it is not a travelling packet of water but a hydraulic jump: fast, shallow water meets slower, deeper water and has to gain height abruptly. The water flows through, the shape stays put as long as discharge and head remain the same.

What is the Froude number and why does it matter?

It describes the ratio of flow velocity to water depth. If it is above 1, the flow is "supercritical", and the jump forms at the transition to "subcritical" flow. Just above 1 the wave stays smooth and without push; at high values it becomes a roller; only a narrow window in between is surfable, and it depends on the design and the water depth.

Can a wave be built at any weir?

No. It needs enough discharge, sufficient head at the drop and a tailwater of suitable depth. Whether a site offers that is only shown by a hydraulic calculation or a model test – which is why every project starts with the feasibility study.

How much water does a surfable wave need?

That depends on width and drop height. The Almkanal wave in Salzburg runs on around 5 m³/s over 4.5 metres of width; Nuremberg quotes values between 4.5 and 10 m³/s for 8 metres; Thun needs over 100 m³/s because the wave forms without any structure in a large river.