Are river waves actually waves?

There are a few ways to look at this problem, so let's get to it. At the base level, river waves are waves, but a different kind than what we usually surf. For this article, we’ll only be talking about rivers far from the ocean - we’ll address tidal bores another day.

Ocean waves, at least the surface gravity waves that we surf, are progressive waves generated by wind that propagate across a distance. Relative to the ocean floor, these waves move forward, traveling hundreds or thousands of miles before reaching our shores.

River waves stand still over a specific spot. These standing waves form when water moves over a variable bottom, like a large boulder in the river. Relative to the riverbed, the wave doesn't move, but it's certainly moving with respect to the water.

How is surfing always crowded? Photo by Moon Kim on Unsplash

This brings us to reference frames. In the reference frame of the river water, which is moving over the riverbed, the wave is most definitely propagating. Think about rafting - in the raft, you're moving towards the standing wave downriver. But looking at it differently, the wave is moving towards you at the exact same speed. Physics doesn't care which perspective you choose to believe, the fact is that you and the wave are getting closer.

So yes, in this picture the river wave is actually a wave, just one that moves in a reference frame we are not used to thinking in. You can picture it as a wave that is moving upstream at exactly the same speed of the current, so it just sits still.

An interesting depiction of a standing wave, where it is the sum of two progressive waves moving in opposite directions. By Lucas Vieira - Own work, Public Domain.

Another way to look at the "wave" that river surfers ride is that it isn't a wave at all, but rather a hydraulic jump - a fundamental transition in fluid mechanics where fast-moving shallow water abruptly shifts to slower, deeper flow.

Let's use the LA River as our example - that equal parts concrete, water, sludge, and trash monstrosity whose name we'll whisper only when necessary. With its perfectly smooth bottom and constant width, it's an unfortunate but perfect laboratory for fluid dynamics. When fast-moving water encounters a change in elevation or an obstacle, the flow has to adjust.

An example of a frictional hydraulic jump, where there doesn’t actually have to be a bump. The river waves that people surf are not usually this variety, but it does show the continuity of water that forces the wave face to form. By Williamborg - Own work, Public Domain.

Upstream, the water flows slowly in what we call a subcritical flow - think deep and slow. Over the bump, it gets more shallow and speeds up in what's called supercritical flow. And finally, once the water is past the bump, it has to transition back to subcritical flow - deeper and slower. This sudden transition creates our surfable wave. The water at the surface actually moves back upstream while the bulk of the flow continues downstream. This hydraulic jump stays fixed in place as long as the upstream flow remains constant.

This creates a rippable surface for the mountain folk, holding steady even as countless gallons rush past. Different riverbed formations will change the physics here slightly, but this description works for our idealized case where the river’s bottom before and after the bump is level. At least the LA River gave us something useful besides some cool car chase scenes in Fast and Furious 41.

So is it right to call this a wave? The physics works out either way - wave or hydraulic jump, the water's still flowing and the surfers are still surfing. The decision is left as an exercise to the reader.

Further Reading:

Hydraulic Jump

Standing Wave