Hi Mark Schlutow ,

I'll have a try despite the fact that being unathletic and accustomed to waves with momentum (electromagnetic ones) renders me the least qualified person to answer:

I feel the answer to your main question is yes. From a local point of view, at the front of the wave the normal to the water / air boundary has a vertical component as well as a component in forward direction. The same is true for the normals of the isobars below the surface.

So, if the surfer manages to balance the gravitational force and the vertical component of the force of the water displaced by his surf board in such a way that he remains at a constant height then the force of the displaced water can have a forward component, too, accelerating the surfer in forward direction.

In other words, a surfer descending relative to the water with the same speed as that of the water moving upwards (relative to earth) neither gains nor loses potential energy but is nevertheless accelerated forward by the tilted isobars of the wave.

To be more exact, the acceleration in the forward direction comes to a close if the forward speed of the surfer equals the speed of the wave propagation, of course; afterwards, the forward force balances only the flow resistance. However, by employing the properties of fluid dynamics, acceleration in the horizontal direction perpendicular to the wave propagation is always possible (and common practice, as seen in videos).

Since both energy and momentum are conserved, the wave should lose what the surfer gains. If I understand McIntyre's chapter 4 correctly, the whole deep water system, with the wave at the surface and the backflow below, has zero momentum. So if the surfer causes a decrease of "local" momentum ("pseudomomentum" according to McIntyre?) near the surface, and the momentum of the backflow remains unchanged then the water system has gained momentum in the backward direction, and the momentum of the complete system including the surfer remains zero.

The wave may have zero net linear momentum, but it has angular momentum, and this is also reduced when the wave does work on the surfer, which it does not only in starting the surfer moving, when the surfer gains linear momentum, but also in maintaining his/her height in the same position on the wave. The force on the underside of the surfboard (about 1000 Newtons) is equal to the rate of change of momentum in the water.

There must be an equivalent transfer of momentum between the wave and the bulk water.

I expect that while this is happening the circular motion of the wave and zero linear momentum are disturbed for a while.

On the opposite side, waves are generated by transfer of linear momentum and energy from, for example a speedboat. The linear momentum must then be transferred into the bulk water, which exerts a reverse force resulting in the circular motion of the water at the surface and change from linear momentum of the boat to circular momentum in the wave. The rate of change of momentum is equal to the retarding force on the boat, which is considerable before it reaches planing speed.

When the wave hits shallow water the backward part of the circular momentum (at the bottom of the wave) is transferred to the earth, leaving forward momentum which we see in the waves washing up the beach.

Dear Dr Mark: "More speed equals more possibilities, as most of surfing’s maneuvers require good momentum. At a beginner level, it is already possible to feel how shifting weight over the front foot provides more speed. Intermediate and advanced surfers do more than just shift weight over their feet, they use specific parts of the wave to create their own acceleration". For a deeper insight have a look at the following useful link:

https://barefootsurftravel.com/livemore-magazine/generate-speed-surfboard

Regards--

The wave described in the link above has already changed from being a wave with no linear momentum. It is shown with a crest and a break starting to form. This is usually because of interaction with the sea-floor which removes backward momentum from the bottom of the wave.

The surfer gains momentum both when they are going down the wave and when they are going up, just in different directions. I think the pumping described also allows the surfer to generate speed from their own muscle effort, as a skateboarder can on the flat by weaving. Even this takes linear momentum out of the wave, though. Pumping doesn't work unless there is something to pump against.

Hi Mark, I would say that due to the locality of the forcing, it is more illustrative to look at the problem in a more detail, i.e. to relax the wave formalism and look at the parcels, their trajectories and interaction with the surf. My conclusion would be that the surfer gets momentum in two ways, from going down and from the forcing of adjacent water parcels. Or if he travels quicker than the parcels, at least the friction is much smaller than when considering steady water. Anyway, I am curious how you will conclude this! Cheers, Petr.

Thanks for the many interesting replies. I really like the perspective by Malcolm White. I was thinking about the boat that excites a wave. In the balanced state when the boat maintains its velocity against the wave drag, I believe that the force by the propeller does not induce a linear momentum that is transported away by the wave. It might induce angular momentum though. Rather the work done on the water results in an increase of potential energy of the boat. And it is this energy which is carried away in line with the findings of McIntyre. In conclusion, a boat actually drives upwards all the time; if this makes sense. For the surfer then it is the other way around.

Hi Mark, I think you can find some hints about this issue by googling Stokes drift. Momentum density is not defined for surface waves (or internal waves in a fluid), but momentum flux (product of horizontal velocity perturbations indueced by waves) is nonzero for waves. This momentum flux actually can drift materials, as logs or trashes on ocean eventually reach the shore.