Surfers or surfboards redirect the forces of planing.
Planing is often associated with high-speed boating, but in fact, any partially submerged relatively rigid object in a flow will experience planing.
The phenomena has been explored extensively in water-craft hull design. Savistsky was one of the first to fully explore and approach the subject empirically and, to a lesser extent, analytically - his work remains the basis for much of what followed in the field. Below is his simplified illustration illustrating the forces at play during planing.
Here, Savistsky, in cross-sectional view, illustrates a simple plank moving horizontally over some water (or other liquid surface). (But because such considerations are relative, you could equivalently view the above illustration as water flowing past a stationary plank.) If we consider the case in which the plank is moving, the means by which it is moving is assumed to be via some external power source, say an inboard or outboard engine, or perhaps the plank is being pulled along, or maybe its means of propulsion is from sail.
The key point in the diagram however is that the forces of planing generated by motion through the fluid, or by the fluid flowing past the the object, are up and back (back as in the direction of the flow relative to the plank, generally in the direction which you would associate with the term ‘drag’. An engineer would say that the resulting force of planing will have both ‘lift’ and ‘drag’ component.)
But this diagram needs to be altered for surfing. Surfers generally don’t have a outboard engine (though admittedly now kites and sails are often used.) The fix however is simple, just rotate the diagram.
Surfboards derive their means of propulsion by getting in the way of flow.
The inclined surface is now the face of the wave, the water moving both upward, as well as forward. And now it can be seen that the planing forces have a component in the direction of that forward motion. (The diagram here doesn’t include the forces that arise from the interaction with the forward movement of water, which is actually decelerating, see “The Decelerating Wave-form”.) But more importantly, there is no need to assume the existence of an external power source like an outboard engine. The flow is the result of the upward acceleration of water of the shoaling wave – and the propulsion comes from the surfboard surfaces redirecting the forces of planing. And it is this redirection of the forces of planing that makes surfboards go, at least under most conditions. In fact, it is what separates the sport from other more ‘gravity reliant’ sports, like skiing. Skiers don’t move by getting in the way of snow, however surfboards move by getting in the way of flow.
How these forces are redirected is up to the surfer, and will depend on how he chooses to present the bottom surfaces of his surfboard to the flow. In particular, its quite possible to redirect them so that there is a component of force in the transverse direction. (To visualize this, think of your flatten hand subject to a jet of water. What direction would you hand tend to move if you angled it relative to the on-coming jet?)
Gravity plays a crucial role in the dynamics.
A surfer’s connection to his board is tenuous, unless he’s strapped in, or has grabbed a rail. Under most circumstances, if orientated correctly, gravity will keep him on his surfboard. This isn’t to say, that at other times, other forces aren’t contributing.
However, gravity also plays additional roles. It is used as a counter force against the forces of planing, keeping the surfer in the upward flow on the wave face. In addition, it is also used as a means of acquiring additional kinetic energy, by sliding down the wave-face, much like a sleigh sliding down a hill. And though gravity plays a crucial ever-present role in the dynamics, for the most part, gravity itself isn’t what allows a surfer to move transversely across the face. Those forces come from the redirecting the forces of planing (as described above.)
Modern surfing styles tend to utilize gravity as an additional source of kinetic energy (dropping then climbing then dropping etc.) far more than the more classical styles in which ‘trimming’ (moving transversely) was emphasized. But even in the modern style, trim is utilized extensively, especially when the surfer is able to get himself in the “pocket” or “barreled”. Climbing and dropping isn’t really an option is these situations, all of his propulsion comes from the redirection of the forces of planing.