I'm still working on my 'physics of bowl carving' video (name TBC) but in the meantime I made these diagrams which might be of interest (also following on from the discussion that emerged from the 'blood in urine thread' elsewhere on here!).
Excluding such things as hips, waterfalls, circular ends (which are different to cornered ends) and looking solely at the standard 'cornered' bowl you commonly see, there are two distinct ways of getting speed by pumping. You can use one or the other, or you can combine them for maximum speed. I'm not sure whether there's already an established name for these but I'm going to call them "perpendicular pumping" and "parallel pumping".
A lot of skaters just learn this stuff intuitively, and know it by feel, and to a certain extent that is the only way to learn it. However if you have an analytical mind you might find it helps to understand the physics behind what you're aiming for.
1) Perpendicular Pumping:
This is pumping up/down using the curve that you would see in a 'side elevation' of the transition. This is often called the transition radius and is the same as the transition radius of a half-pipe. Perpendicular pumping is exactly the same technique you would use on a half-pipe, and for this reason it's often how beginners to bowl carving start out.
There are 2 "pumps". Each pump consists of a compress & expand phase - i.e. you crouch then straighten. Crouching itself adds no speed, but allows you to straighten (expand) at the correct moment, which speeds you up.
You can see perpendicular pumping in these two diagrams - note that the actual corner isn't being used at all and you might as well be on a half-pipe:
Note that the 'tuck' always comes in anticipation of an upcoming change in curvature of the surface (either flat-to-curve, or curve-to-flat), and the 'expand' occurs *across* that change in curvature. i.e. across the change from flat to curve, or from curve to flat.
The reason this isn't very fast is that at the peak of each curve, you slow down to turn around and change direction.
These can be combined, to navigate the corners of a bowl, but it's being done as three separate perpendicular pumps, like this:
However, you can get a lot more speed by using...
2) Parallel Pumping
This is using the *other* curve in the bowl, in a different dimension to the "transition curve". The parallel curve is what you see if you follow the line of the coping in an overhead view. And therefore if you are riding a straight-wall section of the bowl, with straight coping, you can't use parallel pumping on that section - only the corners (although you can prepare for the upcoming corner by compressing on the straight bit leading up to the corner). By using parallel pumping you can extract speed from corners without needing to change height.
Here is a line which shows *only* parallel pumping:
Even though you're using a different 'dimension' of the bowl, the 'tuck' still always comes in anticipation of an upcoming change in curvature of the surface (either flat-to-curve, or curve-to-flat), and the 'expand' occurs *across* that change in curvature. i.e. across the change from flat to curve, or from curve to flat. The difference is now we're calling the start of this run (which is riding *along& the transition wall) 'flat' because you are riding along the transition parallel to the coping, so from your frame of reference it's essentially flat, and the 'curve' is now the corner.
However, it gets fun if you combine the two, so you are making use of both the transition radius (the perpendicular curve) and the corner radius (the parallel curve). This is where you get real speed combined with flexible lines to criss-cross around the bowl at will! This final example has 4 pumps shown in the line that, in sequence, are: Perpendicular (up the wall), Parallel (1st corner), Parallel (2nd corner), Perpendicular (down the wall).
Hope this helps some of you more nerdy skaters out there
And I guess this serves as another sneak peek as to what will be in my video (if I ever finish it)