The two issues I run into with waterfalls are:
Jumping in front of the waterfall vs. jumping into the base of the waterfall vs. letting the waterfall take you along with it. The first would be the equivalent to any other jump into water, such as from a bridge, and @Lookingforabus outlines the physics very well above. Would the other two options not mess with the velocities and such?
They would, yes. Due to even simple fluid dynamics being such a hard problem, you're not going to be able to get precise numbers, but we can determine the direction of effect, even if we can't say the magnitude. One of the Millennium Prize Problems in math (million dollar payday in involving 7 unsolved math questions that have been unsolved for decades or centuries) is about fluid dynamics, and relates to the simpler matter of determining smoothness and energy bounds in fluids (Navier–Stokes existence and smoothness problem), which is vastly more simple than figuring out precisely what happens to a body in freefall hitting turbulent water.
In the case of hitting the base of the waterfall, one would be jumping into turbulent water, which would have a lower surface tension (to some degree), and the surface tension is what makes the water behave like a hard surface on impact. How much of difference (if any) that would make I can't say, and I'm not sure anyone else could, either. The effect of impacting a turbulent, incompressible fluid is actually an incredibly complex problem, and is effected by a multitude of geometric factors, such as the depth of the water, the structures underneath the surface, the density and impact forces of something falling into it and so on. No human studies have been perfomed to answer the question. ;) The bigger problem would probably be entering the flow of water in the waterfall before impact, which would significantly decrease the rate of the fall, and result in a slower impact. As you note, the water falling from the waterfall would have the effect of forcing one under the surface of the water, increasing the chances of drowning, but that water also moves downstream, where there is no such pressure, so it seems like a possibility that one would end up swept down stream and floating to the surface, perhaps before drowning. My suspicion is that the magnitude of the effect on surface tension is relatively small, so there wouldn't be much change in impact forces, and this idea would be about as fatal as just impacting the water outside the base of the waterfall, (as long as you don't enter the waterfall before impact) but it's not possible to say for sure.
The second scenario, being swept over the falls by the water is definitely the least fatal option. The water going over the falls is basically falling with only the assistance of gravity, and has a slower speed than a person in feefall owing to resistance forces from the water below it. A human falling with the flow of water would experience more resistance than in air, and have a significantly lower impact speed, in addition to the effects of hitting turbulent water with a lower surface tension. More importantly, however, in this case, the surface of the water would have very little, if any impact force, because you are essentially in a continuous flow of water from top to bottom. Death would only occur by drowning in this scenario. And in fact, the lower freefall speed and drastically reduced impact force in a flow of water is exactly what people relied on when going over Niagra Falls in a barrel was a popular stunt/performance in the early 1900's. (Though one of the people who died in the attempt had his ballast material break the barrel and he was swept under the water, where he drowned, so it's not completely safe, either.)
If it does, is it the falling that would kill you, or would the water pressure keeping you under lead to drowning with broken bones from the fall? Is there a chance to land directly onto rocks, and would that change the math at all? Niagara Falls is the most common site that I've come across, so it's easiest to source data from, but would those statistics be consistent with other waterfalls given different variables apart from height?
Rocks are a mixed bag, relative to a flat, hard surface, but don't seem to make much difference, based on studies done on jumpers who commit suicide having the same rates of fatality for landing on a smooth surface as an uneven one. On the one hand, there's the possibility that an uneven rock surface channels the impact into non-vital parts of the body (think, for example of landing on a rock with the hip, and having that reduce the impact force to your head or chest), while on the other hand, it's also possible to have a rock channel the impact force to a vital area of the body. And there's the consideration that blood loss from a high speed, blunt force impact is usually fatal anyway, and the effect of broken/shattered bones releasing potassium into the bloodstream, possibly causing a fatal heart attack. (Yes, breaking too many bones can kill someone by triggering a heart attack.) From sufficient height, it just doesn't seem to make a difference in fatality rates.
I've also read on here that it's very difficult to change your orientation in the air to land on a specific part of your body, and I would assume that it becomes even more difficult/impossible when engulfed in water on the way down.
It takes practice, but not much. I was on the dive team in prep school, and by the end of the first practice (a couple hours), we'd all figured out how to orient our bodies properly when we hit the water, which we had to do before they'd let us onto the 10 meter diving platforms. In short falls, it's almost entirely about your position and velocity when you jump, as there's only a tiny amount of time to do anything when you're in the air. (About one second off a 10 meter platform, and only 4 seconds or so from ~85 meters). Of course, none of us are perfect, and even with practice, most of us ended up doing a belly flop or back flop off the higher platforms. Painful, but it was motivating to be precise and disciplined about future dives.
That said, it is possible to change your angle of impact with limb movements and twisting your core, which is how skydivers do it. (And elite divers as well, like at the Olympics - if you watch their dives, you can see them pull their limbs in to flip faster, and extend them out to slow their angualr momentum, or move their hips, cores and shoulders to do twists, all in the span of a second or so before hitting the water.) But with only a few seconds, it's hard to do without thousands of hours of elite training, so best to jump/dive properly than try to sort it in a matter of seconds, midair.
In a relatively fast flow of water, it would absolutely be more difficult, and probably impossible to reliably change orientation, owing to the pressure of the water flow, but as there's little to no impact force while in a flow of water, that wouldn't be an issue anyway.
The social aspect, which may not be a problem depending on your personal views. Waterfalls are popular scenic locations and I would feel uncomfortable traumatizing a family just trying to enjoy a nice vacation. Niagara Falls has dozens/hundreds of people around at most times and it seems to be a very public way to go. I argued in the jumping megathread that watching somebody slip into the abyss of mist is not as damaging as witnessing the actual impact, but still not very cool. Not all waterfalls are going to have this level of traffic, obviously, so your mileage may vary.
Indeed. If one were to go this way, one would want to be a decent distance away from others, without anyone looking at the jumper, to be considerate to others.[/list]
