 Let's now consider another example of flow continuity. In this case, a phenomenon known as a hydraulic jump, which you can witness in rivers or streams, or you can witness in a laboratory setting where you can actually create what's called a flume and create the conditions that set up a hydraulic jump. If you haven't already, I'm going to encourage you to go look at some of the videos that are posted available online of hydraulic jumps in action so you can see the actual phenomena before considering this question. So in this particular picture, we're going to look. We're going to assume that there is a hydraulic jump that's been set up in some sort of rectangular flume. In other words, a long, thin channel. And in one part of the flume, you see this velocity that's coming in here. It's a high speed velocity. And then there's a point where there's enough energy that's been lost in the water and the system. And you have a zone of turbulence here. And you move into a zone where the water is flowing at a slower velocity. Now there's a one particular assumption that we can make here that comes from our idea of our flow continuity. That the flow, which we represent with q, that the flow in and the flow out are the same. That ultimately, this system is in equilibrium. q in equals q out. There are actual examples of hydraulic jumps where the jump is moving. Where the jump will either be moving up or moving down as the system is attempting to achieve equilibrium. But if the jump is stationary, and again, they're called a jump because we have two differences in the height here, if that jump stays in the same place, then we can assume equilibrium that the flow is in is equal to the flow out. And we know from our earlier discussion that we can relate the flow to these speeds, the velocities in and the velocities out here. Now I have actual values for the heights of real hydraulic jumps. Here is an example under one set of conditions where the height here is 50 centimeters. And the height after the jump is equal to 2.40 meters. So we have a difference of about just under five times the depth of water after the jump as we did before the jump. Well, let's go ahead and use our relationship q equals va. We can relate our speed to our flow if we recognize that q is equal to va. All right, let's plug that in here. So for our value in, which I believe we're calling 1, and our value out, which we're calling 2, let's see here we have v1 times area 1 is equal to v2 times area 2. Now in this case, we don't actually know what the area is. But because I told the story about this being in a rectangular wet raceway, we're going to think about if I'm looking down the raceway and the water's flowing at my face, we can think about this area here. Let's call this area 2 for this end. This area here is equal to our height, height 2, times some width that we haven't really described yet. But we'll just call it w. So there is our height 2 times some width. We're going to assume the same width applies on the other side over there with v1. So our areas, we can relate to be rectangular area equals width times height. So let's go ahead and replace those areas. v1 is equal to the width times height 1. And then that's equal to v2. And the area is equal to the width times height 2. If we make that assumption that we have consistent width over our cross-section, then you'll notice in this case divide both sides by the width and the width cancels out. And now the relationship between the velocities is equivalent to the relationship between the two heights. Well, if I have an initial velocity here, if I know my initial velocity for the incoming water, I can actually now calculate the velocity for the outgoing water by plugging in the necessary pieces. I'm going to do a little bit of algebra first here, though. I'm going to go ahead and move this v2 to the other side and divide both sides by h2. So I get the relationship v2 equals v1 h1 over h2. In other words, that the second velocity is equal to the first velocity times the ratio between those two heights. When I plug in the two heights, v2 is equal to v1 50 centimeters over 2.4 meters, which I will convert to centimeters, 240 centimeters. And that's equal to, notice the centimeters cancels out. And that value is equal to 0.208. v2 is equal to 0.208 times v1. In other words, again, it's about a fifth of the speed. And if I recognize that v1 is 10 meters per second and plug that in, I get that v2 is equal to 2.08 meters per second. So we have a very different, much slower flow rate. And that flow rate corresponds to that jump in height. So we have the same amount of flow in and flow out. And that is an interesting phenomenon examined using the flow continuity.