h1 is the height in front of the front, in the direction of the front's travel h2 is the height behind the front

The shaded area is the area of water that has moved past point x1 in time $\Delta{}t$

Non-shaded areas show where the water is. Shaded areas show how 'where the water is' has changed.

The point here is that the water between x1 and x3 has changed shape and location. It now occupies $h_{2}v\Delta{}t$ less area on the left, and $(h_{2}-h_{1})u\Delta{}t$ more area on the right. But it's the same amount of water, so the two should be equal.

h1 is the height in front of the front, in the direction of the front's travel

h2 is the height behind the front

The shaded area is the area of water that has moved past point x1 in time $\Delta{}t$

Non-shaded areas show where the water is. Shaded areas show how 'where the water is' has changed.

The point here is that the water between x1 and x3 has changed shape and location. It now occupies $h_{2}v\Delta{}t$ less area on the left, and $(h_{2}-h_{1})u\Delta{}t$ more area on the right. But it's the same amount of water, so the two should be equal.