I have a question which I do not fully understand about conservation of momentum of the shallow-water equation in conservative form. We are given a velocity $u(x,t)$ and the height $h(x,t) = d + l(x,t)$ where $l(x,t)$ is the water level and $d$ the depth of the water. Then the conservative equations of the shallow-water equations are:

$$\frac{\partial h}{\partial t} + \frac{\partial (hu)}{\partial x} = 0 $$ $$\frac{\partial (hu)}{\partial t} + \frac{\partial}{\partial x} \Big(hu^2 + \frac{1}{2}gh^2 -\nu h \frac{\partial u}{\partial x}\Big) = 0. $$

Next, I have to prove that by taking the integral over the 1D domain $(0, L)$ that in the first equation a quantity is conserved (and explain what this quantity represents). Secondly, for the second equation I have to prove conservation of momentum. Thus far I have found this:

$$\int_0^L \Big(\frac{\partial h}{\partial t} + \frac{\partial (hu)}{\partial x} \Big) dx= 0 \rightarrow \int^L_0\frac{\partial l}{\partial t} dx + d\cdot \big[u\big]^L_0 + [u\cdot l - u\cdot l]^L_0 = 0 $$ and $$\int_0^L \Big(\frac{\partial (hu)}{\partial t} + \frac{\partial}{\partial x} \Big(hu^2 + \frac{1}{2}gh^2 -\nu h \frac{\partial u}{\partial x}\Big) \Big) dx = 0 \rightarrow \int_0^L \frac{\partial hu}{\partial t}dx + \Big[ hu^2 + \frac{1}{2}gh^2 - \nu h \frac{\partial u}{\partial x}\Big]^L_0= 0. $$

From this how can I see that in the first equation a quantity is conserved/ see that momentum is conserved?


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