Conservation of Mass Fluids Wave

[member 428835]

Hi PF! Suppose we have a water wave with mean depth $H$ with disturbance $\zeta$ above/below $H$ propagating through a channel of thickness $b$. The book parenthetically remarks that the continuity equation becomes $$\partial_t(b(H+\zeta))+\partial_x(bHu)=0.$$ However, when I try deriving this I write $$\partial_t(b(H+\zeta) \Delta x )=ub(H+\zeta)|_x-b(H+\zeta)u|_{x+\Delta x}\implies\\
\partial_t(b(H+\zeta))+\partial_x(b(H+\zeta)u)=0$$ which is not quite what they have. Any idea what I'm doing wrong?

 

[Charles Link]

What you wrote appears correct to me. This is just a standard continuity equation $\nabla \cdot \vec{J}+\frac{\partial{\rho}}{\partial{t}}=0$ if I'm not mistaken. I don't think you can just drop the $\zeta$ in the second term.

 

[member 428835]

What you wrote appears correct to me. This is just a standard continuity equation $\nabla \cdot \vec{J}+\frac{\partial{\rho}}{\partial{t}}=0$ if I'm not mistaken. I don't think you can just drop the $\zeta$ in the second term.

The book does, though they do say the channel is very long compared to height. Still, I agree with what you wrote, though I doubt the book made a mistake dropping the $\zeta$ since further work requires their version. Edit: I figured it out: I believe $\zeta \sim A$ where $A$ is wave amplitude. Also, $u\sim A/P$ where $P$ is wave period. Then $uA\sim O(A^2/P)$ and is very small relative to $O(AL/P)$.