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Conservation of Mass Fluids Wave

  1. Jun 6, 2017 #1

    joshmccraney

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    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?
     
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  3. Jun 7, 2017 #2

    Charles Link

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    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.
     
  4. Jun 7, 2017 #3

    joshmccraney

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    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)##.
     
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