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Clarification on Shallow Water Wave Equation

  1. Jul 11, 2009 #1
    I know that we can find the speed of the wave in shallow water by:
    c^2 = gh
    but how do we derive it?
     
  2. jcsd
  3. Jul 11, 2009 #2

    physicsworks

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    Gold Member

    We derive this using Bernoulli theorem and continuity equation.
    In the reference frame which is moving along with the wave:
    Bernoulli theorem:
    [tex]\frac{V^2}{2}+gh=\frac{(V-\delta V)}{2}+g(h+a)[/tex]
    continuity equation:
    [tex]Vh=(V-\delta V)(h+a)[/tex]
    where [tex]V[/tex] is the speed of the wave, [tex]\delta V[/tex] is a drop of the speed in the water where its level grows from the normal [tex]h[/tex] to [tex]h+a[/tex]
    We suppose that [tex]h<\lambda[/tex] where [tex]\lambda[/tex] is a wavelength.
    Form the second equation one has
    [tex]h \delta V = a V[/tex] (*)
    ([tex]a \delta V[/tex] is very very small). Then from the fiirst we get [[tex](\delta V)^2[/tex] is also very small, so we ignore it]:
    [tex]V \delta V = ga[/tex] and with (*) one has
    [tex]V^2=gh[/tex]
     
    Last edited: Jul 11, 2009
  4. Jul 11, 2009 #3
    awesome,
    thank you
     
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