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Math Help - Wave equation for water waves?

  1. #1
    Senior Member TriKri's Avatar
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    Wave equation for water waves?

    Hi,

    Has there to your knowledge been developed any wave equation for for water waves? Like for ocean waves at arbitrary water depths?
    Last edited by TriKri; October 14th 2010 at 06:06 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by TriKri View Post
    Hi,

    Has there to your knowledge been developed any wave equation for for water waves? Like for ocean waves at arbitrary water depths?
    Have you tried the links from the Wikipedia page?

    CB
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  3. #3
    Senior Member TriKri's Avatar
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    According to a book I have, called Physics Handbook, the speed c of a wave can be given by the formulas

    \begin{array}{ll}<br /> <br />
c=\sqrt{gd} & \text{for } d\lesssim\lambda/12 \\ \\<br /> <br />
c=\sqrt{\displaystyle\frac{g\lambda}{2\pi}+\displa  ystyle\frac{2\pi S}{\rho\lambda}} & \text{for } d\gtrsim\lambda/4<br /> <br />
\end{matrix}

    but at wikipedia, you will only find this formula

    c=\displaystyle\sqrt{\frac{g \lambda}{2\pi} \tanh \left(\frac{2\pi d}{\lambda}\right)}

    Seems like there are some different opinions about what the formula should look like.
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  4. #4
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    Quote Originally Posted by TriKri View Post
    According to a book I have, called Physics Handbook, the speed c of a wave can be given by the formulas

    \begin{array}{ll}<br /> <br />
c=\sqrt{gd} & \text{for } d\lesssim\lambda/12 \\ \\<br /> <br />
c=\sqrt{\displaystyle\frac{g\lambda}{2\pi}+\displa  ystyle\frac{2\pi S}{\rho\lambda}} & \text{for } d\gtrsim\lambda/4<br /> <br />
\end{matrix}

    but at wikipedia, you will only find this formula

    c=\displaystyle\sqrt{\frac{g \lambda}{2\pi} \tanh \left(\frac{2\pi d}{\lambda}\right)}

    Seems like there are some different opinions about what the formula should look like.
    I don't have the time to check but it's possible that your book is using a linear approximation to the tanh function. I suggest you do a literature review of the subject if your interest is sufficiently high.
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  5. #5
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    May I point you in this direction

    "Linear and Nonlinear Waves" by G. B. Whitham.
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  6. #6
    Senior Member TriKri's Avatar
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    Does a linear wave in this case mean that the superposition principle is valid? And I will look the book up, thanks!
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  7. #7
    Senior Member TriKri's Avatar
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    I wonder how correct it is to put a tanh there in the fomula; either they just chose this function because it is sigmoid (a necessarity for making c an even function of lambda, still giving it the desired properties) and is relativelly easy to handle analytically, or they have actually derived this formula from other theory and got tanh there. What do you think?
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