Wave equation for water waves?

**TriKri** · October 14th 2010, 02:19 PM

Hi,

Has there to your knowledge been developed any wave equation for for water waves? Like for ocean waves at arbitrary water depths?

**CaptainBlack** · October 14th 2010, 10:26 PM

Originally Posted by TriKri

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

**TriKri** · October 15th 2010, 11:35 AM

According to a book I have, called Physics Handbook, the speed c of a wave can be given by the formulas

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

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.

**mr fantastic** · October 15th 2010, 12:36 PM

Originally Posted by TriKri

According to a book I have, called Physics Handbook, the speed c of a wave can be given by the formulas

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

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.

**Danny** · October 15th 2010, 01:23 PM

May I point you in this direction

"Linear and Nonlinear Waves" by G. B. Whitham.

**TriKri** · October 15th 2010, 01:33 PM

Does a linear wave in this case mean that the superposition principle is valid? And I will look the book up, thanks!

**TriKri** · October 16th 2010, 07:14 PM

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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