Why doesn't linear wave theory produce phase velocities that agree with each other?

Question

I'm not sure I understand the dispersion relationship for water waves. According to Wikipedia, the wavelength of ocean wave at arbitrary depth is given by:

\begin{equation} \left(\frac{2\pi}{T}\right)^2 = \frac{2\pi g}{\lambda}\times\tanh \left(\frac{2 \pi h}{\lambda}\right) \end{equation}

And the phase velocity is given by:

\begin{equation} C_p=\sqrt{\frac{g}{2\pi/\lambda}\tanh(2\pi/\lambda \times h)} \end{equation}

I believe the phase velocity should also equal:

\begin{equation} C_p=\frac{\lambda}{T} \end{equation}

Using the first equation, I can solve for the wavelength when T=14 and h=40 and get:

\begin{equation} \lambda=239.2848 \end{equation}

But, then why do I get different phase velocities when using the two equations above?

\begin{equation} C_p=\sqrt{\frac{g}{2\pi/239.2848}\tanh(2\pi/239.2848 \times h)}=10.92263 \end{equation}

vs:

\begin{equation} C_p=\frac{\lambda}{T}=\frac{239.2848}{14}=17.09177 \end{equation}

Why don't these two phase velocities agree (10.9 vs. 17.09)?

Answer 1

The mistake is that you used the dispersion relation instead of the phase velocity.

The phase velocity is actually: \begin{equation} C_p=\sqrt{\frac{g}{k}\tanh(k h)} \end{equation}