# Group Velocity of shallow water Stokes wave derivation seems wrong

• K29

#### K29

I have a simple question but I'm putting down the whole derivation as it is relevant. There is a point that I don't understand, or seems wrong to me. This is a derivation of Group Velocity followed by simplifying(approximating it) for long wavelength waves in shallow water. This appears in a pack of notes that I have. I feel like it is wrong. But if it is wrong, I don't see how to get to the right answer.

Group Velocity of Stokes wave in general is derived as such:

(1): $V_{G}=\frac{d\omega(k)}{dk}$ where k is wavenumber, $\omega$ is angular frequency

(2): $\omega ^{2}=gk \tanh(kh_{0})$. This is the dispersion relation for stokes waves. g is gravity, $h_{0}$ is depth.

Differentiating both sides of 2 with respect to k:
$2\omega \frac{d\omega}{dk}=g \tanh(kh_{0})+\frac{gkh_{0}}{\cosh ^{2}(kh_{0})}$

Dividing through by $2\omega$:
$\frac{d\omega}{dk}=\frac{g}{2\omega}\tanh(kh_{0})+\frac{gkh_{0}}{\cosh ^{2}(kh_{0})}$

Substituting in (2) rearranged:
$\frac{d\omega}{dk}=\frac{g}{2\omega}\frac{\omega ^2}{gk}+\frac{gkh_{0}}{\cosh ^{2}(kh_{0})}$

$=\frac{\omega}{2k}+\frac{gkh_{0}}{2\omega \cosh ^{2}(kh_{0})}$

$=\frac{\omega}{2k}[1+\frac{gh_{0}k^{2}}{\omega ^2 \cosh ^{2}(kh_{0})}]$

Using phase velocity $c=\frac{w}{k}$ substituting (2):
$=\frac{c}{2}[1+\frac{gh_{0}k^{2}}{gk \tanh(kh_{0} \cosh ^{2}(kh_{0})}]$

$=\frac{c}{2}[1+\frac{gh_{0}k^{2}}{gk \sinh(kh_{0} \cosh(kh_{0})}]$

Using $2\sinh(\theta) \cosh(\theta)=\sinh(2\theta)$
(3):$V_{G}=\frac{d\omega}{dk}=\frac{c}{2}[1+\frac{2kh_{0}}{\sinh(2kh_{0})}]$

I am fine with all the steps up to there. I do not see any problems there.

Now we consider group velocity $V_{G}$ with long wavelength, shallow water:
$kh_{0}=2\pi\frac{h_{0}}{\lambda}<<1$ since wave number k = 2pi/wavelength

We use the fact that
$\sinh(x)=x+O(x^{3})$ as x→0
(This step seems fine to me)

Using the above 2 facts in (3) we have:
4a.)$V_{G}=\frac{c}{2}[1+\frac{2kh_{0}}{2kh_{0}}(1+O(kh_{0})^{2}]$

This is where I have a problem. I see that a $2kh_{0}$ has been pulled out of $O(kh_{0}^2)$. That is fine. But unless I am missing something the above equation only pops out if $\sinh$ was in the numerator of (3). But it is in the denominator. So IN MY OPINION the above equation should be:
4.b)$\frac{c}{2}[1+\frac{2kh_{0}}{2kh_{0}(1+O(kh_{0})^{2}}]$

However as far as I can see 4.a) yields the correct result:
5)$V_{G}=c[1+O(kh_{0})^2]\approx c$

and 4b) does not.

Are the notes incorrect? If so, how does one arrive at 5.) If not, what am I missing?