Deriving Group Velocity for Gravity Waves

[deejaybee11]

Homework Statement

In a lab experiment about Gravity Waves and Dispersion, one of the preliminary questions is:

Show that for gravity waves the group velocity is:

C$_{g}$ = $\frac{C_{p}}{2}$$\sqrt{1 + \frac{2kh}{sinh(2kh)}}$

Homework Equations

C$_{g}$ = $dω/dk$

ω = $\sqrt{gktanh(kh)}$

where ω is the angular frequency, and
C$_{p}$ = $\sqrt{\frac{gtanh(hk)}{k}}$

The Attempt at a Solution

By using the product rule and the chain rule I get

$d/dk$(gktanh(kh))$^{1/2}$ = $\frac{1}{2}$(gktanh(kh))$^{-1/2}$(gtanh(kh) + $gk/cosh^{2}(kh))$

But I have no idea where to go from here.
Any help would be greatly appreciated.

 

[industrygiant]

Not sure if it was a mistype, but the term with cosh^2 should have an h in the denominator.

Now try to see if there is anything you can take out of the brackets to make C_p in front of them.

 

[deejaybee11]

Yeah that was supposed to have an h in the denominator sorry. I have tried numerous ways of factoring something out but i can't seem to get it to work