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