Derivate group velocity equation v_g = c/2 (1 + 2kh / sinh 2kh) ; where c = w/k

[doliv]

Hello,
I have an equation from water wave dispersion relation:

w² = gk*tanh kh ; where tanh is the hyperbolical tangent

And I need to prove that dw/dk results in the group velocity equation, which is:

v_g = c/2 (1 + 2kh / sinh 2kh) ; where c = w/k

I did some calculus, but got stuck in an expression that I didn't figure out how to manipulate to get to final result:

2w dw/dk = [g sinh kh / cosh kh] + [(gk cosh² kh - sinh² kh) / cosh² kh]


There's anything I did wrong and you can help me?

Tks

 

[sinx]

Hello,
I have an equation from water wave dispersion relation:

w² = gk*tanh kh ; where tanh is the hyperbolical tangent

And I need to prove that dw/dk results in the group velocity equation, which is:

v_g = c/2 (1 + 2kh / sinh 2kh) ; where c = w/k

I did some calculus, but got stuck in an expression that I didn't figure out how to manipulate to get to final result:

2w dw/dk = [g sinh kh / cosh kh] + [(gk cosh² kh - sinh² kh) / cosh² kh]


There's anything I did wrong and you can help me?

Tks

w=[gk*tanh kh]^1/2
now find dw/dk