Group Velocity of shallow water Stokes wave derivation seems wrong

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K29
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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): [itex]V_{G}=\frac{d\omega(k)}{dk}[/itex] where k is wavenumber, [itex]\omega[/itex] is angular frequency

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

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

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

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


[itex]=\frac{\omega}{2k}+\frac{gkh_{0}}{2\omega \cosh ^{2}(kh_{0})}[/itex]

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

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

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

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

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

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

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

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

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

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

and 4b) does not.

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

Answers and Replies

  • #2
938
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In (3), first expand sinh(x) and then 1/(1+x^2+...) using geometric series
 
Last edited:
  • #3
K29
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In (3), first expand sinh(x) and then 1/(1+x^2+...) using geometric series

Thanks a lot .That helped
 

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