How Do You Calculate the Rate of Change of Wave Frequency with Water Depth?

[cunhasb]

Could anyone give a hand in solving this problem?

In shallow ocean water, the frequency of the wave action (number of wave crests passing a given point per second) is given by f= [(g/(2PiL))tanh(2Pih/L)]^1/2, where g is the acceleration due to gravity, L is the distance between wave crests, and h is the depth of the water. Get an equation for the rate of change of f with respect to h assuming g and L are constants.

Well this is what I've gotten so far...I'm not sure if it is right or not...

df/dh=1/2((g/2PiL) tanh(2Pih/L))^-1/2 * ((g/L^2)sech^2(2Pih/L))
=((g/L^2)sech^2(2Pih/L))/2√((g/2PiL)tanh(2Pih/L))

well seems that the final answer is

(gPi/2L^3)^1/2 csch^1/2(2Pih/L)sech^3/2(2Pih/L)

So, I have no idea how this answer was derived, looking at it seems that they integrated it (sech^3/2)... is that so?

It would be of great help you anyone could give a hand on this one...


Thank you so much... guys...

 

[FredGarvin]

I am very rusty on my derivatives. But I will say that it doesn't make any sense to integrate something while looking for the derivative of a function. It appears to be a nasty little hyperbolic function...You may want to repost this in the homework or math sections for a bit more help.

 

[Astronuc]

cunhasb - you seem to be on the right track. Make sure constants are correct, and remember

tanh x = sinh x/ cosh x = sech x/csch x, csch x = 1/sinh x, sech x = 1/cosh x

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let f = $(\frac{g}{2\pi L} tanh (\frac{2\pi h}{L})^{1/2}$

or f = $(a g)^{1/2}$, so

df/dh = f ' = 1/2 a^1/2 g^-1/2 g '

now if g = tanh (bh) , then g ' = b sech² (bh) where b = $\frac{2\pi}{L}$

so

f ' = 1/2 $(\frac{g}{2\pi L})^{1/2} (tanh (\frac{2\pi}{L}))^{-1/2}\,sech^2 (\frac{2\pi h}{L})\,\,\frac{2\pi}{L}$

 

[cunhasb]

thank you guys... Sorry not to thank you before... but I was away for a few days...

Thank you so much...