What Are Waves in a Dispersive Medium?

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Homework Statement

Homework Equations

The Attempt at a Solution

i have no clue about the second part..

 

[unscientific]

bumpp

 

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Am I suppose to superimpose the following waves:

exp i(kx - wt) + exp i((k+Δk)x - w't)

whereby the other wave has a different wavelength such that k becomes k+Δk and w becomes w' (with k+Δk) ?

So I found that at h >> 1/k or h >> λ/2∏

v_g = (1/2) v_g

And at h << 1/k or h << λ/2∏

v_g ≈ v_pI first assume that h >> 1/k or h >> λ/(2∏): But how do i prove this? I can't seem to solve the tanh function to find the value of k..

I got:
0.16≈k*tanh(1000k)

If i assume k≈0.16 since tanh(160) ≈ 1..
i get v_p = 7.9m/s
Time taken = 35 hours..

 

[unscientific]

bumpp

 

[unscientific]

bumpp anyone?