# Vibration & Waves - Exam QuestionWatch

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#1
My question is basically how to do part ii) of this question even with the hint I am confused; the first part was simple; There's no dispersion & vp = vg = sqrt(hg) as required.

But how would I go about the second part, I can't even manage to get a diagram down.

The 1-D wave equation for shallow water waves (wavelength depth) is
[∂2η/∂x2] = [∂2η/∂t2]*1/hg

where η is the longitudinal displacement, h is the depth of the water and g is the
gravitational acceleration.

(i) Explain what is meant by dispersion and what are normal and anomalous dispersion.
Use the solution η = exp(i(ωt−kx))
to derive the dispersion relationship for
shallow water waves and hence show that the phase velocity is vp = sqrt(hg)

What dispersion do these waves exhibit and how would a pulse of such waves change
on propagation? [6 marks]

(ii) Treating the water as an incompressible fluid, show that for ψ h, the vertical
displacement ψ of the surface due to the wave is related to the longitudinal
displacement by
ψ ≈ −h*(∂η/∂x)

(Hint: consider the effect of the wave on the thickness and height of an infinitesimal
element δx and hence write down an expression for the volume of such an
element.)
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