Vibration & Waves - Exam Question Watch
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
(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
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