1. Find and sketch the Fourier transforms of (a) f1(x) = cos 3x
(b) f2(x) = sin 5x
1 −L<x<L forsomelengthL>0 (c) f3(x) = 0 otherwise
2. The Green's function G(x,x0,t) for the heat equation ut = kuxx on the real line is dened for t ≥ 0 by
Gt =kGxx, lim G=0, and G(x,x0,0)=δ(x−x0) |x|→∞
Find the equivalent equations for its Fourier transform Gˆ(k,x0,t) and solve them. Com- pare with the known expression for G(x, x0, t). You may use that
−x2 ˆ √ k2σ2 f(x)=exp 2σ2 ⇔f(k)=σ 2πexp − 2 .
3. Write down the denition of the Green's function G for ut = uxxx, nd its Fourier transform Gˆ, and hence show that G can be written as
exp i[k(x − x0) − k3t]dk.
Do not attempt to do this integral, which cannot be computed in closed form.
G(x, x0, t) = 2π
4. It is shown in uid dynamics that ocean waves in deep water that are propagating
in the positive x-direction are governed by the dispersion relation ω(k) = gk
where k > 0 is the wavenumber and g is gravity. Sketch ω(k), compute the phase velocity cp and the group velocity cg, and indicate them in the sketch. What is the wavelength of a wave with period of ten seconds and what is its group velocity?
5. More generally, water waves in a uid of depth H propagating in the positive x- direction are governed the dispersion relation
ω(k) = gk tanh(kH)
for k > 0. Sketch the last expression and show that it reduces to previously mentioned expression in the deep-water limit where kH ≫ 1. What is the limiting form of the last equation in the opposite, shallow-water limit kH ≪ 1 ? Are the waves in this limit dispersive? Consider a Tsunami that has a wavelength of 100 kilometers in an ocean of depth 4 kilometres, does it behave like a shallow-water wave or a deep-water wave? What is its group velocity?
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