QuestionQuestion

Transcribed TextTranscribed Text

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 de􏰀ned 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 de􏰀nition of the Green's function G for ut = uxxx, 􏰀nd its Fourier transform Gˆ, and hence show that G can be written as 1􏰅∞ 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? 1

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Applied Differential Equations
    $30.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Differential Equations Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats