QuestionQuestion

Transcribed TextTranscribed Text

1. Consider ut = uxxx and let the initial conditions consist of two narrow wavepackets. The first packet starts at x1 = 0 and has wavenumber k1 = 2 and the second starts at x2 = 10 and has wavenumber k2 = 1. When and where will their paths cross? 2. Let f(x) be a smooth function with Fourier transform ˆf(k). What is the Fourier transform of the scaled function f(x/σ), where σ > 0 is a real scaling parameter? 3. Let f(x) be a smooth function and consider the solution u(x, t) = 1 2π Z +∞ −∞ exp(i[kx − ω(k)t]) ˆu0(k)dk for some dispersion function ω(k) to the initial-value problem in which uˆ0(k) = F T{exp(ik0x)f(x/σ)} with k0σ  1. We did this in class with f(x) being a Gaussian, so we had a wavepacket. Replicate the argument in class for this general f(x) and derive that u(x, t) ≈ exp(i[k0x − ω0t])f([x − cgt]/σ) where the constants ω0 and cg are to be related to the dispersion function ω(k). What is the interpretation of previous expression? 4. Consider a dispersion relation with two real roots for the frequency that are equal and opposite, i.e. ω1,2 = ±ω(k) for some function ω(k). Show that the solution to the initial-value problem with u(x, 0) = A(x) and ut(x, 0) = B(x) can be written as u(x, t) = 1 2π Z +∞ −∞ exp(ikx) " Aˆ(k) cos ω(k)t + Bˆ(k) ω(k) sin ω(k)t # dk.

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