1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Green function for forced harmonic oscillator

  1. Oct 27, 2014 #1
    1. The problem statement, all variables and given/known data
    The problem requires to solve the integration to find ## G(t) ## after ##G(\omega)## is found via Fourier transform. We have [tex] G(\omega)= \frac{1}{2\pi}\frac{1}{\omega _{0}^2 - \omega ^2}[/tex]

    2. Relevant equations
    As mentioned previously, the question asks to find ##G(t)##

    3. The attempt at a solution
    It is obvious that calculus of residues is required. To account for causality (## G(t<0)=0 ##), the poles at ## \omega=\pm \omega_{0} ## are shifted to the lower half plane by ## i\epsilon ## and integrated along the contour in the lower half plane. Then,
    [tex] G(t)=-\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{e^{-i\omega t}}{(\omega +i\epsilon)^2 -\omega _{0} ^2}d\omega [/tex]
    After calculating the residues at ## \omega =\pm \omega _{0} - i\epsilon ##, I found
    [tex] G(t)=\frac{i}{2\omega_{0}}\sin \omega_{0}t [/tex]

    Is my answer correct?
     
  2. jcsd
  3. Oct 27, 2014 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The presence of ##i## suggests it's not. G(t) should be real, shouldn't it?

    The sine makes sense. The system is at rest and then you impart an impulse, causing it to oscillate. It has to be sine because it starts from x=0.
     
  4. Oct 28, 2014 #3

    RUber

    User Avatar
    Homework Helper

    It looks like the i and the 2 might both get absorbed into the ##\sin \omega_0 t =\frac{e^{i \omega_0 t }-e^{-i \omega_0 t }}{2i}## term.
     
  5. Oct 29, 2014 #4
    Yes, sorry! I forgot to put the "i" in the exponential form of sine. That should clear the problem.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Green function for forced harmonic oscillator
Loading...