Spreading of a pulse as it propagates in a dispersive medium

  1. Hello everyone!!
    Im studying the spreading of a pulse as it propagates in a dispersive medium, from a well known book. My problem arise when i have to solve an expression.

    Firstly i begin considering that a 1-dim pulse can be written as:


    u(x,t) = 1/2*1/√2∏* ∫A(k)*exp(ikx-iw(k)t) dk + cc (complex conjugate)


    and then i showed that A(k) can be express in terms of the initial values of the problem, taking into account that w(k)=w(-k) (isotropic medium):

    A(k) = 1/√2∏ ∫ exp(-ikx) * (u(x,0) + i/w(k) * du/dt (x,0)) dx

    I considered du/dt(x,0)=0 wich means that the problems involves 2 pulses with the same amplitud and velocity but oposite directions.
    So A(k) takes the form:

    A(k) = 1/√2∏ ∫ exp(-ikx) * u(x,0)

    Now i take a Gaussian modulated oscilattion as the initial shape of the pulse:

    u(x,0) = exp(-x^2/2L^2) cos(ko x)


    Then the book says that we can easily reach to the expression:

    A(k) = 1/√2∏ ∫ exp(-ikx) exp(-x^2/2L^2) cos (ko x) dx



    = L/2 (exp(-(L^2/2) (k-ko)^2) + exp(-(L^2/2) (k+ko)^2)

    How did he reach to this?? How can i solve this last integral???


    Then, with the expression of A(k) into u(x,t) arise other problem. How can i solve this other integral.


    Thank you very much for helping me!!
     
  2. jcsd
  3. that is a trick.you have to write cos(k0x) as Re(eik0x),you will get only exponentials then you will have to complete the square in powers of exponentials and use of a simple gaussian integral.
    0 e(-x2)dx=√∏/2
     
  4. Thank you so much! I could solve it!! It wasnt too hard after all :) thanks again.
     
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