Spreading of a pulse as it propagates in a dispersive medium

[Aguss]

Hello everyone!
I am 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 which 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!

 

[andrien]

that is a trick.you have to write cos(k₀x) as Re(e^ik₀x),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

 

[Aguss]

Thank you so much! I could solve it! It wasnt too hard after all :) thanks again.