View Single Post
Aguss
Aguss is offline
#1
Feb13-13, 09:55 PM
P: 7
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!!
Phys.Org News Partner Physics news on Phys.org
Information storage for the next generation of plastic computers
Scientists capture ultrafast snapshots of light-driven superconductivity
Progress in the fight against quantum dissipation