View Single Post
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
Engineers develop new sensor to detect tiny individual nanoparticles
Tiny particles have big potential in debate over nuclear proliferation
Ray tracing and beyond