1. Feb 23, 2010

Math Jeans

1. The problem statement, all variables and given/known data

So I already finished most of this problem, but I'm having trouble figuring out the very last part second part.

The last part of the problem is:
"Finally, take one additional term in the Taylor series expression of $$\omega(k)$$ and show that $$\sigma$$ is now replaced by a complex quantity. Find the expression of the 1/e width of the packet as a function of time for this case and show that the packet moves with the same group velocity as before but spreads in width as it moves. Illustrate this result with a sketch."

I found the complex quantity, and it is the second part I'm having issues with.

2. Relevant equations

The 1/e width is such that at $$k = k_0 \pm \frac{1}{\sqrt{\sigma}}$$, the amplitude distribution is 1/e of its maximum value $$A(k_0)$$.
The 1/e width is defined as $$\frac{2}{\sqrt{\sigma}}$$.

The complex expression for $$\sigma$$ is $$\sigma - \frac{1}{2}i\omega''_0 t$$

3. The attempt at a solution

Well, the implication of this is that:
$$\frac{2}{\sqrt{\sigma - \frac{1}{2}i\omega''_0 t}}$$

Since this is the 1/e width, I had thought that it should be increasing in order to imply spreading, however, when I graph the real component of this equation with respect to time, I always get a decreasing trajectory for t>0. Would this not imply that it is contracting?

Well, I then went ahead and graphed my wave equation, and I did get some spreading (in that the oscillations remained visible for a larger width, however, the width of each curve was the same, but this is fine due to non-variable frequency).

How do I get my expression for $$\sigma$$ to correctly demonstrate the spreading effect?

thanks,
Jeans

2. Feb 23, 2010

Gokul43201

Staff Emeritus
Please also type out the entire question, so the reader has the correct context.

3. Feb 23, 2010

Math Jeans

Actually, you're timing is impeccable because I just figured it out.

The 1/e width refers to width in terms of wave number, so if spreading is in terms of the x-coordinate, then it will become larger as opposed to smaller.