Cusps in the evolution of closed strings

1. Nov 14, 2013

rbwang1225

1. The problem statement, all variables and given/known data
This is problem 7.7 in Zwiebach's book, 2ed ed.
In (b) he want us to show that near the cusp, $y\sim x^{2/3}.$
In (d), Check that the period of the motion of the closed string is $\sigma_1/4c$. How many cusps are formed during a period?

2. Relevant equations
(b) $\overrightarrow X(t_0,\sigma) = \overrightarrow X_0+\frac{1}{2}(\sigma-\sigma_0)^2\overrightarrow T+\frac{1}{3!}(\sigma-\sigma_0)^3\overrightarrow R \\=\frac{1}{2}(\sigma-\sigma_0)^2 T\hat y+\frac{1}{3!}(\sigma-\sigma_0)^3 R(\cos\theta\hat x+\sin\theta \hat y)$, where $|\overrightarrow T|=T \mbox{ and } |\overrightarrow R|=R$.
(d)$\overrightarrow X(t,\sigma) =\frac{1}{2}[\overrightarrow F(u)+\overrightarrow G(v)] =\frac{\sigma_1}{4\pi}(\sin \frac{2\pi u}{\sigma_1}+\frac{1}{2}\sin\frac{4\pi v}{\sigma_1},-\cos\frac{2\pi u}{\sigma_1},-\frac{1}{2}\cos\frac{4\pi u}{\sigma_1}) =\overrightarrow X(t+T,\sigma)$

3. The attempt at a solution
(b) $y=\frac{1}{2}(\sigma-\sigma_0)^2[T+\frac{1}{3}(\sigma-\sigma_0)R\sin\theta]$
$x=\frac{1}{3!}(\sigma-\sigma_0)^3R\cos\theta$
But, I don't see very clearly why $y\sim x^{2/3}$.
(d) I observe that $\overrightarrow F(u) \mbox{ and } \overrightarrow G(v)$ has periods $\sigma_1/c \mbox{ and }\sigma_1/2c$, respectively. But I don't know why the period of $\overrightarrow X$ is smaller.

Any advice would be very appreciated.

Last edited: Nov 14, 2013
2. Nov 14, 2013

George Jones

Staff Emeritus
These aren't quite correct. Try finding them again. Don't factor out any powers of $\sigma-\sigma_0$.

3. Nov 14, 2013

rbwang1225

I corrected it.
Or, I still have something lost?
Thanks a lot!

4. Nov 14, 2013

George Jones

Staff Emeritus
It looks like you're still missing something in the expression for y.

5. Nov 14, 2013

rbwang1225

Oh, sorry for the mistake!
Now, can I conclude that since $y\sim (\sigma-\sigma_0)^2 \mbox{ and } x\sim (\sigma-\sigma_0)^3$ so $y\sim x^{2/3}$?
I think it's yes, because the factors, T, and x-component of R are just numbers, they don't affect the main feature of the line.
Thank you very much!

6. Nov 14, 2013

George Jones

Staff Emeritus
Roughly, yes.

Maybe better is to solve the $x$ equation for $(\sigma-\sigma_0)$, and substitute this into the $y$ equation, so as to obtain $y$ as a function of $x$. The expression for $y$ should contain two terms that involve $x$. Argue that, near a cusp, one of these terms dominates.

7. Nov 14, 2013

rbwang1225

OK, then I know how to get the answer.
But now I have a question in (d). I don't know how to derive the period of the function of the sum of 2 periodic functions. I thought it was the smaller one between the two, but it seems like something was missing.