Frequency Spectrum of a Vibrating String

[Dustinsfl]

Given $\sin\left[\frac{\pi ct}{L}\left(n + \frac{1}{2}\right)\right]$.

The period is $\tau = \frac{2L}{c\left(n + \frac{1}{2}\right)}$ so the frequency is $\frac{1}{\tau}$, correct?

 

[MarkFL]

Yes:

$\displaystyle f=\frac{1}{\tau}$

 

[Dustinsfl]

Yes:

$\displaystyle f=\frac{1}{\tau}$

Is the period I obtained correct? Is that the yes or is the yes it is the reciprocal?

 

[Ackbach]

If you have a function $\sin(kt)$, find the period $\tau$ by setting $k\tau=2\pi$.

 

[MarkFL]

Sorry for being vague. :)

I agree with the period you found and with the relationship between frequency and period that you stated.

$\displaystyle \tau=\frac{2\pi}{\omega}$

In your case $\displaystyle \omega=\frac{\pi e}{L}\left(n+\frac{1}{2} \right)$

And so:

$\displaystyle \tau=\frac{2\pi}{ \frac{\pi e}{L} \left(n+ \frac{1}{2} \right)}= \frac{2L}{e\left(n+\frac{1}{2} \right)}$

 

[Dustinsfl]

If you have a function $\sin(kt)$, find the period $\tau$ by setting $k\tau=2\pi$.

Is there a difference between frequency and natural frequency (eigenfrequencies)?

 

[Ackbach]

Is there a difference between frequency and natural frequency (eigenfrequencies)?

The term "natural frequency" refers to resonance. So if you have a forced mass-spring system, e.g., and you tune the forcing function to the same frequency as a term in the homogeneous solution, you end up with unstable behavior.

The term "frequency" just refers to what's going on in this thread.

There's also the term "angular frequency", which is represented by $\omega=2\pi f$.

Hope that's as clear as mud.

 

[Dustinsfl]

The term "natural frequency" refers to resonance. So if you have a forced mass-spring system, e.g., and you tune the forcing function to the same frequency as a term in the homogeneous solution, you end up with unstable behavior.

The term "frequency" just refers to what's going on in this thread.

There's also the term "angular frequency", which is represented by $\omega=2\pi f$.

Hope that's as clear as mud.

I am trying to find the natural frequency (eigenfrequency) of $u$.
How would I do that then?
$$
u(x,t) = \sum_{n = 1}^{\infty}\sin\left[\frac{\pi x}{L}\left(n + \frac{1}{2}\right)\right]\left\{A_n\cos\left[\frac{\pi ct}{L}\left(n + \frac{1}{2}\right)\right] + B_n\sin\left[\frac{\pi ct}{L}\left(n + \frac{1}{2}\right)\right]\right\}
$$

 

[Ackbach]

I am trying to find the natural frequency (eigenfrequency) of $u$.
How would I do that then?
$$
u(x,t) = \sum_{n = 1}^{\infty}\sin\left[\frac{\pi x}{L}\left(n + \frac{1}{2}\right)\right]\left\{A_n\cos\left[\frac{\pi ct}{L}\left(n + \frac{1}{2}\right)\right] + B_n\sin\left[\frac{\pi ct}{L}\left(n + \frac{1}{2}\right)\right]\right\}
$$

Well, I could be wrong, but I would say that all of your
$$f_{n}=\frac{c\left(n + \frac{1}{2}\right)}{2L}$$
are the eigenfrequencies. I don't think there's one single eigenfrequency.