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[SOLVED] concept question frequency

dwsmith

Well-known member
Feb 1, 2012
1,673
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?
 
Last edited:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Yes:

$\displaystyle f=\frac{1}{\tau}$
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Indicium Physicus
Staff member
Jan 26, 2012
4,193
If you have a function $\sin(kt)$, find the period $\tau$ by setting $k\tau=2\pi$.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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:

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

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Indicium Physicus
Staff member
Jan 26, 2012
4,193
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.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Indicium Physicus
Staff member
Jan 26, 2012
4,193
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.