Question

Standing waves can be established within a cylindrical volume. However, it is not known if the volume is closed at both ends, open at both ends, or is open at one end and closed at the other end. Three successive resonance frequencies within the volume are , , and . How can a student use the information to determine if each end of the volume is open or closed? Justify your selection.

Answer 1

The frequencies are missing in the question. The three successive resonance frequencies within the volume are $f_1, f_2 \text{ and}\ f_3$.

Solution :

Let the volume be : v

The frequency for one end open and one end closed is given by :

So, $f_n = \frac{(2n-1)v}{4L}$

Therefore,

$f_1 = \frac{v}{4L}$     ,     $f_2 = \frac{3v}{4L}$  ,     $f_3 = \frac{5v}{4L}$

So, $f_2 - 3f_1$  and  $f_3=\frac{5}{3}f_2$

Therefore, the ratio of   $\frac{f_3}{f_2}=\frac{5}{3}$    which is not a whole number.

Now the frequency for open volume and closed at both end

$f_n=\frac{nv}{4L}$

So,    $f_1=\frac{v}{4L}$   ,    $f_2 = 2f_1$ ,  $f_2=3f_1$

From above formulae we can see that  ratio of     is not a whole number that is a identification for the frequency of the volume at one end open and one end closed.

Also ratio of the consecutive frequency of the volume at open from both side and closed from both side is always a whole number.