Chapter 17, Problem 16TP

To determine

(a)

Whether the air column is open or closed on each end.

Answer to Problem 16TP

The air column is closed at one end.

Explanation of Solution

Given:

The data length of an air column and a resonant frequency present when the column is struck is given in the table below:

Length (m)	Resonant Frequency (Hz)
$1$	$85.75$
$2$	$43$
$3$	$29$
$4$	$21.5$

Table (1)

Formula used:

The frequency of a tube closed at one end is given as,

  $f_{\text{n}}=n\frac{v_{\text{w}}}{4L}$

The frequency of a tube open at both ends is given as,

  $f_{\text{n}}=n\frac{v_{\text{w}}}{2L}$

Calculation:

The frequency of a tube closed at one end is given as,

  $f_{\text{n}}=n\frac{v_{\text{w}}}{4L}$

Let $n=1$.

On simplifying,

  $\begin{array}{l}{f}_{\text{n}}=\left(1\right)\frac{v_{\text{w}}}{4L}\\ \frac{f_{\text{n}}}{v_{\text{w}}}\left(L\right)=0.25\end{array}$

Similarly, the frequency of a tube open at both ends is given as,

  $f_{\text{n}}=n\frac{v_{\text{w}}}{2L}$

Let $n=1$.

On simplifying,

  $\begin{array}{l}{f}_{\text{n}}=n\frac{v_{\text{w}}}{2L}\\ \frac{f_{\text{n}}}{v_{\text{w}}}\left(L\right)=\frac{1}{2}\end{array}$

To determine whether the tube is closed at one end or open at both ends, find the ratio of product of frequency by length to the speed of sound.

  $\begin{array}{c}\frac{f_{\text{n}}}{v_{\text{w}}}\left(L\right)=\frac{85.75\text{ Hz}}{331\text{m}/\text{s}}\left(1\text{ m}\right)\\ =0.25\end{array}$

This result satisfies the equation of frequency of a tube closed at one end.

Check the ratio of product of frequency by length to the speed of sound on other values of the experiment.

For the length of $2\text{m}$,

  $\begin{array}{c}\frac{f_{\text{n}}}{v_{\text{w}}}\left(L\right)=\frac{\text{43 Hz}}{331\text{m}/\text{s}}\left(\text{2 m}\right)\\ =0.25\end{array}$

For the length of $3\text{m}$,

  $\begin{array}{c}\frac{f_{\text{n}}}{v_{\text{w}}}\left(L\right)=\frac{\text{29 Hz}}{331\text{m}/\text{s}}\left(\text{3 m}\right)\\ \approx 0.25\end{array}$

For the length of $4\text{m}$,

  $\begin{array}{c}\frac{f_{\text{n}}}{v_{\text{w}}}\left(L\right)=\frac{\text{21}\text{.5 Hz}}{331\text{m}/\text{s}}\left(\text{4 m}\right)\\ =0.25\end{array}$

Conclusion:

Since the ratio of product of frequency by length to the speed of sound for all the lengths corresponds to the ratio for an open tube, the air column is closed at one end.

To determine

(b)

The resonant frequency of the column at a length of $2.5\text{ meters}$.

Answer to Problem 16TP

The resonant frequency of the column at a length of $2.5\text{ meters}$ is $33.1\text{ Hz}$.

Explanation of Solution

Formula Used:

The frequency of a tube closed at one end is given as,

  $f_{\text{n}}=n\frac{v_{\text{w}}}{4L}$

Calculation:

The resonant frequency of the column at a length of $2.5\text{ meters}$ is calculated as,

  $\begin{array}{c}{f}_{\text{n}}=n\frac{v_{\text{w}}}{4L}\\ =\left(1\right)\frac{\left(331\text{m}/\text{s}\right)}{4\left(2.5\text{ m}\right)}\\ =33.1\text{ Hz}\end{array}$

Conclusion:

The resonant frequency of the column at a length of $2.5\text{ meters}$ is $33.1\text{ Hz}$.