Chapter 12, Problem 30P

(a)

To determine

The length of an open organ pipe that emits middle C (262 Hz) when the temperature is 21⁰C.

Answer to Problem 30P

The length of open organ is 0.655 m.

Explanation of Solution

Given:

Frequency ( f )  = 262 Hz,

Temperature ( T )   = 21⁰C,

Determine the length of an open organ pipe ( l )=?

Formula used:

  $\lambda =\frac{v}{f}\text{ m}$

Calculation:

The speed of the sound wave depends on the temperature and it is given as 21⁰C,

So, it is given by the formula,

  $v=331\text{ m/s+(0}\text{.6}T)\text{ m/s}\text{.}$

  $\begin{array}{l}v=331\text{ m/s+(0}\text{.6}\times \text{21})\text{ m/s}\text{.}\\ v=343.6\text{ m/s}\end{array}$

The frequency of the wavelength is 262 Hz.

Now, determine the wavelength

  $\lambda =\frac{v}{f}\text{ m}$$\begin{array}{l}\lambda =\frac{\text{343}\text{.6 m/s}}{\text{262 Hz}}\text{ m}\\ \lambda =1.31\text{m}\end{array}$

  $\begin{array}{l}\lambda =\frac{\text{343}\text{.6 m/s}}{\text{262 Hz}}\text{ m }\\ \lambda =1.31\text{ m}\end{array}$

Both the ends of open organ pipe have a node at the middle, and two anti-nodes at each end, the length of the pipe ( l ) is equal to 1/2 λ,  = $\frac{1}{2}\times 1.31\text{ m}$ =0.655 m

The length of the open organ pipe that emits middle C is 0.655 m

Conclusion:

The length of the open organ pipe that emits middle C is 0.655 m

(b)

To determine

The wavelength and frequency of fundamental standing wave in the tube.

Answer to Problem 30P

  $\lambda =1.31\text{ m}$ , f =262 Hz

Explanation of Solution

Given:

Length (distance between the nodes), L = 0.655

Determine the wavelength ( λ ) =?

Frequency ( f )  =?

Formula used:

  $\lambda =2L$

  $f=\frac{v}{\lambda }\text{ Hz}\text{.}$

Calculation:

If the distance between the nodes are known, calculate the wavelength ( $\lambda$ )

  $\lambda =2L$ m.

  $\lambda =2\times 0.656\text{ m}$

  $\lambda =1.31\text{ m}$

Now, determine the frequency

  $f=\frac{v}{\lambda }\text{ Hz}\text{.}$

  $f=\frac{343\text{ m/s}}{1.31\text{ m}}$ Hz.

  $f=261.8=262\text{ Hz}$

The frequency and the wavelength of the tube are 262 Hz and 1.31 m respectively.

Conclusion:

The frequency and the wavelength of the standing wave in the tube are 262 Hz and 1.31 m respectively.

(c)

To determine

Determine λ and f in the travelling sound wave produced in the outside air.

Answer to Problem 30P

  $\lambda =1.31\text{ m}$ , f =262 Hz

Explanation of Solution

Given:

Length (distance between the nodes),

L = 0.655

Determine the wavelength outside the air ( λ ) =?

Frequency in the outside air   ( f )=?

Formula used:

  $\lambda =2L$

  $f=\frac{v}{\lambda }\text{ Hz}\text{.}$

Calculation:

If the distance between the nodes are known, calculate the wavelength ( λ )

  $\lambda =2L$ m.

  $\lambda =2\times 0.656\text{ m}$

  $\lambda =1.31\text{ m}$

Now, determine the frequency

  $f=\frac{v}{\lambda }\text{ Hz}\text{.}$

  $f=\frac{343\text{ m/s}}{1.31\text{ m}}$ Hz.

  $f=261.8=262\text{ Hz}$

The frequency and the wavelength in the outside air are 262 Hz and 1.31 m respectively.

Conclusion:

The wavelength and the frequencyin the outside are 1.31 m and 262 Hz respectively.