# 15.2 Resonance and sound quality

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## Resonance

Resonance is the tendency of a system to vibrate at a maximum amplitude at the natural frequency of the system.

Resonance takes place when a system is made to vibrate at its natural frequency as a result of vibrations that are received from another source of the same frequency. In the following investigation you willmeasure the speed of sound using resonance.

## Experiment : using resonance to measure the speed of sound

Aim:

To measure the speed of sound using resonance

Apparatus:

• one measuring cylinder
• a high frequency (512 Hz) tuning fork
• some water
• a ruler or tape measure

Method:

1. Make the tuning fork vibrate by hitting it on the sole of your shoe or something else that has a rubbery texture. A hard surface is not ideal as you can more easily damage the tuning fork. Be careful to hold the tuning fork by its handle. Don't touch the fork because it will damp the vibrations.
2. Hold the vibrating tuning fork about 1 cm above the cylinder mouth and start adding water to the cylinder at the same time. Keep doing this until the first resonance occurs. Pour out or add a little water until you find the levelat which the loudest sound (i.e. the resonance) is made.
3. When the water is at the resonance level, use a ruler or tape measure to measure the distance ( ${L}_{A}$ ) between the top of the cylinder and the water level.
4. Repeat the steps  [link] above, this time adding more water until you find the next resonance. Remember to hold the tuning fork at the same height of about 1 cm above the cylinder mouth and adjust the water levelto get the loudest sound.
5. Use a ruler or tape measure to find the new distance ( ${L}_{B}$ ) from the top of the cylinder to the new water level.

Conclusions:

The difference between the two resonance water levels (i.e. $L={L}_{A}-{L}_{B}$ ) is half a wavelength, or the same as the distance between a compression and rarefaction. Therefore, since you know the wavelength, and you know the frequencyof the tuning fork, it is easy to calculate the speed of sound!

## Interesting fact

Soldiers march out of time on bridges to avoid stimulating the bridge to vibrate at its natural frequency.

A 512 Hz tuning fork can produce a resonance in a cavity where the air column is 18,2 cm long. It can also produce a second resonance when the length of the air column is 50,1 cm. What is the speed of sound in the cavity?

1. $\begin{array}{ccc}\hfill {L}_{1}& =& 18,2\phantom{\rule{4pt}{0ex}}\mathrm{cm}\hfill \\ \hfill {L}_{2}& =& 50,3\phantom{\rule{4pt}{0ex}}\mathrm{cm}\hfill \\ \hfill f& =& 512\phantom{\rule{4pt}{0ex}}\mathrm{Hz}\hfill \\ \hfill v& =& ?\hfill \end{array}$

Remember that:

$\begin{array}{ccc}\hfill v& =& f×\lambda \hfill \end{array}$

We have values for $f$ and so to calculate $v$ , we need to first find $\lambda$ . You know that the difference in the length of the air column between two resonances is half a wavelength.

2. $\begin{array}{ccc}\hfill {L}_{2}-{L}_{1}& =& 32,1\phantom{\rule{4pt}{0ex}}\mathrm{cm}\hfill \end{array}$

Therefore 32,1 cm = $\frac{1}{2}×\lambda$

So,

$\begin{array}{ccc}\hfill \lambda & =& 2×32,1\phantom{\rule{4pt}{0ex}}\mathrm{cm}\hfill \\ & =& 64,2\phantom{\rule{4pt}{0ex}}\mathrm{cm}\hfill \\ & =& 0,642\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \end{array}$
3. $\begin{array}{ccc}\hfill v& =& f×\lambda \hfill \\ & =& 512×0,642\hfill \\ & =& 328,7\phantom{\rule{4pt}{0ex}}{\mathrm{m}.\mathrm{s}}^{-1}\hfill \end{array}$

From the investigation you will notice that the column of air will make a sound at a certain length. This is where resonance takes place.

## Music and sound quality

In the sound chapter, we referred to the quality of sound as its tone. What makes the tone of a note played on an instrument?When you pluck a string or vibrate air in a tube, you hear mostly the fundamental frequency. Higher harmonics are present, but are fainter. These are called overtones . The tone of a note depends on its mixture of overtones. Different instruments have different mixtures of overtones. This is why the same note soundsdifferent on a flute and a piano.

Let us see how overtones can change the shape of a wave: The quality of a tone depends on its mixture of harmonics.

The resultant waveform is very different from the fundamental frequency. Even though the two waves have the same mainfrequency, they do not sound the same!

## Summary - the physics of music

1. Instruments produce sound because they form standing waves in strings or pipes.
2. The fundamental frequency of a string or a pipe is its natural frequency. The wavelength of the fundamental frequency is twice the length of the string or pipe when both ends are fixed or both ends are open. It is four times the length of the pipe when one end is closed and one end is open.
3. When the string is fixed at both ends, or the pipe is open at both ends the first harmonic is formed when the standing wave forms one whole wavelength in the string or pipe. The second harmonic is formed when thestanding wave forms 1 $\frac{1}{2}$ wavelengths in the string or pipe.
4. When a pipe is open at one end and closed at the other, the first harmonic is formed when the standing wave forms $\frac{11}{3}$ wavelengths in the pipe.
5. The frequency of a wave can be calculated with the equation $f=\frac{v}{\lambda }$ .
6. The wavelength of a standing wave in a string fixed at both ends can be calculated using ${\lambda }_{n}=\frac{2L}{n-1}$ .
7. The wavelength of a standing wave in a pipe with both ends open can be calculated using ${\lambda }_{n}=\frac{2L}{n}$ .
8. The wavelength of a standing wave in a pipe with one end open can be calculated using ${\lambda }_{n}=\frac{4L}{2n-1}$ .
9. Resonance takes place when a system is made to vibrate at its natural frequency as a result of vibrations received from another source of the same frequency.

## Waveforms

Below are some examples of the waveforms produced by a flute, clarinet and saxophone for different frequencies (i.e. notes):

## End of chapter exercises

1. A guitar string with a length of 70 cm is plucked. The speed of a wave in the string is 400 m $·$ s ${}^{-1}$ . Calculate the frequency of the first, second, and third harmonics.
2. A pitch of Middle D (first harmonic = 294 Hz) is sounded out by a vibrating guitar string. The length of the string is 80 cm. Calculate the speed of the standing wave in the guitar string.
3. The frequency of the first harmonic for a guitar string is 587 Hz (pitch of D5). The speed of the wave is 600 m $·$ s ${}^{-1}$ . Find the length of the string.
4. Two notes which have a frequency ratio of 2:1 are said to be separated by an octave. A note which is separated by an octave from middle C (256 Hz) is
1. 254 Hz
2. 128 Hz
3. 258 Hz
4. 512 Hz
5. Playing a middle C on a piano keyboard generates a sound at a frequency of 256 Hz. If the speed of sound in air is 345 m $·$ s ${}^{-1}$ , calculate the wavelength of the sound corresponding to the note of middle C.
6. What is resonance? Explain how you would demonstrate what resonance is if you have a measuring cylinder, tuning fork and water available.
7. A tuning fork with a frequency of 256 Hz produced resonance in an air column of length 25,2 cm and at 89,5 cm. Calculate the speed of sound in the air column.

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