RESONANCE OF A CLOSED AIR COLUMN

EXPERIMENT 12

Experiment conducted 12/2010

PHY 152: General Physics

April 7, 2011

TABLE OF CONTENTS

Discussion______________________________________________________________2

Theory__________________________________________________________________2

Equipment______________________________________________________________4

Experimental Procedure___________________________________________________4

Experimental Procedure___________________________________________________4

Setting up Resonance Tube_____________________________________________________4

Finding Points of Resonance____________________________________________________4

Calculations_____________________________________________________________6

Graphing the Data____________________________________________________________6

Determining Wavelength_______________________________________________________7

Determining the Relationship Between Wavelength and Frequency___________________7

Converting from cm/s to m/s____________________________________________________7

Determining the Speed of Sound from Room Temperature__________________________7

Analysis________________________________________________________________8

Determining the Percent Difference Between Speed of Sound________________________8

Conclusion______________________________________________________________8

Questions_______________________________________________________________9

References______________________________________________________________9

1

RESONANCE OF A CLOSED AIR COLUMN

DISCUSSION

The principle of resonance can by utilized to determine the wavelength of a sound wave. Sound waves are produced by the compression of a substance by vibrations caused by objects such as vocal cords, guitar strings, organ pipe, loud speaker, and so forth (Physics of Sound, para. 1). A resonance frequency exists when a system is driven by its natural frequency. Pendulums and swings demonstrate only one natural frequency (Spalding, 2010, pp. 3-15).

Conversely, a column of air can demonstrate several natural frequencies which are determined by the amount of wavelength segments that can be found within a given tube length. When a tuning fork possesses the same natural frequency as the air column, the vibrations from the tuning fork will push the air particles at a frequency that will cause the air column to vibrate.

Resonance is said to have occurred when the amplitude of sound increases from the sound waves of the tuning fork reinforces the sound waves from the air column. By determining points of resonance, the velocity of a wave can be determine by the equation v=λ∙f.

THEORY

A sound wave can be created when a tuning fork is held over an open portion of a tube similar in respect to how sound is created when a person blows air into a bottle. In a closed air column, the open end is termed the antinode, while the closed end is termed the node. When a tuning fork is placed over the open region of the tube, a disturbance is created by the alternative compressions produced by the sound wave. A standing wave is produced when the alternate compressions and rarefactions of the sound wave down the tubing are reflected at the tubing’s closed end into the opposite direction therefore creating the propagation of waves in the same region although in opposing directions.

The nature by which a standing wave is produced creates a condition where the wave has fixed maximum and minimum points.

The shortest length of a tube to resonate at a given frequency is satisfied by a tube that is a quarter of the wavelength or L=1/4λ. This is because half of the wavelength is represented by the distance from one node to another node, and the tube will resonate at the tuning fork’s frequency. Successive resonant lengths can be noted when the tube length represents an odd number of quarter wavelengths including L=1/4λ, 3/4λ, 5/4λ, and so forth (see Figure 1).

2

Figure 1. Standing waves of L=1/4λ and L=3/4λ respectively drawn using Microsoft Paint.

In equation form, L=nλ/4 when n=1, 3, 5, 7…, and λ=4L/n. When v=λ∙ f, then f =nv/4L when n=1, 3, 5, 7, and so on.

f =nv/4L n=1, 3, 5, 7,….. (1)

In other words, the three parameters that determine the condition of resonance in an air column include f, v, and L where f is the frequency or number of vibrations per minute, v is the velocity of sound in air in meters/second, and L is the length of the column of air in meters.

More wavelength segments fit into the resonance tube when the length of an air column increases when the node-antinode requirements are met. An observer will experience resonance by noting an increase in the amplitude of sound heard when the air column reaches certain lengths that meet the aforementioned conditions. The length of the air column can be manipulated by either adding or draining water from the air column. An increase in amplitude will be heard when the antinode is the open end of the tube, and no increase in

amplitude will be heard when the node is at the open end of the resonance tube. The fundamental frequency or f is noted by the lowest frequency, while higher frequencies are termed overtones.

The length for the resonance can be determined with a resonance that begins at λ/4 with subsequent increments of λ/2 by the

equation L=λ2

n-λ4

where n= 1 2, 3, and etc.

L=λ2

n-λ4

n= 1 2, 3,…

(2)

The equation of a line y=a+bx (where x represents the independent variable, the y represents the dependent variable, and the represents the intercept) can be used to plot the resonance in a closed air column. Therefore, the wavelength of a wave in a resonance tube in a closed air column can be determined by λ=2∙slope.

λ=2∙slope (3)

Additionally, when the wavelength and the frequency of sound are known, the velocity of sound in air can be determined by the equation vs=λ ∙ f .

vs=λ∙f (4) The relationship between the speed

of sound and temperature is given by the equation vs=331.4+0.6 T c v s=331.4+0.6 T c (where temperature is represented in degrees Celsius) since the speed of sound in air is temperature dependent.

vs=331.4+0.6 T c v s=331.4+0.6 T c (5)

3

L=1/4λ L=3/4λ

For instance, at room temperature or T c=¿20̊ C ¿, the speed of sound is 343 m/s or 34, 340 cm/s.

EQUIPMENT

The materials of this experiment consisted of a resonant air column; two vertical metal support rods with metal bases; a tuning for each of the frequencies at 512 Hz, 1024 Hz, and 2048 Hz; a rubber mallet; a room thermometer; a glass reservoir; and tap water.

EXPERIMENTAL PROCEDURE

Setting up the resonance tube

A resonance tube apparatus was set up according to the diagram in Figure 2 by securing a resonance tube (with centimeter division marks) by a clamp to a vertical metal support rod on one metal base and attaching a connecting hose from the resonance tube to a water reservoir can that was clamped to a second vertical support rod on a metal base. The room temperature of the air was determined using a thermometer and was measured at 20 ºC.

Figure 2. Resonance Apparatus (Resonance, 2011, para. 1).

Finding points of resonance

The glass resonance tube was then filled with tap water. A 512 Hz tuning fork

was then hit by a rubber mallet in an area away from the resonance tube, and then the tuning fork was held approximately one centimeter above the open end of the air column. The height of the water column was then adjusted by decreasing the height of the water reservoir can. When the height of the water column was adjusted rapidly, resonance could be heard when the height of the water column reached certain levels.

When changing the height of the water column it was necessary to repeatedly strike the tuning fork with the mallet in order to maintain the vibration. It was also sometimes necessary to pour excess water into the glass reservoir. The positions of the water column when the maximum sounds were heard were labeled as n=1, 2, 3, and so forth and the height in centimeters were then recorded in data table 1.

Special attention was paid to avoid recording relatively faint tones also known as overtones that relate to frequencies that are double and quadruple the frequency being evaluated in the experiment. Those tones suspected as being overtones were not recorded. This procedure was repeated for the 1024 Hz and the 2048 Hz tuning forks.

4

Table 1. Resonance of Tuning Forks at 512 Hz, 1024 Hz, and 2048 Hz with a room temperature of 20 ºC.

CALCULATIONS

5

6

Tuning Fork 1Frequency (f)=512 Hz

Tuning Fork 2Frequency

(f)=1024 Hz

Tuning Fork 3Frequency

(f)=2048 HzResonance (n)

Position (cm)

Resonance(n)

Position(cm)

Resonance(n)

Position(cm)

1 15 1 7 1 32 49 2 24 2 113 81 3 41 3 20

4 57 4 285 74 5 376 91 6 45

7 538 629 7010 8011 87

Graphing the dataA graphical anaylsis of the each tuning fork was completed using Excel. The graph of the

data was drawn in order to obtain the slope of the lines and to facilate in the subsequant calculations. A plot of a graph of length versus the sequential number of the resonant column was created for the tuning forks at 512 Hz, 1024 Hz, and 2048 Hz. A regression line and slope were determined for each of the tuning forks at 512 Hz, 1024 Hz, and 2048 Hz (see Figure 3).

Figure 3. Graphical analysis of the air resonance column at 512 Hz, 1024 Hz, and 2048 Hz.

Determing wavelength

From the slope of the lines of each frequency, the wavelength was determined using the equation λ 2∙ slope(see Table 2).

Worked example:

λ=2∙33.00 cm/nλ=66 cm

Table 2. Determing the wavelength of sound using the slope of a line.

Frequency (Hz)

Wavelength of Sound (cm)

512 Hz λ¿ 66 cm1028 Hz λ¿ 33.48 cm2048 Hz λ¿16.91 cm

Determining the relationship between wave speed and frequency

The relationship between the speed of a wave and its frequency was determined using the wavelength of sound found in Table 2 and the equation vs=λ ∙ f (see Table 3).

Worked example:

vs=66 cm ∙512 Hz

vs=33792cms

Frequency (Hz)

Velocity (cm/s)

512 Hz 33792 cm /s1028 Hz 34284 cm /s2048 Hz 34632 cm /s

Table 3. Determing the wavelength of soundusing the slope of a line.

Converting from cm/s to m/s

The speed of sound was then convert from cm/s to m/s with the equation

vscms

∙ 0.01ms=vs m /s(see Table 4).

Worked example:

vs=33792cms

∙0.01 m /s

vs=337.92 m /s

Table 4. Converting from cm/s to m/s.

Determinging the speed of sound from room temperature

The speed of sound was determined using the temperature found in the room where the experiment was conducted by the equation vs=331.4+0.6 T c (see worked equation).

Worked equation:vs=331.4+0.6 ∙20 ºC

vs=343.4 m/ s

7

Frequency (Hz) Velocity (m/s)

512 Hz 337.92m /s

1028 Hz 342.84m /s

2048 Hz 346.32m /s

ANALYSIS

Determining the percent difference of the speed of sound

The percent difference of the speed of sound for each of the experimental values was then

determined using the equation % Diff = exp−knownknown

∙100 (see Table 6).

Worked example:

% Diff =337.92 m /s−343.4 m / s343.4 m /s

∙100

% Diff =−1.59

Table 6. Determining the percent difference of the speed of sound.

CONCLUSION

It has been shown that by knowing the frequency of a tuning fork and the position of water at resonance points, the spee of sound can be determined.The speed of sound for the tuning forks at the frequencies of 512 Hz, 1028 Hz, and 2048 Hz were determined to be 337.92 m/s, 342.84 m/s, and 346.32 m/s at room temperature.

The speed of sound was determined to be within less than five percent error compared to the known speed of sound at room temperature of 343.4 m/s. The percent difference was greatest for the frequencies of 512 and 2048 Hz and lowest for the frequency at 1028 Hz. The percent difference range from negative percent differences to positive percent differences going from the lowest to highest tested frequencies repectively.

Sources of error for this experiement may consist of mistaking overtones as resonance and therefore including overtones in the data. Another source of error could include missing points of resonance and therefore not including resonance points into the data. Furthermore, the length of the air column could have been incurrectly interpreted when holding the tuning fork above the air column and adjusting the the length of the air column.

8

Frequency (Hz)

% Diff

512 Hz −1.596

1028 Hz −0.163

2048 Hz +0.850

QUESTIONS

1. What is the most likely source of error in each experiment if the tuning forks and electronic tuning fork are accurate to within 0.5%?

The most likely source of error would be to misidentify overtones as points of resonance.

2. Is there any indication that the velocity of sound depends on its wavelength? Explain.

The velocity of sound is mathematically related to the wavelenth in the equation vs=λ ∙ f . In this experiment, the percent difference was negative for the lower frequency trialed and positive for the higher frequency trialed.Therefore, there were minimal differences in the speed of sound found using a range of frequencies. Usually, however, the medium by which the sound travels determines the velocity of sound, and the wavelength of sound determines pitch.

3. The adult male larynx is quite a bit larger than that of the female larynx. What are the acoustical consequences of that?

The human larynx produces changes in pitch and tone of vocalizations by adjusting the length and tension of the vocal folds. The larger male larynx produces a lower pitch or apparent frequency, and a smaller female larynx produces a higher pitch or apparent frequency. Pitch can be manipulated in other objects by adjusting the tension of a string.

4. In a guitar, how would the pitch vary as µ changes? What about F?

By tightening a guitar string, the pitch and frequency increase, Likewise, by loosening a guitar string, the pitch and frequency decrease.

REFERENCES

Resonance. (2011). Retrieved March 25, 2011, from http://www.enasco.com/product/SB16555M#

Spalding, Roger. (2010). General Physics Laboratory Book. Monroe County Community College.

The Physics of Sound. (nd). Retrieved March 17, 2011, from homepages.wmich.edu/~hillenbr/206/ac

Water Resonance Experiment (nd). Retrieved March 18, 2011, from http://hyperphysics.phy-

astr.gsu.edu/hbase/waves/clocol2.html.doc

9