[Solved] proving the standing waves equation using the rubens tube

Proving the standing waves equation using the Ruben’s tube

Abstract

Ruben’s tube is based on the concept of longitudinal waves taking into consideration the fact that sound waves are longitudinal in nature.[1] Often in cases involving longitudinal waves, the medium vibration particles are set parallel to the propagated waves direction.[2] In this research’s experimental design, pressure will be used in generation of standing waves. This research thus investigated the relation between sound wave frequency changes on standing waves. In essence the research sought to prove that frequency, f is directly proportional to velocity, V and inversely proportional to the wavelength, l as represented by the equation

Various experimental variables were kept constant in order to improve the accuracy of the results. These include; maintaining the same type of gas in the tube throughout the experiment, and use of the same tube throughout the experiment. The loudspeakers used also remained unchanged through the whole experiment in addition to maintaining their position to ensure that the distance between the speaker and the tube does not differ.

Data analysis revealed that wavelength and frequency resulting from the experiment are directly proportional to each other. It was noted that as the number of nodes and antinodes in the experiment increased, the respective wavelengths decreased while the frequency increased. The research also looked into possible sources of error and possible solutions.

Introduction

Observing a loud speaker reveals that a sudden change in the magnitude of the sound produced is accompanied by change in the movements of the membranes in an outwards direction. [3] This phenomenon is believed to be a result of a sudden change in frequency of the sound waves. While many people have observed such occurrences in their daily life, most have not been keen enough to try and understand the phenomena.

However, it is worth noting that various experiments have previously been done on the phenomenon with the aim of adequately understanding the phenomenon of sound waves. One of the interesting experimental set up is the Ruben’s tube which has been used to investigate the relation between velocity, wavelength and frequency of waves. [4] The tube takes a similarity to the loudspeakers in relation to changes in frequency. It employs the study of the effect that sound waves have on standing waves.[5] This paper will focus on the study of the relation that exists between frequency, wavelength and velocity of sound waves and thus establish the validity of the equation relating them.

Background information

Rubens tube refers to a flame tube used in demonstration of standing waves. Basically the experiment has long been used in exploring of the relation that exists between the pressure of air and the sound waves.[6]The tube is usually made up of a long tube that has both its ends sealed. The top of the pipe is however perforated to create escape roots for air. While one end is attached to a speaker, the other is connected to a gas supply. Gas is supplied into the tube and the perforations are lit to inflame the leaking gases. Upon lighting of these gases, standing waves with constant frequency are formed. Turning on the speaker, results into formation of nodes and anti-nodes in the tube.[7]The flames rise and drop depending on the pressure changes as a result of the sound waves.[8] Increased pressure results into higher points of flames while lower pressures produce low points of flames.

The tube was first used in 1904 by Rubens Heinrich who took a 4m long tube and drilled 200 small holes on its surface at an interval of 200cm. he then filled the tube with flammable gases and upon lighting realized that the flames lighting from the escaping gases rose to equal heights.[9] After making several experimental trials on the set up, Ruben Concluded that production of sound at one end of the tube results into formation of standing waves similar in wavelength to that of the sound.[10] Borrowing from August Kundt’s 1866 experiment, Ruben noted that on placing cork dust in the tube resulted into the lining up of the dust as nodes and anti-nodes as the sound waves fluctuated in order to form standing waves.[11]

Experimental design

The experiment aimed to prove the equation relating frequency, wavelength and velocity of sound waves as observed with the help of a tube with one end closed while the other end is connected to the frequency generator connected through a loudspeaker. The tube used is referred to as the Ruben’s tube. The frequency generator was chosen over music for generation of sound waves due to its continuity. Music would have produced continuous changes in frequency and even taking of momentary pictures of changes would result into inaccurate data. Plugging in a frequency generator to the loud speakers thus proved more practicable and applicable for yield of more accurate data. Given the longitudinal characteristic of the waves, longitudinal waves form in the tube and are observed as standing waves as witnessed by the lit fire. Low and high pressure regions would be formed in relation to the wave generators frequency. With the amplitude kept constant, a sine waveform is formed.  In the experiment, it was expected that the high points of flames would be the high frequency points.

The particles back and forth vibration though the sound waves motion media takes a characteristic form of sound waves. The characteristic movement of sound waves from side to side displaces the air in a similar pattern as the sound wave energy moves through it. The longitudinal motion of air results into creation of two regions namely; compressed region and the spread-out region. High pressure regions results into compressions while the low pressure regions produce rarefactions. The pressure fluctuations undergo periodic intervals at time intervals that are regular. The sine curve’s peak points correspond to compressions while the low points correspond to the rarefactions as illustrated by the diagram below;[12]

                Sound is represented as a pressure wave with a sine curve nature.

The C’s represent compressions while the R’s represent the rarefactions.

Given that the Ruben’s tube is considered closed at one end and opens at one end, the wave begins with a node and completes with an anti-node. (Jihui 1110) This is a result of the air pressure oscillation at the closed side with the greatest amplitude and thus the anti-node.

Experimental procedure

List of apparatus to be used

–          1″ x 8″ x 6′ pine board

–          4″ x 60″ galvanized steel vent pipe

–          4″ vent pipe wall strap

–          Air duct sealant

–          0.025″ thick latex rubber sheet

–          4″ band hose clamp

–          4″ diameter speaker

–          Frequency generator

–          Lighter

–          Gas burner

–          Thermometer

Experimental set-up

Steel tube is used to minimize heat conduction which may result into melting of the rubber stopper. Circular holes of diameter, 2mm are equidistantly drilled on the surface of the tube leaving a distance of 10mm from the respective ends. The holes are drilled at a distance of 15 centimeters apart and in a straight line. One of the tubes end is closed using the rubber sheet in such a way that no gas can escape through it while the gas supply is attached and sealed with the help of an air duct sealant and the pipe wall strap. The stands are then used to hold the tube, rubber stopper and the rubber membrane. Once this set up is complete, the system is examined fro any sources of leak which might compromise the experiments validity or the experiment’s safety. The gas is then turned on a left to spread out across the tube for fifteen seconds before the circular perforations are lit. Relight the perforations if some fail to get lit. The loudspeaker is then connected to a frequency generator and placed close to the end closed with a rubber membrane in this case, Audacity software is used as the frequency generator. The frequency is varied until such time that the resulting standing waves are perfect in nature. The corresponding room temperature for each different frequency is recorded as this would aid in calculation of the speed of sound. The above mentioned steps are continuously repeated for various frequencies to allow collection of a spectrum of data.

Controlled variables

As different conditions are bound to have an effect on the experiments outcome some variables must be controlled to limit inaccurate data collection. Frequency acts as the independent variable in the experiment while its effect on the generated standing wave forms our dependant variable. The controlled variables in the experiment included the following;

a.      Room temperature

Despite the room temperature being kept constant the use of fire may result into slight temperature variations thus temperatures are controlled prior to each trial and the calculations appropriately revised. Speed of sound is temperature dependant and therefore temperature will be important in its calculation. Its calculation will be based on the formula below:

V= (331.5 ± 1.0m/s) + (0.6 ± 0.1m/s) ×Tc  [13]

b.      Length of the tube

Enclosed wave mediums can easily produce standing waves.[14] Various calculations need to be performed in order to obtain the frequency values in the experiment. The wavelength is directly proportional to the length of the tube and thus the equation below is employed in calculation of frequency:

,  and

As the length of the tube changes, the wavelength of the tube too changes and hence the relevant frequency values also change while the length of the tube used is 2m only 1.8m is considered operational as the 10cm are left on both ends as a precautionary measure.

c.       Gas pressure

The experiment uses natural gas which has no direct effect on data calculation in the experiment. For successful production of standing waves, the experiment is done within a laboratory room with the air conditioning turned off the all the windows closed. Initial trials fail to produce standing waves successful. However, after adjusting the gas pressure to appropriate levels the waves are successfully produced. The pressure is then constantly maintained throughout the experiment. The gas pressure in the tube is considered a dependant variable given that it changes as the frequency changes as may be observed by the high and low flames produced.

4. Number of Nodes and Anti-Nodes

The number of nodes and the anti-nodes are directly proportional to the frequency of the waves produced.[15] An increase in the generated frequency is accompanied by a decrease in wavelength and thus the nodes and the anti-nodes correspondingly increase as more standing waves are produced. However in he experiment the number of nodes and antinodes were easily observed and hence no calculations were required for the same.

N/B: the value of n used in the calculation was determined by calculating the total number of nodes and antinodes less 1.

Collected and calculated data

Room temperature = 20 ± 1m/s

Speed of sound = 343 ± 0.1m/s

The length of the tube = 1.8 ± 0.01m

Table 1:

Nodes + Anti-Nodes
1+1
2+2
3+3
4+4
5+5
6+6
7+7
8+8
9+9
10 + 10
Wavelength,
7.20
2.40
1.44
1.03
0.80
0.65
0.55
0.48
0.42
0.38
Estimated Frequency, Hz
47.7
143.0
239.0
334.0
429.0
525.0
624.0
715.0
817.1
903.2
Best standing wave’s general frequency
45.4
137.0
228.0
318.0
410.0
502.0
605.0
702.0
808.0
889.0
% error
4.19
4.20
4.60
4.79
4.43
4.38
3.04
1.82
1.10
1.57
Average % error
3.41

Data calculation

V= (331.5 ± 1.0m/s) + (0.6 ± 0.1m/s) ×Tc

Where Tc = temperature in °C

Where V= speed of sound, l is the wavelength and f is the frequency

Data analysis

The average % error is calculated in relation to the calculated data and the observed frequencies assigned to the best standing waves. Its value is thus set at 3.41%

The resulting graph of the input data is reflected below;

Figure 1: Graph of frequency against wavelength as per the values in table 1

The graph reveals and in inverse proportionality between the wavelength and frequencies of the resultant waves. From the graph an increase in frequency is accompanied by a decrease in wavelength. Considering the proportional inverse relation between wavelength and number of nodes and anti-nodes, it then follows that frequency is directly proportional to the number of nodes and antinodes. The same can be directly observed from the table where an increase in the number of nodes and anti-nodes is accompanied by a corresponding increase in frequency.

The calculated values according to the tables reveal that

But

Þ

Conclusion

The data collected and the graph verify the validity of the equation relation wavelength and frequency of sound waves. It in essence confirms the earlier stated hypothesis as we note that frequency affects wavelength according to the equation;

It is also important to note that in one end closed tubes, the wave reduced by sound begins wit a node and ends with an anti-node thus the formula used n value as being one less the total number of nodes and anti-nodes. The L value used too involved deduction of the un-holed areas which included the 10cm provision from both ends and thus L was considered as being 1.8m. The experiment in addition revealed that sound is also a pressure wave as the nodes and the anti-nodes were produced in accordance with pressure regions. Increase in frequency resulted into an increase in height of the flames further confirming the relation between pressure and sound waves. This is based on the principle that increase in frequency results into pressing together of gas particles resulting into increased forces that heighten the flame.

Validity of the experiment

The % error of the experiment is set at 3.41%. Also worth noting is the fact that the graph did not direct fit and a best line of fit had to be obtained. Various errors account for these deviations. Such include;

–          The formula used in determining the sound velocity did not take into consideration the humidity in the room. Correcting this would require revising of the formula as well as availing a hygrometer to ensure that such is taken care of.

–          Temperature variations due to use of heat in observing flames required a more sensitive thermometer in order to effectively note them.

–          The expansion of the tube as result of eating was considered negligible while it could have had minimal effects on the experiment.

Future similar experiment should thus take into consideration the above mentioned factors in order to produce even more accurate results.

Works cited

Ficken, G.W. and Stephenson, F.C. “Rubens flame tube demonstration,” The physics Teacher17 (1), 1979:306-310

Gardner, M. D., Gee, K.L. and Dix, G. “An investigation of Rubens flame tube resonances,. Journal of Acoustic Sciences, 125(1), 2009:1285-1292

Hart, R (2003 Sept 18) “Standing Waves applet” Retrieved February 26, 2009, Web site: http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html

Jihui, D and Wang, T.P. “Demonstration of Longitudinal Sound waves in pipe revisited” Journal of Physics 53(4), 1985:1110

Spagna, G. “Rubens Flame Tube demonstration: Closer look at the flames” Journal of Physics 51(2), 1983: 848

[1] Ficken, G.W. and Stephenson, F.C. “Rubens flame tube demonstration,” The physics Teacher17 (1), 1979:306

[2] Gardner, M. D., Gee, K.L. and Dix, G. “An investigation of Rubens flame tube resonances,. Journal of Acoustic Sciences, 125(1), 2009:1285

[3] Ficken, G.W. and Stephenson, F.C. “Rubens flame tube demonstration,” The physics Teacher17 (1), 1979:307
[4] Gardner, M. D., Gee, K.L. and Dix, G. “An investigation of Rubens flame tube resonances,. Journal of Acoustic Sciences, 125(1), 2009:1285

[5] Ficken, G.W. and Stephenson, F.C. “Rubens flame tube demonstration,” The physics Teacher17 (1), 1979:308
[6] Ibid, 309
[7] Gardner, M. D., Gee, K.L. and Dix, G. “An investigation of Rubens flame tube resonances,. Journal of Acoustic Sciences, 125(1), 2009:1287
[8] Ficken, G.W. and Stephenson, F.C. “Rubens flame tube demonstration,” The physics Teacher17 (1), 1979:309
[9] Gardner, M. D., Gee, K.L. and Dix, G. “An investigation of Rubens flame tube resonances,. Journal of Acoustic Sciences, 125(1), 2009:1288
[10] Spagna, G. “Rubens Flame Tube demonstration: Closer look at the flames” Journal of Physics 51(2), 1983: 848
[11] Gardner, M. D., Gee, K.L. and Dix, G. “An investigation of Rubens flame tube resonances,. Journal of Acoustic Sciences, 125(1), 2009:1289
[12] Jihui, D and Wang, T.P. “Demonstration of Longitudinal Sound waves in pipe revisited” Journal of Physics 53(4), 1985:1110

[13] Hart, R (2003 Sept 18) “Standing Waves applet” Retrieved February 26, 2009, Web site: http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html
[14] Jihui, D and Wang, T.P. “Demonstration of Longitudinal Sound waves in pipe revisited” Journal of Physics 53(4), 1985:1110

[15] Jihui, D and Wang, T.P. “Demonstration of Longitudinal Sound waves in pipe revisited” Journal of Physics 53(4), 1985:1110

 

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