Oct 23 2014

Speed of Sound in Air

EXPERIMENT 4 LELAND JANSEN

PHYS 130 SECTION EN02 KIRSTY GARDNER (TA) Introduction

A key property of sound waves (longitudinal compression waves) are their ability to be generated through vibrations in a medium. Knowing the speed through which a sound wave travels through a medium has a wide variety of applications including the construction of concert halls, ultrasound imaging, and sonar.

This speed is dependent on the bulk modulus and density of the medium through which the wave travels, as described in Equation 1.

� (1) � = √ �

Therefore, the medium through which the wave travels is responsible for determining the speed at which sound will propagate. It can be concluded the speed of sound in air is not the same value everywhere due to variances in the bulk modulus and density as a result of factors such as temperature, humidity, and atmospheric pressure. Given that this data is not readily available, an alternative method of finding the speed of sound can be described as the product of the frequency of a particular sound and its wavelength as shown in Equation 2. The wavelength of the sound varies depending in part on the properties of the medium as described above, and can be determined by measuring its resonance wavelength in a pipe. A pipe will resonate with maximum sound intensity when standing waves are created.

� = �� (2)

The purpose of Laboratory 4 is to determine the speed of sound in air using resonance in a pipe. This value will be calculated using regression analysis of experimentally determined data, and compared to a theoretical value based on temperature. Additionally, the end correction distance �0 will be determined and compared to its theoretical value. It is required that the resonance lengths of maximum intensity in a pipe be calculated.

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Experimental Method

An apparatus was set up in such a way that a plunger was inserted inside a tube with its end extended all the way through the length of the tube. The tube now will now not be open at either end.

A tuning fork of frequency 348Hz was then hit with a mallet and positioned at the end of the tube opposite to the plunger’s handle as shown in Figure 1. The plunger was then slowly retracted until a standing wave was formed. This phenomena was obvious because maximum sound intensity was heard. It was required that the tuning fork be hit again when retracting the plunger so that the intensity of the tuning fork’s sound does not noticeably fade.

When maximum intensity inside the tube was heard, the retraction distance of the plunger was recorded alongside the associated frequency of the tuning fork shown in Figure 2. This distance corresponds with the effective length of the tube (closed at one end).

A complete apparatus setup is featured in Figure 3.

The tuning fork was again hit and the plunger further retracted until a second sound intensity maximum (i.e. loudest possible sound) was heard. This value was also recorded. The process was repeated until the plunger has been retracted entirely thus making the determination of the subsequent sound intensity maximum unattainable due to the limiting length of the plunger and tube (i.e. the distance the plunger must be retracted exceeds the length of the tube). The entire process was then repeated for three additional tuning forks with frequencies 426.7Hz, 480Hz, and 512Hz.

Finally, results were analyzed to calculate the speed of sound in air and the end correction distance.

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Figure 1 Figure 2

Positioning of tuning fork Measurement of plunger retraction distance

Figure 3

Complete apparatus set up

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Results

Table 1

Constants and other measurements

Item Value Tube length (m) 1.010±0.001 Tube diameter (m) 0.038±0.001 Temperature (°C) 19±1

The measured temperature is an estimate based on a reference temperature of 20°C and has been assigned an estimated error of ±1°C.

Table 2

Measured tube length for frequency’s first, second, and third resonance

Length of tube for nth resonance (±0.01 m) Note Frequency (Hz) 1 2 3 G 384 0.205 0.645 A 426.7 0.198 0.591 0.983 B 480 0.161 0.523 0.866 C 512 0.158 0.499 0.835

It should be noted that the third resonance could not be found due to the restriction posed by the length of the pipe. Specifically, the third resonant is expected to occur at a tube length greater than 1.01m (the length of the tube).

Period can be calculated using Equation 3. Results are tabulated in Table 3 and graphed in Figure 4.

2� − 1 � = (3) �

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Table 3

Length of tube required for resonance of a given period n Frequency (Hz) Period (s) Length (±0.01 m) 1 384 0.000651 0.205 2 384 0.001953 0.645 1 426.7 0.000586 0.198 2 426.7 0.001758 0.591 3 426.7 0.002929 0.983 1 480 0.000521 0.161 2 480 0.001563 0.523 3 480 0.002604 0.866 1 512 0.000488 0.158 2 512 0.001465 0.499 3 512 0.002441 0.835

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Figure 4

Length of tube required for resonance with respect to period

1.00 y = 339.6x - 0.0087 R² = 0.9992 0.80

0.60

Length (m) Length 0.40

0.20

0.00 0.000 0.001 0.001 0.002 0.002 0.003 0.003 Period (s)

Statistical data for the least-squares regression in Figure 4 calculated using the Microsoft Excel LINEST function is displayed in Table 4. The equation of the regression is shown in Equation 5.

Table 4

Least-squares regression line statistical data output for length of tube required for resonance with respect to period

LINEST Value Slope 339.5978 Error in slope 3.103856 Correlation coefficient 0.999249 F-test overall value 11970.89 Regression sum of squares 0.916286 Intercept -0.00866 Error in intercept 0.005464 Error of the regression 0.008749 Degrees of freedom 9 Residual sum of squares 0.000689

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Regression analysis

2� − 1 � = � − � (4) � 4� 0

The theoretical model described by Equation 4 and can be written the form � = �� + � where:

� = �� � = � = 340�⁄� 2� − 1 � = 4� � = −�0 = −0.009

The least-squares regressing line can be calculated using the data LINEST data tabulated in Table 4 to yield Equation 5.

� = 340� − 0.009 (5)

For every one second increase in the independent variable Period, the linearized model predicts an average increase of 340m in the dependent variable length. Additionally, with no Period (� = 0�), the model predicts an end correction constant of length 0.009m.

The error in the regression’s slope and intercept are calculated using the Microsoft Excel LINEST function and are featured in Table 4. The values are iterated below.

� = 340 ± 3�⁄� �0 = 0.009 ± 0.005�

It should be noted that the coefficient of determination of the least-squares regression line is �2 = 0.9992. Therefore, 99.92% of the variation in the Length can be explained by the linear relationship with Period, and the least-squares regression line is a very good fit to these data.

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Theoretical versus calculated speed of sound

The theoretical speed at which sound propagates through air and the calculated error in speed can be found using Equations 6 and7, respectively, where � is the temperature (kelvin) and �0 is the speed of sound in dry air at 273.15K.

� (6) � = � √ 0 273.15�

�0�� (7) �� = 2�

Theoretical speed of sound at 19±1°C:

(19 + 273.15)K � = (343�⁄�)√ 273.15� � ≈ 354.7�⁄�

(343�⁄�)(1�) �� = 2(19 + 273.15)� �� ≈ 0.6�⁄�

∴ � = 354.7 ± 0.6�/�

Measured speed of sound:

� = 340 ± 3�⁄�

The theoretical and measured values of the speed of sound agree within four times their respective error. It should be noted that the theoretically determined value assumes perfectly still and dry air, a condition not necessarily present when conducting the experiment.

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Theoretical versus calculated speed of sound

The theoretical approximation of �0 and the calculated error can be calculated using Equations 8 and 9, respectively, where � is the radius of the tube.

�0 ≈ 0.6� (8)

��0 = 0.6�� (9)

Theoretical value of end correction �0 at a diameter of 0.038±0.001m: 0.038� � ≈ 0.6 ( ) 0 2 �0 ≈ 0.011�

��0 = 0.6(0.001�) δ�0 = 0.006�

∴ �0 = 0.011 ± 0.006�

Measured value of end correction �0:

�0 = 0.009 ± 0.005�

The theoretical and measured values of the end correction agree given that they are within their respective error. It should be noted that Equation 8 only provides theoretical approximation of �0.

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Discussion

Given that a transverse waves reflect at a solid end, the wave cannot have maximum amplitude at that point. Therefore, a node must be present at the closed end of a pipe.

Diagrams of the first and second harmonic of a transverse wave inside a pipe closed at both ends are depicted in Figure 5.

Through Figure 5, the ratio of between �1 and �2 can be determined:

�2 � = � ÷ = 2 �1 2

Figure 5

First and second harmonics in a closed pipe

�1 = �⁄2 �2 = �

Diagrams of the first and third harmonic of a transverse wave inside a pipe closed at one end are depicted in Figure 6. Given that a node cannot occur on an open pipe end, the second harmonic does not exist.

Through Figure 6, the ratio of between �1 and �3 can be determined:

�3 3� � = ÷ = 3 �1 4 4

Figure 6

First and third harmonics in a pipe open at one end

�1 = �⁄4 �3 = 3�⁄4

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The ratio of two consecutive resonances of a pipe open at one end is a factor of 1.5 times greater than the ratio of two consecutive resonances of a pipe closed at both ends.

�3 �2 3 ÷ = = 1.5 �1 �1 2

The lowest frequency that causes the pipe to resonate can be derived from Equation 2 to yield Equation 10. The wavelength of the fundamental frequency is equal to four-times the length of the pipe plus the end correction measurement as shown in Equation 11.

� � = (10) �

� (11) �1 = 4(� + �0)

Experimental results are used in the following calculation to determine the lowest resonant frequency that can occur in the pipe. 340�⁄� � = 1 4(1.010 + 0.009)� �1 ≈ 83��

�� �� + ��0 ��1 = �1 ( + ) |�| |� + �0| 3�⁄� 0.001� + 0.005� �� = (83��) ( + ) 1 |340�⁄�| |1.010� + 0.009�|

��1 ≈ 1��

∴ �1 = 83 ± 1��

The lowest possible frequency that can be achieved in the pipe is 83±1Hz, or a frequency between musical notes E2 and F2.

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Potential sources error include inaccuracies in instrument error, human observation, and apparatus inaccuracies.

When determining the maximum amplitude, the limits of human hearing acuity likely resulted in slight differences between the true maximum amplitude achievable and that which was heard. To compensate, a consensus was achieved amongst all three group members each time a maximum amplitude was chosen.

Given the height distance between the plunger handle and the top of the pipe, the horizontal displacement of the plunger was difficult to measure. To more accurately measure this distance, a straight object was placed perpendicular to the end of the plunger handle in such a way that the distance could accurately be measured.

Apparatus inaccuracies and generalized assumptions also potentially lead to inaccurate data. For example, the lab was assumed to be conducted in “ideal conditions”. In other words, variances in the positioning of the tuning fork could lead to variation in the data. A conscious effort was made to attempt to avoid this. Additionally, it was assumed that the tube was perfectly cylindrical with no deformations.

These errors were not represented in the calculations performed in the analysis.

Conclusion

The speed of a sound wave is dependent on the medium’s properties through which it is propagating. When a tuning fork vibrates at a given frequency, the resulting wavelength varies depending in part on the physical properties of the air such as temperature. The speed of sound in air was investigated in Experiment 4 as a function of the sound wave’s resonance length and frequency as described in Equation 2. This value was compared to a theoretical value based on temperature. Additionally, the end correction length was calculated and compared to its theoretical value based on the radius of the tube. The experimentally determined values for speed (�) and the end correction (�0) are iterated below.

� = 340 ± 3�⁄� �0 = 0.009 ± 0.005�

Experiment 4 investigates the properties of sound waves and determines the relationship between speed, frequency, and wavelength. Specifically, the speed of sound and end correction distance were calculated. Furthermore, the applications of the speed of sound are investigated, including those in Engineering.

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