Introduction

When investigating the speed of sound in air, the measurement called ‘end correction’ is included to make the measurements more accurate. During IB physics HL lessons, our teacher did not show us how the end correction could be modelled as an experiment, or how the end correction of a pipe was found.

What was fascinating was that the end correction was absolute, and given as 0.3d where ‘d’ was the diameter of the pipe. I was not comfortable with this value of 0.3d as no proof or experiment was shown. This is when I decided that I wanted to investigate this relationship, and verify this value of 0.3d. I designed my own experiment by using several glass and PVC pipes of varying diameters. I carried out the same experiment one would use to find the speed of sound in air, using a resonance tube. My independent variable was the pipe diameter, and the dependent variable was the distance between the end of the resonance tube and the resonating tuning fork when there is maximum amplitude or when the sound was loudest. The research question I modelled my experiment around was ‘how can L=0.3d be modelled as a linear equation y=mx+c?

Hypothesis

My experiment data was collected using 6 tubes. Each tube had a different internal diameter. For each internal diameter, the height above the tube for which there was loudest sound was recorded. Since the equation for end correction is L=0.3d, it is similar to the straight line equation y=mx+c in which the L is the height above the resonance tube, gradient m = 0.3, x is the diameter and c, the y-intercept, is zero. According to the equation, it should be a straight line that goes through the origin. If that is true, then my values for L on the y-axis should give me a straight line graph through the origin. The gradient of the graph should be 0.3 or close to that, if everything in the experiment is done well and if the equation for end correction is in fact L=0.3d.

Background information

As is taught in the IB physics course, the equation to find end correction is relatively simple. The equation is as follows:

L=0.3d

Where L is equal to the end correction and d is equal to the diameter of the tube. This formula therefore suggests a linear relationship between the end correction of a tube and its diameter. This means that tube length, material from which the tube is made and sound frequency do not affect the end correction of the tube at all.

Using this equation, it can be modelled such that it fits into the linear equation of y=mx+c. In this case the gradient m would be equal to 0.3, c would be equal to 0, d against L.

Designing the experiment

For this experiment, the two main things that were taken into investigation were the diameter of the tubes and the end correction. Ideally, the frequency would have been investigated as well, but to keep the experiment constant and fair I used the same frequency throughout the experiment.

Variables

Apart from d being the independent variable and L being the dependent variable, all other variables such as equipment used, air temperature and pressure, etc. were assumed constant and therefore controlled variables.

Materials

3 open ended glass tubes and 3 open ended PVC tubes of varying diameter

Tuning fork of 512Hz

Body of water

Vernier calliper

Measuring ruler

Clamp stand

Slow-motion camera

Procedure

Firstly, the six open ended glass pipes were measured by the Vernier calliper and their diameters were recorded. Then, one was taken (smallest diameter first) and placed in the body of water so that in effect the length of the tube was maintained constantly at 20cm. This tube was held up by a clamp stand. A second clamp stand was put in place so that the measuring ruler in was held in place just above the end of the tube where the investigation would take place. Using the tuning fork of frequency 512Hz, the tuning fork gently struck the table so that a wave was emitted from the fork. A tuning fork was held about 2 centimetres above the end of the tube, the tuning fork was slowly brought down until a loud sound was heard. This was done for every tube three times to avoid random error.

Data & processing

Table of raw results for first experiment:

Tube diameter (m) Distance of resonance above tube (m)

Trial 1 Trial 2 Trial 3 Average ±0.001

0.020 ± 0.0005 0.005 0.006 0.006 0.00567

0.025 ± 0.0005 0.008 0.007 0.007 0.00734

0.030 ± 0.0005 0.010 0.010 0.008 0.00934

0.035 ± 0.0005 0.010 0.012 0.009 0.01034

0.040 ± 0.0005 0.013 0.012 0.012 0.01234

0.045 ± 0.0005 0.012 0.015 0.014 0.01367

Graphs of averages:

N.B: for an accurate presentation of the data, all values in the table above were converted to millimetres.

As shown on the graph below, the gradient m was equal to 0.32 and the y-intercept (c) was equal to -0.6167. This means, when applying the equation y=mx+c, the equation transforms into y=0.32x-0.6167. This is not equal to, but is roughly similar to the original end correction equation of L=0.3d+0. Here is a graph showing lines of best fit passing through error bars:

Error in slope:

(Maximum slope gradient-minimum slope gradient)/2

(0.4323-0.207)/2

Therefore the error in slope is equal to: 0.11265≈0.1

Sample uncertainties calculations:

To calculate error bars along the and uncertainties for the average end correction data the following formula was used: ∆L=∆d/d× L where ∆d=0.5 (as that was the uncertainty for the diameter calculated using the Vernier Calliper) and d was equal to the diameter in the first data set.

First uncertainty: 0.0005/0.02×0.00567=0.001425 ≈0.001

Second uncertainty: 0.0005/0.025×0.00734=0.001468≈0.001

Third uncertainty: 0.0005/0.03×0.00934=0.0015567≈0.001

Fourth uncertainty: 0.0005/0.035×0.01034=0.0014771 ≈0.001

Fifth uncertainty: 0.0005/0.04×0.01234=0.0015425 ≈0.001

Sixth uncertainty: 0.0005/0.045×0.01367=0.001519≈0.001

Conclusion

The original research question was whether the equation for end correction verifiable was via experiment. From the graphs shown above, the slope (gradient) of it was approximately 0.32 ±0.1. therefore the range of the correct values found for end correction L in my experiment ranged from 0.22d →0.42d. The true value for end correction L is known to be 0.3d, which falls perfectly in the range of my results. I can successfully conclude that end correction can be found and verified via experiment.

Evaluation

Reflecting on the experiment done, I think the data produced was accurate and precise for the all values. Ideally, in this experiment I would have also varied the frequency for each resonance experiment using a frequency emitter and the equation C=(v/2f-L)÷2 where L is the length of the tube, v the speed of sound in air, f the frequency and C being the end correction. This would have allowed me to better derive and explore the relationship between end correction and tube diameter but a frequency emitter was not available at the time so the experiment and the premise of the internal assessment had to be modified to allow the use of tuning forks instead.

Two problems I noticed with my experiment were

Getting the exact position and reading of the resonating tuning fork. This happened because the tuning fork was constantly vibrating so getting an exact reading was very difficult. This was hard to fix, but if can be improved by taking more measurements and using a higher quality camera and using an interface with sound input to pinpoint exactly where the resonance occurred and where the reading could take place.

There was a slight systematic error, as shown in the graph, the slope did not go through the origin but instead passed through -0.6. This could have been caused by a systematic error. More accurate data and better calibrated measuring tools could have prevented this.

When investigating the speed of sound in air, the measurement called ‘end correction’ is included to make the measurements more accurate. During IB physics HL lessons, our teacher did not show us how the end correction could be modelled as an experiment, or how the end correction of a pipe was found.

What was fascinating was that the end correction was absolute, and given as 0.3d where ‘d’ was the diameter of the pipe. I was not comfortable with this value of 0.3d as no proof or experiment was shown. This is when I decided that I wanted to investigate this relationship, and verify this value of 0.3d. I designed my own experiment by using several glass and PVC pipes of varying diameters. I carried out the same experiment one would use to find the speed of sound in air, using a resonance tube. My independent variable was the pipe diameter, and the dependent variable was the distance between the end of the resonance tube and the resonating tuning fork when there is maximum amplitude or when the sound was loudest. The research question I modelled my experiment around was ‘how can L=0.3d be modelled as a linear equation y=mx+c?

Hypothesis

My experiment data was collected using 6 tubes. Each tube had a different internal diameter. For each internal diameter, the height above the tube for which there was loudest sound was recorded. Since the equation for end correction is L=0.3d, it is similar to the straight line equation y=mx+c in which the L is the height above the resonance tube, gradient m = 0.3, x is the diameter and c, the y-intercept, is zero. According to the equation, it should be a straight line that goes through the origin. If that is true, then my values for L on the y-axis should give me a straight line graph through the origin. The gradient of the graph should be 0.3 or close to that, if everything in the experiment is done well and if the equation for end correction is in fact L=0.3d.

Background information

As is taught in the IB physics course, the equation to find end correction is relatively simple. The equation is as follows:

L=0.3d

Where L is equal to the end correction and d is equal to the diameter of the tube. This formula therefore suggests a linear relationship between the end correction of a tube and its diameter. This means that tube length, material from which the tube is made and sound frequency do not affect the end correction of the tube at all.

Using this equation, it can be modelled such that it fits into the linear equation of y=mx+c. In this case the gradient m would be equal to 0.3, c would be equal to 0, d against L.

Designing the experiment

For this experiment, the two main things that were taken into investigation were the diameter of the tubes and the end correction. Ideally, the frequency would have been investigated as well, but to keep the experiment constant and fair I used the same frequency throughout the experiment.

Variables

Apart from d being the independent variable and L being the dependent variable, all other variables such as equipment used, air temperature and pressure, etc. were assumed constant and therefore controlled variables.

Materials

3 open ended glass tubes and 3 open ended PVC tubes of varying diameter

Tuning fork of 512Hz

Body of water

Vernier calliper

Measuring ruler

Clamp stand

Slow-motion camera

Procedure

Firstly, the six open ended glass pipes were measured by the Vernier calliper and their diameters were recorded. Then, one was taken (smallest diameter first) and placed in the body of water so that in effect the length of the tube was maintained constantly at 20cm. This tube was held up by a clamp stand. A second clamp stand was put in place so that the measuring ruler in was held in place just above the end of the tube where the investigation would take place. Using the tuning fork of frequency 512Hz, the tuning fork gently struck the table so that a wave was emitted from the fork. A tuning fork was held about 2 centimetres above the end of the tube, the tuning fork was slowly brought down until a loud sound was heard. This was done for every tube three times to avoid random error.

Data & processing

Table of raw results for first experiment:

Tube diameter (m) Distance of resonance above tube (m)

Trial 1 Trial 2 Trial 3 Average ±0.001

0.020 ± 0.0005 0.005 0.006 0.006 0.00567

0.025 ± 0.0005 0.008 0.007 0.007 0.00734

0.030 ± 0.0005 0.010 0.010 0.008 0.00934

0.035 ± 0.0005 0.010 0.012 0.009 0.01034

0.040 ± 0.0005 0.013 0.012 0.012 0.01234

0.045 ± 0.0005 0.012 0.015 0.014 0.01367

Graphs of averages:

N.B: for an accurate presentation of the data, all values in the table above were converted to millimetres.

As shown on the graph below, the gradient m was equal to 0.32 and the y-intercept (c) was equal to -0.6167. This means, when applying the equation y=mx+c, the equation transforms into y=0.32x-0.6167. This is not equal to, but is roughly similar to the original end correction equation of L=0.3d+0. Here is a graph showing lines of best fit passing through error bars:

Error in slope:

(Maximum slope gradient-minimum slope gradient)/2

(0.4323-0.207)/2

Therefore the error in slope is equal to: 0.11265≈0.1

Sample uncertainties calculations:

To calculate error bars along the and uncertainties for the average end correction data the following formula was used: ∆L=∆d/d× L where ∆d=0.5 (as that was the uncertainty for the diameter calculated using the Vernier Calliper) and d was equal to the diameter in the first data set.

First uncertainty: 0.0005/0.02×0.00567=0.001425 ≈0.001

Second uncertainty: 0.0005/0.025×0.00734=0.001468≈0.001

Third uncertainty: 0.0005/0.03×0.00934=0.0015567≈0.001

Fourth uncertainty: 0.0005/0.035×0.01034=0.0014771 ≈0.001

Fifth uncertainty: 0.0005/0.04×0.01234=0.0015425 ≈0.001

Sixth uncertainty: 0.0005/0.045×0.01367=0.001519≈0.001

Conclusion

The original research question was whether the equation for end correction verifiable was via experiment. From the graphs shown above, the slope (gradient) of it was approximately 0.32 ±0.1. therefore the range of the correct values found for end correction L in my experiment ranged from 0.22d →0.42d. The true value for end correction L is known to be 0.3d, which falls perfectly in the range of my results. I can successfully conclude that end correction can be found and verified via experiment.

Evaluation

Reflecting on the experiment done, I think the data produced was accurate and precise for the all values. Ideally, in this experiment I would have also varied the frequency for each resonance experiment using a frequency emitter and the equation C=(v/2f-L)÷2 where L is the length of the tube, v the speed of sound in air, f the frequency and C being the end correction. This would have allowed me to better derive and explore the relationship between end correction and tube diameter but a frequency emitter was not available at the time so the experiment and the premise of the internal assessment had to be modified to allow the use of tuning forks instead.

Two problems I noticed with my experiment were

Getting the exact position and reading of the resonating tuning fork. This happened because the tuning fork was constantly vibrating so getting an exact reading was very difficult. This was hard to fix, but if can be improved by taking more measurements and using a higher quality camera and using an interface with sound input to pinpoint exactly where the resonance occurred and where the reading could take place.

There was a slight systematic error, as shown in the graph, the slope did not go through the origin but instead passed through -0.6. This could have been caused by a systematic error. More accurate data and better calibrated measuring tools could have prevented this.