How can the relationship between the velocity of a buzzer, varied between 3.3 and 10.1 ms^{-1}, and its perceived frequency, measured between 3046 and 3233 Hz, be used to calculate the speed of sound?

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The Doppler Effect is a fascinating physical phenomenon which occurs in everyday life, for example when hearing the sound of a police car’s siren approaching and then receding. Similarly, circular motion is an area of Physics which I see regularly in action, for example with a car turning on a roundabout. For me, the topics I learn in Physics which have a real-life application I can see are the most interesting, because they show how Physics is not just some abstract science, but it has practical value.

I had experienced the Doppler Effect many times in my life without knowing about it, and now that I have learnt what it is all about, I would like to further my knowledge of it in the laboratory.

I chose to explore a method involving both the Doppler Effect and circular motion because, being an aspiring engineer, I would like to develop my understanding of these fundamental concepts in Physics; specifically, the method I chose allows me to model the work of an engineer in a safe yet very useful way.

The Doppler Effect is the change in the observed frequency of a wave which happens when there is relative motion between the source of the wave and the observer of the wave.

Consider a source emitting a sound wave with frequency *f*. Then, the maxima of the amplitude of the sound wave occur at intervals given by the period \(T=\frac{1}{f}\) . If the source is not moving, an observer would perceive the maxima spaced by time intervals equal to *T*. Let the maxima be separated by a wavelength *λ = Tv*, where *v* is the speed of sound.

We now consider, instead, the source moving with a velocity *u _{s}* . After emitting one maximum, it would cover a distance

The second maximum will be perceived by the source as having a frequency

\(ftowards\, =(\frac{v}{v-u_s})f\) (1)

or

\(λtowards\, =(1-\frac{u_s}{v})\lambda\) (2)

This formula shows that if the source is approaching, the perceived frequency is increased by a factor of \((\frac{v}{v-u_s})\). If the moving source were travelling away from a stationary observer with velocity *u _{s}*, the distance between consecutive maxima would be

The perceived frequency is given by

\(faway = (\frac{v}{v+u_s})\) (3)

or

* **\(λaway =(1+\frac{u_s}{v})\lambda\)** *(4)

This formula shows that if the source is receding, the perceived frequency decreases by a factor of \((\frac{v}{v+u_s})\).

Since the buzzer will be moving in circular motion, it is crucial to examine some fundamental principles, to better understand its motion. The buzzer is assumed to be a point particle; the buzzer is being swung in circular motion by a rope, which will be considered to be a one-dimensional line.

If a line changes orientation with time, suppose that it rotates through an angle Δ*θ* in a time Δ*t*. Then, its average angular velocity ω is defined to be

\(\omega=\frac{\Delta\theta}{\Delta t}\) (5)

ω is a quantity which represents the number of radians swept by the line per unit time.

The period T for one revolution of the line is the time interval taken by it to sweep an angle of *2π* radians. Thus, rearranging equation (5), *T = 2π ω*. The frequency *f* of rotation is given by \(f=\frac{1}{T}\), therefore *ω = 2πf*.

Let the buzzer, which is a point particle, be performing circular motion at a radius *r*. It will travel a distance *2πr* in a time *T* with speed *u _{s}*, hence \(T=\frac{2\pi r}{u_s}\), and

*u _{s} = ωr *(6)

In order for a body to perform circular motion, a force, known as a centripetal force, must be applied to it. This must mean that there is a radial acceleration. This is calculated as shown below.

The two triangles are similar, therefore

\(\frac{|\Delta v|}{v}=\frac{|\Delta r|}{r}\) (7)

Dividing equation (7) by Δ*t* gives-

\(\frac{1}{v}|\frac{\Delta v}{\Delta t}|=\frac{1}{r}|\frac{\Delta r}{\Delta t}|\) (8)

Therefore, \(\frac{|a|}{v}=\frac{|v|}{r}\), and

\(a=\frac{v^2}{r}=\omega^2r\) (9)

This acceleration is not uniform because it is constant in magnitude but not in direction. The direction of the acceleration of the buzzer is directed radially inwards, towards the centre of the circle its motion describes. (In this last section of Background Information, *u _{s}* has been replaced by

As the velocity of the buzzer (the moving source) increases, the frequency I will perceive as a stationary observer will depend upon whether it is coming towards me or away from me.

This can be shown mathematically from equations

\(f^′towards = (\frac{v}{v-u_s})f\) (10)

\(f^′away = (\frac{v}{v+u_s})f\) (11)

Equation (10) shows that if the buzzer is coming towards me and its velocity, *u _{s}* , increases, I will perceive a higher frequency. If its velocity decreases, I will perceive a lower frequency. On the other hand, equation (11) shows that if the buzzer is going away from me and its velocity increases, I will perceive a lower frequency, and if its velocity decreases, I will perceive a higher frequency. However, I predict that although these variables will have a linear relationship, they are not directly (or inversely, depending on the case) proportional, because there will be a

I also predict that as the buzzer swings, the radius of its motion will stay practically constant; similarly, its speed should stay constant.

**Dependent variable**

The perceived frequency of the sound coming from the buzzer. This was measured using a combination of equipment including Audacity. This variable was measured twice, first for the perceived frequency when the buzzer was approaching the laptop and then for when it was receding from it. The measurement on Audacity involved using a spectrogram to clearly see the frequency of the buzzer. The spectrogram showed the frequency in bands with difference of ±5 Hz, hence, since it is an analogue measurement, the uncertainty on all perceived frequencies is set to be Δ*f′* = ± 2 Hz.

**Independent variable**

The velocity of the buzzer. It has been shown that the perceived frequency depends upon the velocity, hence by changing the velocity a range of frequencies can be observed. The velocity was changed manually by swinging the rope to which the buzzer was tied at faster or slower speeds; the source of the emitted sound waves was a buzzer which will move in circular motion. There were five different velocities, with five different trials for each velocity. The velocity of the buzzer was calculated using the formula discussed in *Background Information,* \(u_s=\frac{2\pi R}{T}\). The uncertainty on the velocity of the buzzer is variable, since it depends upon the perceived frequency.

**Control variables**

- The same buzzer was used throughout the experiment. This was controlled because different buzzers might have had different natural frequencies, therefore by using more than one buzzer the results would have been inconsistent-
- Only Audacity was used as a means of recording the sound of the buzzer, because each different piece of technology might have displayed the frequencies of the sound emitted by the buzzer with different uncertainties and thus make the experiment inconsistent; for the same reason, the same timer was used to measure the time period.
- The same spherical container was used for the buzzer. This is because other spherical containers might have been too small, and therefore the buzzer, while swinging, could have collided with its container, and Audacity would then have picked up a sound irrelevant to this experiment.
- The stationary observer was the same person, since different persons might have swung the buzzer in different ways.
- The experiment was conducted in the same location throughout; this was important because different locations might have been at different temperatures or be subject to different levels of background noise, and it was fundamental that Audacity exclusively recorded the sound of the buzzer. The temperature in the room was measured at 15°C using a digital thermometer. Temperature was an important factor in this experiment because it contributed to the literature value for the speed of sound, which varies with temperature.
- Even though, strictly speaking, this is not a control variable, the buzzer was swung using the same

length of rope, or otherwise the value of*R*used in the formula to calculate its speed would be

inconsistent. In the experiment,*R*= 1 m, measured using a meter stick, was chosen for simplicity,

keeping calculations tidy.

- My iPad, on which several videos of myself while spinning the buzzer were taken. These were later used to calculate the time for one rotation of the buzzer at different velocities;
- Audacity, an audio recording program available on my laptop which was used to visualise the change in frequency of the buzzer when it was receding from me and when it was coming towards me; this program worked like a microphone for recording the sound emitted from the buzzer;
- A piezoelectric buzzer: this acted as the moving source; furthermore, the buzzer was equipped with a button which when pressed produced the sound that was picked up by Audacity;
- A hollow sphere with some holes on its surface; it was used to put the buzzer inside of it (the sphere can be opened up in half), and a rope was tied to one of its holes with a knot;
- A 2.48 m red rope made of nylon, to which the buzzer was tied. A rope was used to allow the buzzer to perform circular motion and hence change its velocity, since we will assume that the buzzer moves in a perfect circle. I swung the rope.
- Digital thermometer, used to measure the temperature of the room in which the experiment was performed, in order to get a value for the speed of sound. The uncertainty on the thermometer recording is Δ
*T*= ± 0.1 °C; - A ruler used to measure the length of the rope. Its uncertainty is Δ
*l*= ± 0.5 mm; - The timer on my smartphone. This was used to measure the time period of one rotation of the buzzer in seconds. The uncertainty on the time period is Δ
*T*= ± 0.01 s.

**Obtaining values for the perceived frequencies**

- It is important to underline that this experiment was conducted in a quiet place, away from as many sources of sound as possible.
- A 2.48 m rope made of nylon was tied to a sphere. The buzzer was encapsulated by and attached to a sphere with some holes in it, to which the rope was tied with a knot.
- The laptop was placed with Audacity open at a close distance from myself of around 2 m.
- Audacity was set to record.
- The buzzer was set to go.
- The buzzer performed circular motion. The buzzer was swung for around 30 seconds.
- The perceived frequency of the signal rose and fell as the buzzer first approached and then receded from the laptop, respectively
- The buzzer was turned off by pushing on the button again.
- Audacity was stopped from recording.
- Steps 4–9 were repeated other four times, swinging the buzzer with approximately the same speed as in the first trial; this gives a total of five trials for each speed
- Steps 4–10 were repeated other four times spinning the buzzer at different speeds, ranging from very slow to very fast (relative to me). This gave five different speeds with five trials for each.

**Obtaining values for the time period of the buzzer’s motion**

- The buzzer was handled from the string attached to it, which was reduced to a length of 1 m. This was the radius of the buzzer’s motion.
- The iPad was set to record.
- The buzzer swung, performing circular motion. This continued until a steady rhythm was reached.
- Step 3 was repeated other four times, each time increasing the speed of the buzzer.
- The Timer on a smartphone was used to measure the time period by looking back at the video.

**The recording on Audacity, done on a laptop, and the video recording, done on a tablet, were performed simultaneously.**

Hazard | Risk | Control |
---|---|---|

Rotating sphere and buzzer | Hurting myself, for example due to the sphere hitting my face. | Handle the buzzer with care. Do not swing the buzzer with excessive velocity. Be sure to have a solid grip on the rope. |

Sound waves | Damage to the ear due to the high frequencies of sound used. | Do not stand very close to the sound of the waves. Cover your ear if necessary; this can reduce the chances of being hurt by a shock wave produced by the high frequency sounds. |

Water, especially if in contact with electricity | Electrocution; slipping on the floor. | Ensure that any sources of water are far away from the electrical equipment. If water is on the floor, dry the floor using paper towels. |

General obstacles and obstructions, such as bags on the floor. | Slipping and hurting myself. | Ensure that the experiment is being conducted with a free room for moving myself. Bags, stools, and other obstructions must be placed under the desk or, better, must not be there during the experiment. |

The natural frequency of the piezoelectric buzzer was calculated by performing six trials during which the buzzer was placed at a distance of approximately 2 m from the laptop, with Audacity recording the frequency. The results were then averaged to get an average natural frequency, f-

\(f=\frac{3144+3139+3139+3138+3138+3138}{6}=3139Hz\)

The uncertainty in f is found by considering the range of values obtained, and halving this value-

\( \Delta f'_{towards}=\frac{3144-3138}{2}=\frac{6}{2}=\pm3Hz\)

However, there is a better method which was used for the uncertainty in all the perceived frequencies (both when approaching and when receding); considering the fact that the perceived frequency was measured using the spectrogram mode on Audacity, it is known that the difference between the frequencies that I could detect with my eyes was of ± 5 *Hz*. Since the scale used on the Audacity program is an analogue one, the uncertainty in the frequency is half of this, and, using the odd-even rounding rule-

Δ*f' _{towards}* = Δ

The following example shows how the velocity of the buzzer was calculated: \(u_s=\frac{2\pi }{T}\), where *R* is the radius of the circle described by the motion of the buzzer, in this experiment 1 m, and *T* is the time period for one revolution. For the lowest speed buzzer: \(u_s=\frac{2\pi}{1.88}=3.34ms^{-1}\)

When finding the uncertainty on the velocity of the moving source, i.e. the buzzer, the formula used for finding it must be considered: \(u_s=\frac{2\pi R}{T}\)

The formula used for this is-

\(\Delta u_s=u_s\sqrt{(\frac{\Delta R}{R})^2+(\frac{\Delta T}{T})^2}\)

Considering the results obtained for *u _{s}* = 3.34

\(\Delta u_s=3.34\sqrt{(\frac{0.5.10^{-3}}{1})^2+(\frac{0.01}{1.88})^2}=\pm0.02ms^{-1}\)

To better understand the meaning of the uncertainty on the gradient and on the y–intercept, it is useful to refer back to the equations used in this experiment.

In particular, \(f'=\frac{fv}{v-u_s}\). After some rearrangement: \(f'=\frac{fv}{v(1-\frac{u_s}{v})}\). Using the Taylor series up to the first order for this expression gives an approximation of the relationship between *f′* and *u _{s}*-

\(f'\approx f+\frac{fu_s}{v}\). Therefore, this formula is now in the form *y = mx + c*, where \(\frac{f}{v}\) is the slope of the graph and *f* is its y–intercept.

The uncertainty on the gradient for the experiment was calculated using the formula-

\(\Delta m= \frac{m_{max}-m_{min}}{2}\)

Substituting the values for the approaching buzzer-

\(\Delta m=\frac{4.0835-3.1625}{2}=\pm0.5m^{-1}\)

Therefore, *m* = (4.0 ± 0.5) *m*^{-1}.

Similarly, for the receding buzzer-

\(\Delta m=\frac{-4.3436-(-5.1543)}{2}=\pm0.4m^{-1}\)

Thus, *m* = (−5.0 ± 0.4) *m*^{-1}.

The uncertainty on the y–intercept is obtained in a similar way-

\(\Delta c=\frac{c_{max}-c_{min}}{2}\)

For the approaching buzzer-

\(\Delta c=\frac{3199.6-3195.2}{2}=\pm2Hz\)

Therefore,*c* = (3197 ± 2) *Hz*.

For the receding buzzer-

\(\Delta c=\frac{3136.4-3079.1}{2}=\pm3Hz\)

So,

*c* = (3090 ± 3) *Hz*.

The speed of sound can be shown to be calculated as follows: \(v=\sqrt{\frac{\gamma RT}{M}}\), where γ is the adiabatic constant, which, in air: *γ* = 1.4, R is the universal gas constant, equal to 8.31 *J* *mol*^{-1}*K*^{-1}, *M* is the molecular mass of gas, which for air: *M* = 0.02895 *kg* *mol*^{-1} and *T* is the temperature in Kelvin.

Substituting the values-

\(v=\sqrt{\frac{1.4×8.31×288}{0.02895}}=340ms^{-1}\)

Referring to the Taylor expansion used above, the speed of sound v can be calculated by using the formula-

\(v=\frac{f}{slope}\)

For the approaching buzzer,

\(v=\frac{3139}{4.0}=340ms^{-1}\)

For the receding buzzer,

\(v=\frac{3139}{5.0}=600ms^{-1}\)

The uncertainty in the experimental values of the speed of sound was calculated using the formula-

\(\Delta v=\sqrt{(\frac{\Delta f}{f})^2=(\frac{\Delta slope}{slope})^2}\)

For the approaching buzzer,

\(\Delta v=800×\sqrt{(\frac{3}{3139})^2+(\frac{0.5}{4})^2}=\pm100ms^{-1}\)

For the receding buzzer,

\(\Delta v=600×\sqrt{(\frac{3}{3139})^2+(\frac{0.4}{-5})^2}=\pm ms^{-1}\)

Considering the data obtained for the approaching buzzer, the first piece of information which can be established is that as the velocity of the buzzer increased, the perceived frequency also increased. Conversely, for when the buzzer was receding, it has been established that as the buzzer’s velocity increased the perceived frequency decreased.

However, velocity and perceived frequency (in both cases) are not directly (or inversely) proportional, because there is quite a big y–intercept for both graphs, approximately correspondent to the natural frequency of the buzzer. This is because it is not possible to fit a straight line which passes through all error bars and through the origin.

The uncertainties in my experiment came from a combination of random and systematic errors. A random error in this experiment could be the fact that the velocity of the buzzer was not constant throughout one whole revolution. This could have been because although the rope was swung at a length of 1 m, it is possible that its length may have changed during the experiment, as perhaps my grip on the rope changed. This would have had various effects on the relationship between perceived frequency and velocity, depending on the case analysed; however, it was found that the uncertainty on the velocity is not very big at all.

This leads to an analysis of the uncertainty of the gradient, which is quite significant compared to the values obtained. Indeed, it can be seen from the visual display of data that the trendline passes through all error bars for only one graph, indicating that at least this type of error was not very significant to the outcome of the investigation, but for the receding buzzer, the trendline does not pass through all error bars. Indeed, the errors associated with this part of the experiment are generally bigger. This might have been because when swinging the buzzer, I found it easier to swing when in front of me rather than when it was behind me.

From the data analysis above, it can be seen that the value obtained for the speed of sound considering the gradient of the graph is at more than one experimental uncertainty away from the accepted value. One factor affecting this is the inaccurate value of the natural frequency (y −intercept), for both cases with the approaching and receding buzzer; indeed, the natural frequency *f* = 3139 *Hz*, and the approaching frequency = 3197 Hz, while the receding frequency = 3090 Hz. As can be seen from the data analysis above, the experimental uncertainties for both figures do not overlap with the natural frequency obtained by using Audacity.

This is quite a big error, which shows that although it is possible to combine the gradient of the graph of *f′* against *u _{s}*, this method does not ensure great accuracy when determining the speed of sound; this is further reinforced by the fact that although the uncertainties on the gradients of both graphs are noticeable, they do not, however, overlap with the accepted values, and by the fact that the linear relationship between the two variables is only approximate, therefore the experimental value of the speed of sound constant could be much closer or much farther from the accepted value.

The above analysis strongly suggests that my hypothesis was correct; I was able to correctly predict the relationship and correlation between my variables. However, I feel like to make an even stronger hypothesis I could have used a more convincing mathematical argument in support of the correlation between the velocity of the buzzer and the perceived frequency. This could have been done by using the technique of Maclaurin expansion to higher orders.

In detail; we consider the geometric series \(\frac{1}{1-x}\), where for this instance \(x=\frac{u_s}{v}\). This is a function which has Maclaurin series: \(\Sigma^\infty_{n=0}x^n\). In this case, this expression is multiplied throughout by *f*. This would yield the expression -

\(f'=f((\frac{u_s}{v})^0+(\frac{u_s}{v})^1+(\frac{u_s}{v})^2+(\frac{u_s}{v}+(\frac{u_s}{v})^3+...+(\frac{u_s}{v})^\infty)⟹f'=f+\frac{fu_s}{v}+\frac{fu_s^2}{v^2}+\frac{fu^3_s}{v}+...+(\frac{u_s}{v})^\infty \). A Maclaurin series expression taken to higher orders would have been better because it

considers more terms, and thus should produce more accurate results.

The procedure I followed also has its merits. First of all, having two different methods of looking at the perceived frequency (when the buzzer was approaching the laptop and when it was receding) was an advantage because it allowed for comparison of the method and shed more light on the errors involved in this kind of experiment. Moreover, I believe that taking five trials for each velocity, and having five different velocities, improved my method because this allowed for a substantial amount of data to be used to produce graphs and it reduced random errors, and indeed there don’t seem to have emerged anomalies from the data collected. Other strong points of the method I used include the fact that although it had some potential for risks, I didn’t find any risks for my safety throughout the experiment, since it was easy to put into practice the controls.

Nevertheless, the method used suffered from a lot of weaknesses. First of all, something which must be considered is that there is some ambiguity in this experiment. It was chosen to select the reference point of approach to be the laptop, however, this could have been any other point; furthermore, the way in which the buzzer was swung is also quite ambiguous, and indeed there could be various interpretations as to how the buzzer should be swung, for example by keeping my arm still and only allowing my wrist to move. It is also not clear what radius should be used for the swinging of the buzzer, although possibly a quite large radius, such as that used in this experiment, could be considered to be suitable. Another point of ambiguity arises from the fact that it was not specified in my method the path which should be followed by the buzzer– for example, it is unclear whether it would have been better for me to swing it above my head or around me; in the latter case, the method would have changed because I would have had to rotate about myself to follow the buzzer’s motion.

Considering the limitations of the method, four fundamental aspects emerge. Firstly, there were some slight problems with the choice of setting, because sometimes there were background noises such as voices of people, and although it is very difficult to determine whether Audacity picked up on these or not, certainly it is difficult to remove them and, if present, they would have skewed the results of the frequencies upwards.

Another important point to be made is in regard to the quality of the sound emitted by the buzzer; this is because it was noticed that after some time (around 30 seconds or less) the sound emitted by the buzzer started changing, and of course, emitting a different frequency, the buzzer would have made the results inconsistent.

Also, the software used to measure the frequencies had its limitations, because although the frequency could be zoomed in quite far, in practice it was difficult to distinguish between differences in frequency of 1 Hz. This means that there were human error resulting from my inability to always correctly read the frequency of the buzzer from the spectrogram on Audacity.

A final consideration must be given to my limitations when looking back at the video of me swinging the buzzer on my iPad. It was very difficult to find a correct value for the time period, since it was difficult to start and stop the video at points where the buzzer would exactly be in the same position as when it started. This error could have been reduced by taking more trials for each velocity.

In terms of possible alternatives to this method, a different source of sound with a higher or lower natural frequency could be used, to extend the results obtained from this experiment and to confirm or disprove the validity of the results obtained for this method and whether the trends established are true for different frequencies. Furthermore, the equipment could be changed, for example by using a protective capsule made of some softer material, such as rubber, which would reduce the risks of injury due to the sphere hitting the stationary observer.

Another possible change to the equipment could involve measuring the perceived frequency using the “Analyse → Plot Spectrum →Frequency Analysis” functions on Audacity and selecting the points of minimum and maximum frequency; then the perceived frequency would be the peak of the selected areas. It could be useful to compare these two methods, and ultimately reach a conclusion on which of the two yields the better results.

Something else which could be explored is the effect of the radius of the circle (in which the moving source is moving) on its velocity, and hence on the perceived frequency both when approaching and when receding. This method would involve a different method, possibly with a more resistant rope: the buzzer could be made to swing in various radii but with the same speed throughout the whole experiment; its velocity of approach and of recession would then be affected to some degree, and this would be the focus of such an experiment.

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- 11.5: Taylor Series. University of Berkeley, math.berkeley.edu/~scanlon/m16bs04/ln/16b2lec27.pdf.
- A Summary of Error Propagation. Harvard University, 2007, ipl.physics.harvard.edu/wp- uploads/2013/03/PS3_Error_Propagation_sp13.pdf.
- Chattopadhyay, D., and P. S. Rakshit. Electronics: Fundamentals and Applications. New Age Publishers, 2006.
- Cheung, Dr. A.C.H. Circular Motion. Trinity College, Cambridge, www.tcm.phy.cam.ac.uk/~achc2/circ2010.pdf.
- “Development of the Adiabatic Condition.” Total Internal Reflection, hyperphysics.phy- astr.gsu.edu/hbase/thermo/adiabc.html#c1.
- Doppler Effect.” Centripetal Force, hyperphysics.phy-astr.gsu.edu/hbase/Sound/dopp.html.
- General Dictionary - Optical Laws: Refraction and Reflection, spaceguard.rm.iasf.cnr.it/NScience/neo/dictionary/kepler.htm.
- Low Pass Filter- Explained, www.learningaboutelectronics.com/Articles/R-squared- calculator.php#answer.
- Schwartz, Matthew. “Lecture 21: The Doppler Effect.”
- “Sound Speed in Gases.” Total Internal Reflection, hyperphysics.phy- astr.gsu.edu/hbase/Sound/souspe3.html.
- TR52.00.01-2006_Recommended_Environments_for_Standards_Laboratories.pdf.
- Tsokos, K. Andreas. Physics for the IB Diploma. Cambridge University Press, 2014.
- US Department of Commerce, and NOAA. “Speed of Sound Calculator.” National Weather Service, NOAA's National Weather Service, 4 Aug. 2018, www.weather.gov/epz/wxcalc_speedofsound.