Experimental Determination of the Speed of Sound in Air

Introduction

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Sounds are vibrations that spread as waves and are carried through mediums, such as, solids, liquids or gasses, this idea was first established by Sir Isaac Newton. Frequency is the number of waves or complete revolutions that happen in one second (Vorderman,2012). The aim of this experiment is finding the various wavelengths at which resonance occurs when changes in the tuning forks are taking place to find the speed of sound and the end correction. In this experiment the tuning forks will be changing to determine the different wavelengths at different frequencies to make it a fair test.

Therefore, the data found can result in finding the speed of sound by using the wavelength equation.

Theory

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Sound is produced due to having waves that are in contact with either solids, liquids, or gas. Sound is a prime example of longitudinal waves (International AAS Level). According to the editors of CGP (2018), a wave in which the vibrations are in the direction of spreading energy.

The speed of sound is determined using the following equation:

λ (unit: m/s)

Equation 1: Wavelength Equation

Where (λ) is the speed of sound or wave speed and is calculated when multiplying the frequency (f) in hertz and wavelength (λ) in meters. A wavelength (λ) is the distance between two crests or the length of an entire or complete wave cycle (Vorderman, 2012). The equation used to find the wavelength when the length of the fundamental harmonic is known, the following equation shown below is used where the length is multiplied by 4 to find the wavelength (λ).

λ = 4L (units: m)

Equation 2: Length to Wavelength Equation

Frequency is calculated through the number of wave cycles per second. In this experiment, the frequency is constant depending on the tuning fork that is being used. Resonance is the sound that radiates from an object when it vibrates at the same frequency rate as another object (CGP, 2018). This experiment will have an end correction (c) that will need to be calculated due to resonance and how it will create a standing wave, the wave’s antinode is hypothesized to be at the open end of the pipe but it is, in fact, at the outside of the pipe (small distance outside the pipe). The end correction (c) can be determined using the velocity as shown in the following equation:

v/4f = L + C (units of C: m)

Equation 3: End Correction Equation

The wavelength data that were taken are only from the first harmonic. Some of the assumptions that have been made are: the water inside the graduated cylinder was at ~25 degrees Celsius; the graduated cylinder was held still by the stand, boss, and clamp and was at a 90-degree angle; if the experiment was done in a room with no outside noise-influence it will have better and accurate data; the wavelength of the sound wave for each harmonic should be constant.

Once the results of the experiment had been recorded, they were plotted as a wavelength/period graph where the wavelength was on the y-axis and the period on the x-axis. The period can be found using the equation T = 1/f or the inverse of the frequency. Once the results are plotted a line of best fit is drawn, and an equation can be derived with its gradient; the gradient found is the speed of sound.

Health and Safety:

Health and safety percussions had been taken, such as, paying attention to the amount of water that is in the graduated cylinder and avoiding overload to avoid spilling, making sure that there are no cracks in the graduated cylinder or any of the pipes to also avoid spillage which can lead to someone slipping causing serious injuries, beware of any sharp edges of the pipe or graduated cylinder to avoid slicing the finger, avoid dropping any of the tuning forks on the ground because it is made of metal and can be dangerous if it landed on the foot. This experiment does not use any harmful substances or liquids but paying attention is a necessity.

Apparatus:

Table 1: Apparatus Table

Item	Variable	Reason	Precision
Stand	N/A	To hold up the equipment and balance them.	-
Utility Clamp	Holder	Holding the graduated cylinder.	-
Boss	Holder	Balancing and holding the utility clamp and kept at a certain length.	-
Graduated Cylinder	Volume	It is filled with a specific volume of water.	-
Resonance Pipe	Volume	It is filled with part of the water in the graduated cylinder.	-
Tuning Forks	Frequency	Has different frequencies.	(+/- 1Hz)
Meter Ruler	Length	To measure the length where resonance occurs.	(+/- 0.05cm)

Method:

A graduated cylinder was placed on a stand and within the cylinder is a resonance pipe. The graduated cylinder was hooked to a utility clamp which is part of the boss. Once the equipment’s were balanced and well put water was added to 70~80% of the graduated cylinder. Furthermore, a tuning fork was used by hitting it at the end of a table or a special material causing a vibration. The tuning fork was then placed on the top of the resonance pipe while the pipe was moved up and down depending on how strong the sound was. Once the highest resonance was found the point where the sound resonated loudest was held still for the length to be measured using a meter ruler from the top of the pipe till the point where highest resonance occurred. The same process can be repeated by pulling the pipe higher to find the other harmonics. The process is then repeated for the same harmonics 3 times to find an average length. The length is then converted from centimeters to meters, then multiplied by 4 to find the wavelength. The number on the tuning fork is then used to find the speed of the sound wave by multiplying the tuning fork’s frequency with the wavelength. The process is repeated for different tuning forks.

Results:

Table 2: Results Table

Frequency	Period	Length	L1	L2	L3	L(AVG)	Uncertainty	Wavelength	Speed
256Hz	1/256s	32.5cm	32.8cm	32.6cm	32.63cm	+/-0.05cm	130.52cm	334.08m/s
	288Hz	1/288s	30.2cm	30.5cm	30.3cm				30.3cm
	320Hz	1/320s	26.1cm	26.0cm	26.4cm				26.17cm
	341.3Hz	1/341.3s	24.5cm	24.1cm	25.2cm				24.6cm
	384Hz	1/384s	22.3cm	21.2cm	21.9cm				21.8cm
	425.5Hz	1/425.5s	19.2cm	19.0cm	19.5cm				19.23cm
	480Hz	1/480s	17.1cm	17.5cm	17.7cm				17.43cm
512Hz	1/512s	16.4cm	16.4cm	16.3cm	16.37cm	+/-0.05cm	65.4cm	334.8m/s

The uncertainty of the wavelength is found by multiplying the uncertainty of the ruler by 4. 4*0.05cm= 0.2cm.

Table 3: Results in Coordinates

Wavelength (y-axis)	Period (x-axis)
130.52cm	1/256s
121.2cm	1/288s
104.68cm	1/320s
98.4cm	1/341.3s
87.2cm	1/384s
76.92cm	1/425.5s
69.72cm	1/480s
65.4cm	1/512s

Analysis and Errors

In this experiment, we are using the result table to find the speed of sound.

As shown in the first column of the result table, the tuning fork used was at 256Hz. Assuming that there are no uncertainties in any of the tuning forks, the highest resonance found in the fundamental or 1st harmonic was at 32.63cm, which is the average of the first 3 measurements of the lengths taken. To find the wavelength, the equation 2 is used:

λ = 4L

Calculation:

λ = 4(32.63) = 130.52cm

The final calculation was later converted from centimeters to meters.

λ = 130.52cm = 1.3052m

After the wavelength is found, it is inserted into equation 1 to find the speed of sound.

Speed of Sound = 256 * 1.3052 = 334.1312m/s

The underlined number is the speed that was found using the numbers within the first column of the result table. However, uncertainties were found within the ruler (+/- 0.1cm) which can alter the final result by a small fraction.

Finding the End Correction:

There are two ways to find the end correction: using the graph and using the end correction equation. In this experiment, we use the second one.

The end correction was found using the following equation:

End Correction Equation:

c = L - (λ/4)

Calculation:

Rearrange to find the end correction (c):

c = L - (1.3052m/4) = L - 0.3263m

Uncertainties:

To find the uncertainties of the wavelength and the frequency, equation 6 is used:

Δλ = λ * (ΔL/L)

The average wavelength was found:

λ(AVG) = (130.52cm + 121.2cm + 104.68cm + 98.4cm + 87.2cm + 76.92cm + 69.72cm + 65.4cm) / 8 = 95.316cm = 0.95316m

The Uncertainty of the wavelength was found:

Δλ = 0.1cm = 0.001m (for each measurement)

Following, the average frequency was found:

f(AVG) = 1 / ((1/256s + 1/288s + 1/320s + 1/341.3s + 1/384s + 1/425.5s + 1/480s + 1/512s) / 8) = 367.4Hz

The uncertainty of the frequency was found:

Δf = 1Hz (for each measurement)

This led to finding the uncertainty of the velocity:

Δv = (Δλ/λ) * v + (Δf/f) * v = (0.001m/0.95316m) * 334.1312m/s + (1Hz/367.4Hz) * 334.1312m/s = 3.52m/s

Error:

The error % of the average can be calculated using equation 7:

Equation 7: Error Percentage Equation

Error % = (Δv/v(AVG)) * 100% = (3.52m/s / 334.1312m/s) * 100% = 2.101%

The average velocity was found:

v(AVG) = (334.1312m/s + 349.06m/s + 334.98m/s + 335.5m/s + 334.8m/s + 327.29m/s + 334.66m/s + 334.8m/s) / 8 = 337.94m/s

The error % of the average velocity was found: 2.101%

Discussion

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The concluded result of the percentage error of the average velocity of the sound was 2.101% which is a very low percentage error and it can be said that the experiment was a success. The experiment could have approached the book value even more by having better measurements of each harmonic which will lead to a lower percentage error. Many assumptions were made in this experiment and that was part of the reason the percentage error was off by a small factor of the original book value.

Conclusion

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To conclude, the aim of the experiment was to measure the speed of sound in air and the percentage error was at 2.101%, which is a fairly low percentage that proves the experiment was on the track to have approached the original book value. Some assumptions have been made and said that they might affect the final result which it did and thus confirming that with better equipment’s better data.