Calculating Speed of Sound Experiment

As a mechanical wave, sound illustrates many properties as it transfers energy through a material medium. The wave’s period, frequency, wavelength, and speed are the areas of intrigue within this experiment. Specifically, this laboratory exercise will utilize three differing methods of experimentation to calculate values of speed, with the goal of matching these speeds to a pre-determined theoretical value based solely upon the temperature of the classroom. With a calculation of this expected theoretical value yielding a speed of 347.6 m/s, the subsequent experimental values and their corresponding percent errors are as follows: experiment one generated a speed of 372.3 m/s with a percent error of 7.11%, experiment two resulted in a speed of 340.5 m/s with a percent error of 2.01%, and experiment three gave a speed of 344.0 m/s carrying a percent error of 1.04%. As later discussed, numerous experimental errors may be responsible for these varying deviations.

Theory:

As a mechanical wave, sound is defined by oscillations transferring energy through a medium. In opposition to other types of waves, such as electromagnetic, the presence of this material medium is strictly required to successfully propagate and transmit sound. In addition to the above categorical divisions of waves as either electromagnetic or mechanical (also existing is the electron and proton transferring matter waves), waves can be further classified as transverse or longitudinal. As sound involves a parallel displacement of the surrounding air particles associated with the directional travel of its wave, sound is categorized as a longitudinal wave. In contrast, the affiliated particles of a transverse wave oscillate perpendicular to the direction of the transmitting energy, as in an electromagnetic wave. Specific to longitudinal waves, the medium harboring the parallel motions of the particles becomes subject to two alternating changes in pressure density, know as compressions and rarefactions. Compressions are simply areas of high pressure in the medium, whereas rarefactions are areas of low pressure. As with all forms of waves, sound possesses a variety of general characteristics, including a wave length (λ, the distance between two repeating elements of a wave, such as a peak or trough), a period (T, the length of time necessary for the passing of one wavelength), a frequency (f, equivalent to the inverse of the period and measured in hertz), and a speed, v. A generalized equation can be utilized to relate these elements to one another and reads as follows in terms of speed: v = λf. Within this experiment, the medium through which sound will be transmitted is the surrounding air (composed of a great variety of gases), and consequently the sole factor influencing the speed will be air temperature (measured in Kelvin, °K). Taken into consideration, the speed of sound can therefore be written in terms of temperature as follows: 20.1?T_k. Throughout this laboratory exercise, the value of the speed of sound obtained from this formula will be used as a means of theoretical comparison for the three calculated experimental speeds.

The first of the three experiments entails a direct measurement of the speed of sound via storage oscilloscope, a device that converts waves into electrical signals. The production of an intense sound (in this case the clapping of a clip board’s fastener) near a microphone connected to the oscilloscope will generate a pulse on the device’s screen at a specific time, denoted t₁. A spherical mirror mounted on the oscilloscope will then reflect this sound toward the wall at the terminating end of the hallway of experimentation, which itself will generate a rebounding echo back to the oscilloscope. The microphone will detect this echo and transmit it to the oscilloscope, which itself will again produce a visual pulse displayed on the instrument’s screen at the time, t₂. These two recorded times were able to be quantified due to a pre-calibration of the oscilloscope to measure the time difference between pulses. Through employment of the traditional definition of speed as change in distance over time

(v = Δx / Δt), the velocity of this sound can then be obtained, using the following formula, v = 2x / (t₂-t₁), where 2x is equivalent to the sum of the distances traveled by sound to the terminating wall (distance x) and back to the oscilloscope (again distance x). Using a percent error calculation (percent error = [| velocity_{theoretical –} velocity_experimental | / ( velocity_theoretical )] * 100) this theoretically calculated velocity can then be compared to the theoretical value of v previously discussed.

The second experiment within this laboratory exercise will draw upon the properties of resonance to find an experimental velocity of sound. Resonance involves a specific frequency of a sound wave bringing about oscillations of maximum amplitude within a system, that are often marked by an audible vibrating, humming noise. In this experiment, a wave of known frequency is generated by striking a tuning fork, which when held over the top of a glass air column containing an exact volumetric length of air will transmit it’s waves into the column resulting in the creation of resonance. This length of air, known as the resonant length, marks the distance traveled by the wave propagated via tuning fork from the top of the air column to the junction of the air and the water filling the remainder of the column. As will be later discussed, by calculating two differing resonant lengths for the frequency of the tuning fork, the use of a defined formula along with simple algebra can be employed to determine the wavelength of the tuning fork’s wave. As the wave enters the column, it will travel the resonant length in a sinusoidal manner to the air-water junction, and reflect back towards the top of the column in an identical and opposite sinusoidal motion. As the displacements of the original wave entering the column along with those of the reflective wave will superimpose upon one another, constructive interference at the top of the air column will result, which further classifies the two waves as standing waves. Distinct among standing waves is the occurrence of nodes, points of zero displacement along the wavelength where the surrounding air is effectively immobilized, and anti-nodes, points of maximum displacement along the wavelength at which the surrounding air experiences its largest freedom of movement. Together, the nodes and anti-nodes constitute the mode of oscillation of the wave, whose order plays a large role in determining the resonant length. For example, the first resonant length obtained in this experiment is associated with the first order mode of oscillation, where the anti-node is located at the top of the column and the node is found at the intersection of the water and air. The second resonant length obtained corresponds to a second order mode, possessing a greater number of nodes and anti-nodes, and therefore has a wavelength covering a greater distance within the air column. Specifically, the first resonant length, L, will cover a distance equivalent to ¼λ, while the second resonant length, L¹_, is equal to ¾λ . In order to derive these wavelength values, the following formula can be employed, L + c = (2m -1) * ¼λ, where L is the resonant length, and c represents a correction factor accounting for the difference between the actual location of the anti-node and it’s assumed placement at the top of the column. With this in consideration, the above mentioned wavelengths and their associated resonant lengths can be represented as follows: L + c = ¼λ and L¹ + c = ¾λ . These two equations can then be related to state that L¹ – L = ½λ, which can itself be manipulated to state: λ = 2 (L¹ – L). Substitution of the two resonant lengths into this formula will yield an experimental wavelength value, that when placed into the formula, v = λf, will generate an experimental speed which can then be compared to the theoretical value via percent error calculation.

The final experiment within this laboratory exercise will involve the use of a dual-trace oscilloscope (function generator) to produce an ultrasonic signal whose wavelength will be determined via usage of transducers. The above mentioned system involves the production of an electrical signal (that when transduced will create a wave possessing a wavelength of 40,000 Hz) by the oscilloscope that will first be recorded and displayed on one of the device’s channels as the oscillating signal reaches the first of the two transducers, known as the transmitter. This transmitter will convert the electronic signal into an ultrasonic wave, that will then be detected by the second transducer, known as the receiver, and again be converted back into an oscillating electric signal. This signal will be recorded and displayed on the other channel of the oscilloscope, which will allow the wavelength of the wave to then be quantified. Following the methods described in the procedure section, the wavelength can be subsequently calculated and placed into the formula, v = λf, to find an experimental speed upon which a percent error calculation can be performed for comparison.

Procedure:

Experiment 1

Before experimentation begins, a theoretical value for the speed of sound must be determined. Utilizing the formula mentioned within the theory section, record the temperature present in the classroom and use it’s value to calculate a theoretical speed for experimental comparison.

Obtain a storage oscilloscope along with the device’s associated microphone and spherical mirror, and position this system into a corridor of appropriate setting so that it is properly aligned upon a floor tile facing the terminating wall of the hallway.

Ensuring that the oscilloscope has been calibrated to the specifications outlined within the theory section, use a clip board to produce a sharp, intense sound into the microphone by clapping the metal fastener upon the board.

As discussed in the theory section, two times will be generated by the oscilloscope corresponding to the moment of detection of the original sound wave produced from the clipboard, as well as the resulting echo of the sound wave reflecting from the far wall. Record these two times, as they will be used to calculate the speed of the sound wave.

Starting from the tile directly beneath the oscilloscope system, count the number of square-foot tiles spanning the distance to the terminating wall, convert this distance to meters, and double it’s value to account for the total distance that sound has traveled. Use this value along with the change in time to find the experimental velocity of sound and calculate the percent error.

Experiment 2

To simplify experimentation, obtain a tuning fork that generates a frequency of above 380 Hertz when struck. With a rubber hammer, strike the tuning fork and hold it slightly above the lips of a glass air column containing water.

As the first resonance length is located between a 15-28 cm displacement from the top of the air column, use the capillary action device mounted upon the column to vary the lengths of the air within this specified interval.

While maneuvering the length of the air, listen attentively for the distinctive vibration and humming noise that is characteristic of a resonance length. Upon hearing this noise, record the displaced length of the air from the top of the column as resonant length, L.

Repeat the above mentioned steps in search for a second resonant length, L¹_. As this length corresponds to ¾ wavelength (whereas the first resonant length corresponded to ¼ wavelength), the location of L¹should therefore be approximately equivalent to three times the displacement from the top of the air column of length, L.

Use the two resonant length measurements along with the frequency specified by the tuning fork to calculate the experimental speed of sound, and obtain a percent error.

Experiment 3

Before beginning experimentation, pre-calibrate the function generator so that electrical signals will be produced that possess a wave with a frequency of 40,000 Hertz when transduced.

Grasp the receiver and move the device along it’s supportive track until it’s corresponding movable peaks align with the stationary peaks displayed upon the screen of the generator. Utilizing the ruler adjacent to the supportive track, note this initial position of the receiver.

Carefully slide the receiver along it’s track away from the transmitter, paying close attention to one of the stationary peaks. Every re-alignment of the next sequential movable peak to this specified stationary peak corresponds to displacement of one wavelength. Allow ten such sequential peaks to come into alignment, and record the total distance traveled by the receiver over this 10 wavelength span.

Find the average value of one wavelength by dividing the total distance displaced from the receiver’s initial to final position by 10, and record this figure.

Utilizing the formula discussed within the theory section, determine the experimental velocity of sound and compare this value to its theoretical counterpart with a percent error calculation.

Sample Calculations:

Theoretical Velocity: v_th

v_th = 20.1 * ?(T_c + 273)

v_th = 20.1 * ?(26 + 273)

v_th = 347.6 m/s

Experiment 1: Total Distance, x

x = 2 (x_feet) * (x _{meters / foot})

x = (86.3 ft) * (0.33 m/ft)

x = 56.96 m

Experiment 1: Velocity, v

v = Δx / Δt

v = (56.96 m) / (0.153 sec)

v = 372.3 m/s

Experiment 1: % Error

% Error = [| velocity_{theoretical –} velocity_experimental | / ( velocity_theoretical )] * 100)

% Error = [| 347.6 m/s – 372.3 m/s | / (347.6 m/s )] * 100)

% Error = [| -24.7 m/sec | / (347.6 m/s )] * 100)

% Error = 7.11%

Experiment 2: Wavelength, λ

λ = 2 (L¹ – L)

λ = 2 (0.604 m – 0.205 m)

λ = 2 (0.399 m)

λ = 0.798 m

Experiment 2: Velocity, v

v = λf

v = 426.7 Hz * 0.798 m

v = 340.5 m/s

Experiment 2: % Error

% Error = [| velocity_{theoretical –} velocity_experimental | / ( velocity_theoretical )] * 100)

% Error = [| 347.6 m/s – 340.5 m/s | / (347.6 m/s)] * 100)

% Error = [| 7.1 m/s | / (347.6 m/s)] * 100)

% Error = 2.01%

Experiment 3: Wavelength, λ

λ = (total displacement of receiver) / (total number of wavelengths)

λ = (0.592 m – 0.506 m) / 10

λ = (0.086 m) / 10

λ = 0.0086 m

Experiment 3: Velocity, v

v = λf

v = 0.0086 m * 40,000 Hz

v = 344 m/s

Experiment 3: % Error

% Error = [| velocity_{theoretical –} velocity_experimental | / ( velocity_theoretical )] * 100)

% Error = [| 347.6 m/s – 344.0 m/s | / (347.6 m/s)] * 100)

% Error = [| 3.6 m/s | / (347.6 m/s)] * 100)

% Error = 1.04%

Conclusion:

As a mechanical wave of longitudinal nature, sound is characterized by waves that create a parallel displacement of the surrounding air particles associated with it’s direction of travel through a material medium. In addition to these properties, sound also possesses the general trademarks of all waves, including a wave length, a period, a frequency, and a speed, the area of interest in this laboratory exercise. If a sound wave happens to be propagated in a medium of air, as is the case in this experiment, the speed can be defined simply as 20.1T_k, and this corresponding value can then be used as a theoretical comparison for speeds calculated through experimentation. As eluded to, the goal of this experiment will be to calculate the speed of sound employing three different methods, with the aim of matching these experimental speeds with the expected theoretical speed discussed above. For the first series of experiments involving the manipulation an oscilloscope to produce a distance-over- time ratio for speed, the resulting velocity of sound was 372.3 m/s, marking a percent difference of 7.11% from the theoretical value of 347.6 m/s. The next experiment involving the properties of resonance to calculate a velocity yielded a speed of 340.5 m/s, with a percent error of 2.01%. Finally, the third experiment, which used the generation of an ultrasonic signal to calculate the velocity of sound, produced a speed of 344.0 m/s, with a percent error of 1.04%. Within the first experiment, one possible source of error includes a skewed value for the distance of 2x, due to the estimation of the length of the final, incomplete tile that marked the end of the corridor. If this value was indeed inexact, the speed calculated through the distance-over-time formula would as a result be subsequently erroneous. Also within this experiment, the theoretical basis for the speed of sound was calculated using the temperature present in the classroom. As this experiment took place within the hallway, any deviation in temperature between the two locations would result in differing theoretical values, thus skewing the percent error calculation for this experiment. In the second series of experiments for this laboratory exercise, sources of error stem from the estimation and subsequent inadequacies of human judgment associated with both interpreting the exact ruler readings for the resonant lengths as well as listening for the appropriate vibrating noise that marks the production of resonance at a particular resonant length. As both of these errors are either directly or indirectly associated with the value for the resonant length, the value for the wavelength produced through resonant length calculations will be skewed, thus influencing the obtained velocity. Finally, sources of error present within the third experiment again involve errors of human judgment, with the first again related to the estimations involved in using a ruler (specifically to measure the distance traveled by the receiver), and the second associated with the estimation of proper peak alignment between the stationary and movable peaks at their beginning and final locations. As both of these potential errors involve the wavelength, a subsequent calculation of the speed from this value will yield a flawed speed measurement.