The Effects of External Factors on the Motion of Standing Waves in Strings

Kareem Ibrahim, David Diop, and Vincent Sangpo

The City College of New York

 

Author’s Note:

This paper was prepared for English 210, taught by professor Susan Delamare.

 

Table of Contents

 

Abstract ………………………………………………………………………………………p. 3

Introduction …………………………………………………………………………………..p. 4

Materials and Methods ……………………………………………………………………….p. 6

Results and Discussion ……………………………………………………………………….p. 7

Conclusions …………………………………………………………………………………..p. 12

References …………………………………………………………………………………….p. 14

Appendix ……………………………………………………………………………………..p. 16

 

Abstract

Background:

Waves are a common occurrence in nature. They can be defined as a pattern of motion, travelling through a medium. One particular type of wave is a standing wave. Standing waves are similar to regular sinusoidal waves, differing only in that these waves appear to be still. The basic structure of each wave begins with a node, a point that doesn’t move. From the node, the wave rises until it reaches its maximum amplitude at a point known as an antinode. From there the wave descends until it reaches another node as the cycle repeats (“Standing Waves”, 1999). This process occurs when two opposite and equal forces reflect each other from certain fixed ends. This is observed naturally in bridges and oceanic movements, therefore providing a necessary stimulus to study standing waves.

Aim/Objective:

This experiment extends previous research concerning strings with two fixed ends. This research identifies how differing factors such as frequency, string length, and pressure affect the movement and arrangement of the standing waves produced.

Method:

Three tests were conducted. The first test involved adjusting the frequency of the oscillating machine connected to the string while keeping the mass of the weights unchanged. The second test involved adjusting the mass weights while keeping the oscillating frequency at 120 Hertz. The third test involved a different string used, while keeping the mass weight adjustment patterns and oscillating frequency of the second test unchanged.

 

Results:

The results of the three experiments were used to determine how the factors of mass and frequency individually influenced the vibration of the string. The wavelength proved to be twice the length of the string divided by the number of antinodes, areas of maximum displacement from the original axis, on the string. The value of the acceleration of gravity was also measured to be near the accepted value, at 9.80 m/s2 (meters per second per second), based on the slope of the resulting graphs comparing mass of the pan and tension on the string. This result was constant regardless of the string used in the experiment.

Conclusion:

The results of this experiment can be used to infer how standing waves may interact and affect other mediums besides strings, including structures like short-span bridges. The measurement of gravity represents how accurate our data is towards the physical world. Further experimentation that is more mathematically inclined and to scale with real world applications such as bridges is needed to provide more accurate results.

Introduction

In a string with both ends fixed at specific points, energy can travel along the string in the form of waves. These disturbances on the string cause particles to move up and down, perpendicular to the direction that the waves appear to travel. However, at certain frequencies, string lengths, and amounts of pressure applied on the string, a phenomena known as standing waves occur (Shankar, 2014). Unlike a regular sinusoidal transverse wave—a wave pattern that appears to move along the axis of propagation—the wave pattern in standing waves appears to be still with the location of its nodes (its unmoving points) and antinodes (its points of largest amplitude) being unchanged. Standing waves are an application of the idea of resonance—a wave in a natural frequency of vibration—to a string-wave situation (“Standing Waves”, 1999).

Ocean waves are seen as an example of the application of standing waves as they can be classified based on their frequency of motion. Continental shelf waves, for instance, emit a low resonant frequency due to the shape of the surrounding terrain in the ocean. The nature of this “trapped wave” has led to explanations for coastal upwelling—the process of water moving upwards on a sea bank—and boundary currents along certain shorelines (Leblond and Mysak, 1979).

The applications of these waves also extend into areas like music. One major example is the use of standing waves in music with a guitar. A guitar makes sound when each string is plucked, thereby producing a standing wave that produces a harmony. The guitarist changes the sound of a string by adjusting the tension of the string at its fixed end. In addition to this, standing wave manipulations appear in oscillations observed in space. Varying the length of the object at which a standing wave propagates through changes the speed of the wave (Kuridze, Verth, Mathioudakis, Erdelyi, Jess, Morton, …, Keenan, 2013).

        The aim of this research was to analyze how a few external factors, such as the mass and frequency of a string, would affect a string’s oscillatory motion. Standing waves occur only at specific frequency levels depending on the string used and other factors such as tension. Understanding the factors that affect standing wave movement is vastly important and applicable. By studying the ways that standing waves can be influenced, we can make certain assumptions on how these changes could affect current research and study into this topic.

This research will focus on two parts of string manipulation: the first way will show when a string’s linear mass density and frequency change and the second way will show just a change in mass. The effects will be observed and monitored, and will further describe the ways that a standing wave can be manipulated and changed. For the lab involving a stretched string exhibiting wave motion, it was hypothesized that a larger tension on the string would lower the number of nodes or antinodes (mode number) that would appear on the string during oscillation, due to a lower mass weight and knowledge of transverse wave functions.

Materials and Methods

Our experiments involved numerous variables: the mass at the end of the string, the  frequency and vibration of the wave along the string, and the type of string used. A mass was attached to the end of the string on the side of the pulley. This in turn reflected the amount of tension produced on the string.

The first experiment was conducted with a constant 0.200kg mass and string, with only the frequency of the wave being changed. Measurements were made through the number of antinodes, n, that appeared at certain frequencies. The wavelength was recorded and we continued experimenting until n=5. The length of the string would be measured twice: once for the segment where the waves occurred, and another for the string’s linear mass density.

The second experiment involved a constant frequency while the mass of the string’s fixed ends was changed. This time we recorded the mass used and the mode number—the number of antinodes present on the string. We would then use a different type of string and repeat this experiment.

We measured the acceleration due to gravity throughout this experiment in order to determine how close this experiment resembled the physical world. Appendix C shows the relation between some of the measured quantities in the experiment and the variable representing acceleration due to gravity (g).

While all of the experiments involved the same type of string material used, which was most likely cotton, the third experiment involved a string with a different length, unlike the first two experiments, which used the same length of string. In a string-wave system, the length of the string can affect the amount of tension that influences the string’s oscillatory motion, based on the given mass weights.

Diagram of experiment

General layout of experiment set-up, Standing Waves. (n.d.). Retrieved from          https://physicslabs.ccnysites.cuny.edu/labs/208/208-vibrating-strings/vibrating-strings.php

Results

The string’s linear density, the frequency of the buzzer used for the string’s vibration, and the mass of the weight pans shown in the diagram above were used. Several formulas in physics that define waves and their propagations were used to calculate any trends between measured quantities and approximations for gravity acceleration (see Appendix C).

Frequency is set at a constant 120 Hz for Experiment 2.

Entire length of string in Experiment 1 and Part 1 of Experiment 2: 1.183 m

Part used in experiment (where waves occurred): 1.148 m

0.05 kg pan used in Experiment 1

Mass of first string used: 5.511 g or 0.00511 kg

Linear density calculated for first string type:

μ=0.00511 kg1.183 m=0.004658kgm or 4.658gm

Entire length of second string in Experiment 2-Part 2: 1.03 m

Part used in experiment only: 0.847 m

Mass of second string used in Experiment 2-Part 2: 4.46 g or 0.0046 kg

μ=0.00446 kg1.03 m=0.00433kgm or 4.33gm

Experiment 1: Adjusting frequency (refer to appendix A for data table)

Slope given by 1/2L (L denotes length of string used in experiment in m)

Experiment 2: Constant frequency, adjusted mass (refer to appendix B)

Tension of string calculated for each value based on the formula

(f*2Ln)2*μ , with f still at 120 Hz but with different L at 0.847 m. See appendix B for data table and coordinates for the graph.

Calculated value for g: 9.83 m/s^2

Experiment 2-Part 2: Different string used

See appendix B for data table and coordinates for the graph.

Calculated value for g: 9.82 N/kg or m/s^2

Measured values

Based on the data, the wavelength was proven to be twice the length of the string used in the experiment divided over the number of antinodes. Additionally, the magnitude of acceleration due to gravity came out around 9.82 to 9.83 m/s^2, close to the accepted value of 9.80 m/s^2. The mode number (antinodes present in the standing wave) seemed to decrease as the tension and mass of the pans increased.

Discussion

Bridge construction analysis

Analysis of these experiments echo applications towards short-span bridge design and construction. Short-span bridges act similarly to the string in our experiment; they have two fixed points at each end, connecting two paths. A short-span bridge has a wide but brief length and may be influenced by the movement of wind and ground vibrations. A common occurrence in bridge collapse is due to a lack of support on the bridge. One such situation occurred with the Showa bridge where an earthquake produced severe resonance on the structure, leading to its collapse (Bhattacharya, Tokimatsu, Goda, Sarkar, Shadlou, Rouholamin, 2014). Factors that led to the collapse were the weight and linear density of the bridge, both of which are factors in our experiment. By increasing the mass of the end points of the bridge, we can decrease the mode number of the standing waves produced from an external factor. With respect to short-span bridge design, increasing the tension on the ends of a bridge, and thus its weights, will lower the level of vibration that the bridge could experience, thus stabilizing the design.

Guitar tuning

Considering that guitar strings have two fixed ends and they vibrate in a standing wave pattern, our experiment may directly apply to the usage of guitars. A guitarist tunes his or her’s guitar by adjusting the pitch, or frequency, of each string to achieve an optimal sound. Another way an optimal sound can be produced is by using a guitar capo—a clip that applies pressure on various strings and changes the position of the endpoints of those strings, thus changing its tension. Our experiment directly shows that an adjustment of mass on an endpoint of a string directly changes tension and frequency of any standing wave produced on the string. Therefore, lower or higher pitches may be sought after by adjusting the tension of a guitar string.

Limitations

The experiment at hand has numerous limitations. Our research was on a limited scale since only a thin string with a low mass per unit length was used, and thus any extra friction was neglected. In the larger scope of bridges, the factors that influence their design, and the materials involved in construction were also neglected. The closest bridge that stems from our results is likely the short-span bridge. Therefore, our experiment does not apply to suspension or beam bridges. We also do not account for other external factors that influence bridge movement, such as wind, earthquakes, and extra friction caused by the materials used in bridge construction, such as reinforced concrete or steel. We merely discuss how direct impulses affect a string’s movement. Finally, we lack an appropriate mathematical model. This experiment is meant to extend previous ideas of standing wave motions and resonance, therefore we do not produce a mathematical model to indicate ways construction can be improved.

Most of the errors in the calculation of gravity came from the measurement of the string. There exists numerous knots on the string and instead of measuring the exact length of the total string beforehand, the string’s length was approximated with the knots, which may have compromised the calculated value for the string’s linear density. Also, the friction caused by the pulley on the string is assumed to be negligible when it in fact does exist. Friction affects the experiment by distorting the string’s tension, which in turn influences the final calculated value for gravitational acceleration.

Acceptance of the hypothesis

Our results are consistent with our initial hypothesis: increasing the tension of the string will reduce the amount of nodes that appear in a standing wave oscillation of the string (See Appendix B). A negative correlation is shown between the tension on the string caused by weights and nodes on the string.

 

Conclusions

The purpose of this experiment is to show the various factors that influence the propagation of standing waves on a string. Our results showed that as the tension of a string increases, the number of nodes that appear on the string decrease. This experiment has several implications. One major implication is in the construction of bridges. Substituting the string for a bridge could emulate similar results and may lead to improved construction work by reducing wave motions by a bridge. Another implication is the connection to musical instruments like guitars; adjusting the string tensions in a guitar changes the frequency and how one may observe the sound of the guitar. We do lack scale in this experiment, however. We used a small scale string apparatus to retrieve our data, larger models that account for various other factors, such as the wind surrounding the string, should yield more accurate applications. Mathematical modeling is also a necessary component to extend the usability and practicality of this experiment.

 

References

Bhattacharya, S., Tokimatsu, K., Goda, K., Sarkar, R., Shadlou, M., & Rouholamin, M. (2014).

Collapse of Showa Bridge during 1964 Niigata earthquake: A quantitative reappraisal on

the failure mechanisms. Soil Dynamics & Earthquake Engineering (0267-7261), 65, 55–71. https://doi-org.ccny-proxy1.libr.ccny.cuny.edu/10.1016/j.soildyn.2014.05.004

Kuridze, D., Verth, G., Mathioudakis, M., Erdelyi, R., Jess, D. B., Morton, R. J., Christian, J.,

Keenan, F. P.. (2013). Characteristics of Transverse Waves in Chromospheric Mottles.

The Astrophysical Journal, 779.

Leblond, P., & Mysak, L. (1979). Ocean Waves: A Survey of Some Recent Results. SIAM Review, 21(3), 289-328. Retrieved October 21, 2018 from http://www.jstor.org/stable/2029570

“Physics 208 – Lab 1 – Vibrating Strings”. (n.d.). [Figure]. Retrieved from

https://physicslabs.ccnysites.cuny.edu/labs/208/208-vibrating-strings/vibrating-strings.ph

p

Shankar, R. (2014). Fundamentals of Physics : Mechanics, Relativity, and Thermodynamics.

New Haven: Yale University Press. Retrieved from

http://ccny-proxy1.libr.ccny.cuny.edu/login?url=https://search.ebscohost.com/login.aspx

direct=true&db=e000xna&AN=709762&site=ehost-live

Standing Waves on a String. (1999, October 11). Retrieved October 21, 2018, from http://hep.physics.indiana.edu/~rickv/Standing_Waves_on_String.html

Appendix

Appendix A:

Experiment 1: Adjusting frequency of oscillator machine/buzzer

Initial data collection

n	wavelength (m)	frequency (Hz)
1	0.975	13.0
2	0.495	25.5
3	0.325	37.7
4	0.244	51.7
5	0.195	64.4

Data used in the graph for Experiment 1

wavelength (m)	1/n
0.195	0.200
0.244	0.250
0.325	0.333
0.495	0.500
0.975	1.000

Appendix B:

Experiment 2: Constant frequency, adjusted mass
Tension of string calculated for each value based on the formula (f*2Ln)2*, with f still at 120 Hz but with different L at 0.847 m

M (kg)	n	M (kg)	Tension of string (N)
0.120	13.000	0.12	1.519145617
0.155	12.000	0.155	1.782886176
0.190	11.000	0.19	2.121781895
0.230	10.000	0.23	2.567356093
0.291	9.000	0.291	3.169575424

 

Part 2 of experiment 2: Different string used

M (kg)	n	M (kg)	Tension of string (N)
0.210	9.000	0.210	2.208982023
0.260	8.000	0.260	2.795742873
0.320	7.000	0.320	3.651582528
0.390	6.000	0.390	4.970209552

Appendix C:

This lab involves an understanding of standing waves and how they interact with the respective medium of oscillation. In a standing wave, a medium, such as a piece of string, is stretched and held tightly on both ends. An intermittent pulse or vibration is sent from one end of the string, and will reverse direction when it reaches the other end. However, as the pulses are continually produced from the origin, the waves resulting from the pulses will interact with the traveling waves from the opposite end. At specific energy levels associated with the pulses, a phenomenon known as standing waves occur, where the resulting wave pattern seems to only be traveling up and down. The location of the nodes (where the wave seems to stay at the original line) and the antinodes (where the wave reaches maximum displacement from the line) stay the same in a standing wave.

The specific energy levels at which standing waves occur (resonance) are known as resonant frequencies. These values correspond to mode numbers (n), based on the number of nodes that appear in the resulting wave pattern. In a string fixed at both ends, the wavelength at each mode number can be evaluated through 2Ln=, where L is the length of the string used in the experiment. The speed of the wave is calculated through v=f, as well as v=T, where T is the tension in the string caused by the weights, and the density (mass per unit length) of the string, the medium in which the wave travels. Equating the two values for v and substituting value for wavelength, the equation comes out as f*2Ln=T. However, the only tension on the string in this specific lab would be from the weight of the mass pans, so the T could be substituted by M*g. After multiplication and removing square roots, the resulting formula becomes (f*2Ln)2*=M*g. This is very similar to the general linear formula y=k*x on a two-dimensional, Cartesian coordinate system. As the mass of the weights were varied in the experiment, g was essentially the slope of the line, thus resulting in the graphs shown in the previous sections.

Appendix D:

Waves follow a general pattern of alternating between two upper and lower limits in any sort of medium, regardless of the speed that the wave travels, the frequency of the pulses on the wave, or the wavelength. The resulting pattern follows a sine curve, a trigonometric function with maximum and minimum points at A and -A, where A is the amplitude (maximum displacement of the wave from the original axis of propagation). The cycle repeats for each equally spaced time period along the direction of travel. This specific wave is known as a transverse wave, in which elements on the string (like particles) appear to move up and down, parallel to the general direction of the waves. In this experiment, the waves on the string caused by the buzzer are transverse waves.