melde's experiment lab report

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3b22.11 - standing waves in a string - melde's experiment.

Assemble the apparatus as indicated. By varying the amount of mass or tension on the string you can get from 2 to 11 or 13 standing waves. In the constant diameter string these waves are all the same size. In the decreasing diameter (tapered) string the waves will be of different sizes. For instance there will be 3 waves on one half of the string vs. only 2 on the other half, instead of 3 on each half.

This is best seen in the classroom when set high enough to use the blackboards as background.

The reed vibrator in the above instructions may be replaced with the driven tuning fork if desired. 2 to 8 VDC of several amps will be needed to run the driven forks.

The Cenco 60 Hz. mechanical drivers also come with a version of the Melde's experiement using Aluminum, Steel, and Copper wires of different diameters which can easily be set up with any of the other mechanical drivers we have if desired.   

• Peter Riggs, "Revisiting Standing Waves on a Circular Path", TPT, Vol. 59, #2, Feb. 2021, p. 100.
• G. Bozzo, F. de Sabata, S. Pistori, and F. Monti, "Imaging and Studying Standing Waves with a Homemade Melde-Type Apparatus and Information and Communication Technology (ICT)", TPT, Vol. 57, #9, Dec. 2019, p. 612-615.
• Dylan Welsch, and Blane Baker, "Wavelength Variation of a Standing Wave Along a Vertical Spring", TPT, Vol. 56, #3, Mar. 2018, p. 154. 
• Cameron T. Vongsawad, Mark L. Berardi, Tracianne B. Neilsen, Kent L. Gee, Jennifer K. Whiting, and M. Jeannette Lawler, "Acoustics for the Deaf: Can You See Me Now?", TPT, Vol. 54, #6, Sept. 2016, p. 369.
• Paul Gluck, "Standing Waves in a Nonuniform Medium", TPT, Vol. 49, #2, Feb. 2011, p. 76.
• John Welch, "Demonstrating Wavelength Dependency on Medium Density", TPT, Vol. 47, #7, Oct. 2009, p. 476.
• Ana Cros and Chantel Ferrer-Roca, "Excitation of Standing Waves by an Electric Toothbrush", TPT, Vol. 44, #9, Dec. 2006, p. 578.
• Haym Kruglak and Thomas B. Greenslade Jr., "From Our Files: Demonstrations with a Sabre-Saw Oscillator", TPT, Vol. 43, #7, Oct. 2005, p. 474.
• Christine Carmichael and Steven Smith, "Demonstration of Beats With A Double-Driven String", TPT, Vol. 42, #8, Nov. 2004, p. 462.
• David D. Lockhart and Thomas B. Greenslade Jr., "From Our Files: Standing Wave Generator", TPT, Vol. 41, #1, Jan. 2003, p. 54.
• James H. Larson, "Beats on a Vibrating String", TPT, Vol. 37, #6, Sept. 1999, p. 373.
• Mark Graham, "Melde's Experiment with an Aquarium Aerator. Rich Dynamics with Inexpensive Apparatus", TPT, Vol. 36, #5, May 1998, p. 276.
• Walter Connonny, John Abedschan, and Dick Speakman, "Laser- Enhanced Vibrating String", TPT, Vol. 29, #2, Feb. 1991, p. 114.
• Parshotam D. Gupta, "Coloration on a String Vibrating in a Standing Wave Pattern", TPT, Vol. 26, #6, Sept. 1988, p. 371.
• Gerard Lietz, Anthony Behof, and Marshall Ellenstein, "Illuminating Standing Waves", TPT, Vol. 24, #7, Oct. 1986, p. 449.
• C. J. Roddy, "Standing Waves - A New Twist", TPT, Vol. 21, #6, Sept. 1983, p. 408.
• Verner Jensen, "Standing Transverse Waves of Inhomogeneous Strings", TPT, Vol. 21, #1, Jan. 1983, p. 53.
• Sue Gray Al-Salam and Ronald B. Edge, "Why is the String Colored?", TPT, Vol. 18, #7, Oct. 1980, p. 518.
• Samuel Jacobs, "Can Standing Sound Waves Have Zero Amplitude Nodes?", TPT, Vol. 17, #7, Oct. 1979, p. 460.
• David D. Lockhart, "Standing Wave Generator", TPT, Vol. 9, #5, May 1971, p. 283, reprinted in TPT, Vol. 41, #1, Jan. 2003, p. 54.
• Haym Kruglak, "Demonstration with a Sabre - Saw Oscillator", TPT, Vol. 8, #2, Feb. 1970, p. 88, reprinted in TPT, Vol. 43, #7, Oct. 2005, p. 474.
• Marvin Ohriner, "Standing Waves by a Current Carrying Conductor", TPT, Vol. 5, #6, Sept. 1967, p. 287.
• "Problems: Energy Transfer in Vibrating Strings", TPT, Vol. 5, #4, Apr. 1967, p. 175.
• Timothy C. Molteno and Nicholas B. Tufillaro, "An Experimental Investigation Into the Dynamics of a String", AJP, Vol. 72, #9, Sept. 2004, p. 1157.
• Allan Walstad, "The Longitudinal Momentum of Transverse Traveling Waves on a String", AJP, Vol. 72, #7, July 2004, p. 971.
• David R. Rowland, "Parametric Resonance and Nonlinear String Vibrations", AJP, Vol. 72, #6, June 2004, p. 758.
• E. Kashy, D. A. Johnson, J. McIntyre, S. L. Wolfe, "Transverse Standing Waves in a String With Free Ends", AJP, Vol. 65, #4, Apr. 1997, p. 310.
• Donald E. Hall, "Optical Measurement of String Motion", AJP, Vol. 55, #6, June 1987, p. 573.
• John A. Elliot, "Nonlinear Resonance in Vibrating Strings", AJP, Vol. 50, #12, Dec. 1982, p. 1148.
• F. P. Clay Jr., and R. L. Kernell, "Standing Waves in a String Driven By Loudspeakers and Signal Generators", AJP, Vol. 50, #10, Oct. 1982, p. 910.
• Donald F. Kirwan and Jack Willis, "Direct Measurement of Transverse Wave Speed on a Stretched String", AJP, Vol. 43, #7, July 1975, p. 651.
• "W-120: Melde's String - Sonometer", DICK and RAE Physics Demo Notebook.
• Freier and Anderson, "Sa-9, 10", A Demonstration Handbook for Physics.
• S-34, 35: Richard Manliffe Sutton, Demonstration Experiments in Physics.
• Martin C. Sagendorf, "Vibrations in a String",  Physics Demonstration Apparatus, 2009. p. 74.
• A. D. Bulman, "Maintained Fork and Melde Arrangements", Model-Making for Physics, p. 59.
• Roger J. Hanson, Edited by Karl Mamola, "Optoelectronic Detection of String Vibration", Apparatus for Teaching Physics, p. 74.
• Jearl Walker, "1.183, Oscillating Car Antenna", The Flying Circus of Physics Ed. 2, p. 79.
• Charles Taylor, "Demonstration 3.2", The Art and Science of Lecture Demonstration, p. 111.
• Ron Hipschman, "Giant Guitar Spring", Exploratorium Cookbook III, 186.1 - 186.3.
• "Vibrating String", The Exploratorium Science Snackbook, p. 103-1.
• Ron Hipschman, "Vibrating String", Exploratorium Cookbook II, p. 116.1 - 116.2.

Disclaimer: These demonstrations are provided only for illustrative use by persons affiliated with The University of Iowa and only under the direction of a trained instructor or physicist.  The University of Iowa is not responsible for demonstrations performed by those using their own equipment or who choose to use this reference material for their own purpose.  The demonstrations included here are within the public domain and can be found in materials contained in libraries, bookstores, and through electronic sources.  Performing all or any portion of any of these demonstrations, with or without revisions not depicted here entails inherent risks.  These risks include, without limitation, bodily injury (and possibly death), including risks to health that may be temporary or permanent and that may exacerbate a pre-existing medical condition; and property loss or damage.  Anyone performing any part of these demonstrations, even with revisions, knowingly and voluntarily assumes all risks associated with them.

STANDING WAVES IN A STRING:

MELDE’S EXPERIMENT

Click here to see a video of the experiment  

... took a musical string of brass, three quarters of a foot long and stretched it with a weight of six and five-eighth pounds, which I found matched by its vibrations a certain standard note in my organ. I found that a string of the same material and tension, fifteen feet, that is, twenty times as long, made ten recurrences in a second. I inferred that the number of vibrations of the shorter string must also be twenty times as great; and thus such a string must make in one second of time two hundred vibrations...

From the work of MERSENNE

By sound we mean that phenomenon which is capable of stimulating the sensation of hearing. Sound always originates in some type of motion. In many instances the source of sound is a standing wave in some vibrating body -- a drum head, the vocal cords of a guitar string or the air column in an organ pipe. Our goal in this experiment is to learn something about the formation of standing waves in strings and the boundary conditions which determine the pitch (frequency) of the sound produced.

Imagine a long string attached to the wall at one end. We grasp the other end, place the string under some tension, T, and vibrate this end up and down at some constant frequency, f.

Successive crests and troughs are produced by the motion of the hand and are seen to move in succession down the string with constant speed, v. Such a wave is called a running wave. The distance between any two successive points in the wave train which have the same phase is called the wavelength, l . Careful study shows that the wavelength, frequency, and speed are related by the wave equation:

v = λ f                         (1)

Experiment also shows that the speed of the wave through the string is independent of the frequency and amplitude of the wave and depends only upon the characteristics of the media (the string) through which the wave moves. This is a general property of many types of wave motion. Specifically the speed of the running wave in the string is related to the tension in the string, T, and the linear density of the string, m , and (the mass per unit length of the string). The velocity is given by:

When equations (1) and (2) are combined, we can solve for the frequency of vibration.  That is

In this experiment we use an electrically driven vibrator to generate the wave and we are interested in the standing waves that are produced in the string under certain circumstances.

Consider the following situation. A running wave is produced by the vibration of the fork. The wave moves to the right where it encounters the wall. If the wall was not present and the string longer, the wave would have continued beyond the wall as shown by the dotted wave form. The wall is, however, present and the initial running wave (solid wave form) is reflected back into the string as a second running wave; see second sketch. This wave is moving toward the left.

The details of the reflection process are rather involved and will be discussed more fully in class. Note, however, that the wave has "turned over" as a part of the reflection process.  A nice simulation is here: StringWave

The instantaneous shape of the string is found by adding together the displacements that would be produced by the two waves acting independently. The series of sketches to the side show the resultant wave form (shape of the string) at six different times as the incident wave continues to move to the right and the reflected wave continues to move to the left. Note that there are points on the string where the instantaneous displacement of the string is always zero. These points are called nodal points. At intermediate points the wave amplitude builds up to a positive maximum, dies out, and then builds to a negative maximum. Thus, instead of seeing waves move successively down the string one sees the string vibrating in a series of loops -- each loop being just one half wavelength in extent. This type of wave is called a standing wave or a stationary wave.

Standing waves will form in the string only if certain "boundary conditions" are satisfied. Specifically, the length of the string must be some integral number of half-wavelengths for the initial running wave. Since the frequency of the wave is fixed by the vibrator, this means that we can adjust the tension in the string and therefore the wave speed (see equation (2)) until the wavelength (see equation (1)) satisfies this condition. Under these circumstances the string vibrates with the same frequency as the vibrator -- the system is said to be in resonance and the energy flows from the vibrator into the vibrating string. The amplitude of the string builds to quite large magnitudes at resonance.

SPECIFIC OBJECTIVES:

When you have completed this experimental activity you should be able to: (1) write and use the wave equation; (2) distinguish between running waves and standing waves; (3) describe the conditions that lead to standing waves in a string; (4) identify the factors that determine the speed of a running wave in a string; and (5) write and use the velocity equation for transverse waves in a string (equation (2)).

EXPERIMENTAL ACTIVITY:

Connect the equipment as shown in the attached schematic.  Attach the string to a rigid post on one end and pass the string over the pulley to a weight hanger on the other end. Place the vibrator about 6-8 inches from the fixed end.  Place 150 grams total on the weight hanger, including the weight hanger.  Before starting the vibrator, be sure to unlock it.   Start the vibrator and adjust the frequency of the frequency generator until there is a node at the center of the string. 

Adjust the frequency carefully so as to obtain the largest amplitude of the loops. Examine the wave motion carefully. Are all the loops of the same size (length)? Is the point where the string is attached to the vibrator a nodal point? How about the point where the string passes over the pulley? Does the speed referred to in equations (1) and (2) refer to any motion you can observe directly?

Record the number of vibrating segments of the string, n, the tension in the string (T in Newton’s), the wavelength (λ in m), and the frequency. Note that the best value for the wavelength is just twice the length of the loop of the vibrating string on the weight hanger end. Pay attention to the proper use of significant figures.

Repeat the above observations by varying the values of the mass from 150 grams to at least 750 grams in 100 gram increments. Finally, remove the string from the vibrator, untie all the knots, and measure carefully both the mass of the string and its length. Then calculate the linear density, m , by dividing the mass by the length. Record this value for reference.

ANALYSIS OF RESULTS:

Begin your analysis by plotting two graphs: (1) the wave velocity versus tension and (2) the square of the wave velocity versus tension. Draw these graphs properly, paying proper attention to coordinate labels, titles, etc. What conclusions do you draw from looking at these two graphs?

The v 2 versus T graph should be a straight line. Determine the slope of that line, showing your work on the graph paper. The reciprocal of this slope should be the linear density of the string. Compute this value, and compare it with the number you obtained from direct measurement of mass and length. Use percent difference for your comparison.

Also use equation (3) to compute the theoretical frequency for each of the hanging mass values.  Compare these values to the measured frequency readings recorded earlier.  Compute the percent difference.

FINAL SUMMARY:

Report your average value for linear density. How did your two values compare? What is the relationship between the wave velocity and the tension? What additional conclusions can you make from this lab?

Be sure to answer the questions posed earlier in your summary.

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Vibrating Strings Standing Waves-Melde’s Experiment

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Lab Section 22

Professor Boley

Teachers Assistant: Hasib

In the lab, we did an experiment where we used a frequency mechanical wave diver to

demonstrate a mechanical wave. In this experiment we will produce standing waves on a string

clamped at one end and pulled over a rod and attached to a hanging mass at the other end. The

wave source consists of a modified loudspeaker connected to a function generator and power

amplifier to create the wavelengths that we will be seeing. A rod attached to the loudspeaker

presses against the string to wave as the speaker moves back and forth; it produces a periodic

wave disturbance on the string.

The experiment begins with a mass of 200 g hanging from the string to cause tension.

We turned the frequency until we obtained a standing wave of how many loops we needed. We

did this about 10 times to get 10 to 1 loops. We adjusted the amplitude to make sure we have a

close to a perfect standing wave. Then we pitched where the first node was and measured it to

the rod. We adjusted the frequency to 60 Hz where 10 loops could be seen and the

measurement in meters was calculated to be .3814m adjusted the frequency to 54 Hz

where 9 loops could be seen and the measurement in meters was calculated to be .4231m. We

adjusted the frequency to 49 Hz where 8 loops could be seen and the measurement in meters

was calculated to be .4688m adjusted the frequency to 44 Hz where 7 loops could be seen

and the measurement in meters was calculated to be .5271m adjusted the frequency to 38.

Hz where 6 loops could be seen and the measurement in meters was calculated to be .6073m

adjusted the frequency to 32 Hz where 5 loops could be seen and the measurement in meters

was calculated to be .7088m adjusted the frequency to 27 Hz where 4 loops could be seen

and the measurement in meters was calculated to be .8425m adjusted the frequency to 22.

Hz where 3 loops could be seen and the measurement in meters was calculated to be 1.

We adjusted the frequency to 16 Hz where 2 loops could be seen and the measurement in

meters was calculated to be 1. Finally, we adjusted the frequency to 10 Hz where 1 loop

could be seen and the measurement in meters was calculated to be 2.

Then we did the process again using a weight of 300 g hanging from the string. We

adjusted the frequency to 64 Hz where 10 loops could be seen and the measurement in meters

was calculated to be .3838m adjusted the frequency to 59 Hz where 9 loops could be seen

and the measurement in meters was calculated to be .4206m. We adjusted the frequency to 54.

Hz where 8 loops could be seen and the measurement in meters was calculated to be .4668m

adjusted the frequency to 49 Hz where 7 loops could be seen and the measurement in meters

was calculated to be .5291m adjusted the frequency to 42 Hz where 6 loops could be seen

wave divers arm going up and down as the wave is produced.

• Based on the predicted mathematical relationships on page 19, what is the qualitative relationship between the tension “T” and the frequency “f”, that is, when you increase the tension what should happen to the frequency (provided that there is little or no change in the wavelength)? Do the data you obtained in the two sections of the chart agree with the prediction? Why or why not?

When you increase the tension on the string the frequency should go up. The prediction is

supported by the data when looking at both strings of tension 1 and 2.

• Use the mathematical relationships on page 19 along with your data to predict what will happen to the resonant frequency of a standing wave in the same string of uniform linear mass density if the tension is quadrupled while the wavelength is held constant.

The linear mass will decrease if the tension is quadrupled while the wavelength is held constant

the frequency will double.

• Based on the data you obtained, if the tension and the linear mass density are held constant, what happens to velocity? Determine an average value of the velocity for the two different values of the tension using v = f λ. Compare these with the values obtained for the slopes of your graphs.

The string with tension 1 the average velocity was 23 m/sec and the string of

tension 2 the average velocity was 28 m/sec. The velocity shown from the slopes of the

graph resulted in 22 m/sec and 25 m/sec and the percent error was found to be 3% and

10% error with not a big difference.

• Describe a method to experimentally determine the type of mathematical relationship that exists between the mass per unit length (μ) and the wavelength (λ) of the standing wave.

If is constant then the graph that is created of T and μ then you can use “T” to calculate the

constant from the slope in the curve.

• Qualitatively, what would happen to the wavelength of the standing wave if the linear mass density was increased?

If is constant then when μ increase there would be a decrease in frequency causing an increase

in wavelength.

Part 1 – String - Tension 1 N Tension: .200 x 9 = 1 N Wavelength: 190 (10 = 38 = .3814m 190 (9 = 42 = .4231m 187 (8 = 46 = .4688m 184 (7 = 52 = .5271m 182 (6 = 60 = .6073m 177 (5 = 70 = .7088m

168 (4 = 84 = .8425m 157 (3 = 105 = 1 141 (2 = 141 = 1 106 (1 = 212 = 2 Velocity calculated: V= Comparison of % Error between Calculated “v” of String (Tension 1 N) and fitted Graph of Tension (1 N) verses Wavelength “v”: x 100 = 3% error Part 2 – String - Tension 2 N Tension: .300 x 9 = 2 N Wavelength: 191 (10 = 38 = .3838m 189 (9 = 42 = .4206m 186 (8 = 46 = .4668m 185 (7 = 52 = .5291m 179 (6 = 59 = .5970m 175 (5 = 70 = .7036m 166 (4 = 83 = .8305m 157 (3 = 104 = 1 136 (2 = 136 = 1 106 (1 = 212 = 2 Velocity calculated: V= Comparison of % Error between Calculated “v” of String (Tension 2 N) and fitted Graph of Tension (2 N) verses Wavelength “v”: x 100 = 10% error

• Multiple Choice

Course : Physics 115 (Phys 115)

University : western illinois university.

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