A           P           P          A           R           A           T          U           S                           ...
Fig. 3. Standing waves in a string appear colored when viewed in fluorescent light, which emits dif-                      ...
antinodes an odd number of half-               rotation also switches, as a strobo-      the slope is greatest. “Transvers...
erican Association of Physics                  Garelick, Am. J. Phys. 43, 926              20. David Halliday, Robert Resn...
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Melde’s experiment with an aquarium aerator


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Melde’s experiment with an aquarium aerator

  1. 1. A P P A R A T U S FOR TEACHING PHYSICSColumn Editor: Karl C. Mamola, Department of Physics and Astronomy, Appalachian StateUniversity, Boone, NC 28608; mamolakc@appstate.eduMelde’s Experiment with an Aquarium AeratorRich Dynamics with Inexpensive ApparatusMark Graham, Department of Physics and Astronomy, University of Alabama, PO Box 870324, Tuscaloosa, AL35487-0324; mark@voicemall.comM elde’s experiment con- sists of a taut string with a periodic drivingforce applied to it.1-3 With the prop-er conditions for resonance, thestring will vibrate with great ampli-tude. A typical standing-wave pat-tern has large-amplitude vibrationthroughout the string between pointsof no motion called nodes, whichsurprise students so much that theyoften must touch them to believethey are there! Standing waves areusually explained as interferencebetween transmitted and reflectedwaves,4 but especially for those Education majors taught by Dean Stan Jones explore standing waves at University of Alabama.uncomfortable with the oxymoron Left to right: Anna Craft, Lucie Klingler, and Jaime Ludack.“standing wave,” the pattern can alsobe explained as the eigenmodes of in the string has a length correspond- by adjusting the tension on a string.oscillation of a continuous system.5 ing to one-half wavelength of the Less expensive commercial vibratorsThe distance between adjacent nodes wave traveling through the string. are available that oscillate a metal Sometimes plate under an electromagnet driven Melde’s experiment by line current.7 is done with signal Noncommercial vibrators have generators passing been made from loudspeak- current through a ers,8 doorbell-clappers,9 buzzers,10 wire in a magnetic motors,11 hair cutters,3 jigsaws,3 and field, allowing the ac-dc converters,12 but the method I student to tune to describe here came from a childhood resonance by tuning memory of when my older brother the frequency (our Bob showed me the inside of an former method).6 At aquarium aerator. An aerator consists constant frequency, of an electromagnet driven by line it is also possible to current that shakes a magnet glued to tune to resonance by the end of an armature. This in turnFig. 1. Inside an aquarium aerator. Electromagnet (left) drives perma- adjusting the speed drives the bellows that pump the air.nent magnet (middle) on one end of armature. This drives bellows(right) that pump air through spout (right). String tied to end of arma- of the transverse The frequency of the vibration is theture as shown used to create standing waves. wave, accomplished line frequency of the current, which276 THE PHYSICS TEACHER Vol. 36, May 1998 Melde’s Experiment with an Aquarium Aerator
  2. 2. Fig. 3. Standing waves in a string appear colored when viewed in fluorescent light, which emits dif- ferent intensities of color throughout cycle. Viewed from above, standing wave appears blue on the edges; viewed sideways, edges appear orange. weighing a long piece of string of system is near resonance. Second, the known length for the linear density , great tension required to achieve and computing c = T/ . The stan- small numbers of antinodes will dard formula = 2L/n, which in- eventually tug the vibrating arm so cludes the assumption that the length far away from the electromagnet that of the string corresponds to an inte- it is no longer driven. You can pre- gral number of “swells,” should not vent this by clamping the aerator atFig. 2. Left: standing wave in homogeneous be used because the vibrator position an angle so the arm is as much paral-string for which each swell has the same is neither a node nor an antinode.13 If lel to the direction of the string ten-length, even though they appear smaller in the you look closely at one of the stand- sion as possible. Finally (as teachersdistance. Right: tapered fly line grows thicker ing waves made by this device, the at our 1997 summer workshop dis-with distance. Swells in foreground are longer swell nearest the vibrator is shorter covered), if the pump is preventedthan neighbor to left, but swells in distancebecome smaller, implying that wave is slower than the others. The wavelength is from pumping air, the standing wavein thicker end of fly line. measured as twice the distance amplitude will be greatly enhanced. between two nodes.13 Many modes You can glue the aerator’s spoutis 60 Hz in the United States, and is may be obtained, and we have gotten closed or insert one of the backingthe source of the obnoxious drone as few as two antinodes for a length screws into the spout. When all ofyou hear in a pet store. [Note: of string greater than a meter. Upon these techniques were applied, we gotCommercial vibrators produce a fre- graphing wavelength versus speed, antinode widths sometimes of an inchquency of twice the line frequency students measured a line frequency and a half !because the electromagnet drives a with no more than 5% error.steel plate (no permanent magnet) A few techniques will maximize Phenomena with Standingthat is attracted to the electromagnet the amplitude of the standing waves. Waves in a Stringtwice each cycle.] First, the tension can be carefully The criterion for the formation of tuned by using a spoon to deliver a standing wave is that after transmis-Apparatus Setup sand slowly into the bucket when the sion, the reflected wave returns to To use the aerator to drive a string,unscrew the bottom of the pump(unplugged of course) and tie a stringto the end of the armature (see Fig.1). It is prudent to wrap tape aroundthe transformer end to prevent mav-erick fingers from touching electricalconnections. You can immediatelyshow standing waves by just holdingthe pump in one hand and tugging onthe string with the other. For a morecontrolled demonstration, secure theaerator to the table with a C clampand run the string over a pulley. Hanga cup containing gunshot, sand, oreven water on the end and adjust theweight to create the standing wave. Fig. 4. Mode created when typical twirling string sways at half the vibrator frequency. Close-upsThe speed of the waves, c, is found of strings cross section in lower left shows two typical Lissajous trajectories of an element ofby weighing the cup for the tension T, string, photographed by sweeping a laser though the standing wave.Melde’s Experiment with an Aquarium Aerator Vol. 36, May 1998 THE PHYSICS TEACHER 277
  3. 3. antinodes an odd number of half- rotation also switches, as a strobo- the slope is greatest. “Transverse”periods later. This means that a scope reveals. waves might better be referred to asstanding wave can form even if the The picture on the cover was taken planar waves, because the stretchingwave speed is not constant through- by first photographing the vibrating of an element of string means a pointout the string. With the additional string under stroboscopic light at five on the string undergoes both trans-piscine resource of a tapered fly line, times its vibrational frequency. A verse and longitudinal motion. Thisyou can make standing waves with side-by-side pair of red and green means energy can be transmitted lon-varying wavelength, showing that the laser beams were then sliced perpen- gitudinally through the node.22wavelength shortens as the string dicularly through the vibrating string. The stretching of the string makesgets thicker, proving that the speed The resulting picture reveals that the equations of motion nonlinear.decreases with the linear density of when this is repeated at intervals This effect results in exceedingly richthe string14 (see Fig. 2). along the length of the string, the dynamics when driven sufficiently Students are mesmerized when motion of a string particle is indeed hard.23 When the tension is slowlythe device is shown with a strobe circular. changed, you can observe suchlight at the line frequency. (But, stro- A more sophisticated trajectory diverse behavior as beats, slow sway-boscopic light can induce epileptic occurs (see Fig. 4) when an addition- ing and rotation of vibrational planes,seizures, so epileptics should be al mode at half-line frequency devel- switching from planar to rotationalwarned not to watch.) Observers can ops in one plane while the string is motion, and jumps in amplitude.distinctly see the pump arm “slowly” being tuned to an even number of In about an hour and with a totaloscillating back and forth. The strobe swells. Half-frequency vibrations cost of about ten dollars per appara-light reveals that the string vibration occur when a string is being driven tus, teachers at our Advancedis usually circular motion (see cover longitudinally rather than transverse- Placement Physics Teachers Work-photo), just as when children play ly. If you were holding a planar shop24 constructed a standing- wavejump rope, and the rope appears as a standing wave in a rope, from its generator from an aquarium aerator,rigid structure just rotating, not oscil- stretching you would feel the tension took the data, and deduced the vibra-lating. Rotation is natural for a sys- increase above default twice each tion frequency. I hope that othertem being driven only along one period. You might try to excite the teachers will find this an exciting anddirection.15 The twirling string may wave by just varying the tension, one easy laboratory as well.be thought of as two transverse example of parametric excitation.19waves in orthogonal planes a quarter However, by driving your hand longi- Acknowledgmentscycle apart, just as two modes of tudinally back and forth at the same Great thanks are due Jerrypolarization describe circularly frequency of the existing standing Busenitz, Gene Byrd, Ronald Edge,polarized light. wave, you would only increase the Stan Jones, J. W. Harrell, our gradu- The twirling string may be literal- tension above default once each peri- ate students, and the participants ofly viewed this way if it is lit by fluo- od, producing a standing wave at half the Advanced Placement Physicsrescent lighting. You may notice red- frequency. The tendency for the Teachers Workshop for their sugges-dish and bluish tints to the standing string to jump into this mixed subhar- tions. Our Educational Media depart-wave (see Fig. 3), especially if the monic mode can be annoying when ment was invaluable in advising mestring is white and viewed against a you’re taking data for Melde’s exper- how to take the photographs appear-black background.16 Fluorescent iment itself, but can be quelled by ing in this article. I also thank The Finlights do not emit all colors simulta- pinching the string at one of the Inn, John’s Photo, and The Wormneously with equal intensity along swaying would-be nodes of the Shack of Tuscaloosa, Alabama, forthe phase of their line cycle, which is expected pattern. their assistance in the production ofthe same frequency as the vibration The ability to pinch a node while this demonstration. David Burba ofof the string.17 When viewed from leaving the pattern beyond unaffected Vanderbilt University was essentialabove, the wave appears one way, makes transverse waves difficult to in pointing out some of the referencesperhaps red in the middle and blue on believe in. Although planar waves in used in this article.the edges, but when viewed side- a string are presented as the archetypeways, the colors are swapped of transverse waves20 and are even Referencesbecause of the quarter-cycle differ- modeled this way in advanced 1. Richard Manliffe Sutton,ence in phase. When you tune past mechanics books,21 the motion of an Demonstrations Experiments inmaximum resonance of the standing element of string in a planar vibration Physics (McGraw Hill, Newwaves, the color scheme will switch cannot in general be just strictly York, 1938), p. 143. 2. George D. Freir and Frances J.because of the change in phase of the transverse. To vibrate, the string must Anderson, A Demonstrationstring’s vibration with respect to the stretch, and the stretching of an ele- Handbook for Physics (Am-driving force.18 The direction of the ment of string is a maximum where278 THE PHYSICS TEACHER Vol. 36, May 1998 Melde’s Experiment with an Aquarium Aerator
  4. 4. erican Association of Physics Garelick, Am. J. Phys. 43, 926 20. David Halliday, Robert Resnick Teachers, 1996), p. S-6. (1975). and Jearl Walker, Funda- 3. Philip Johnson, Karl Trappe et 11. Wallace A. Hilton, Am. J. Phys. mentals of Physics (Wiley, al., PIRA Demonstration Bib- 20, 310 (1952). 1997), p. 401. liograph (Physics Instructional 12. Daniel Nashol, Am. J. Phys. 33, 21. Alexander L. Fetter and John Resource Association, 1997), 856 (1965). Dirk Walecka, Theoretical 3B22.10–11.. 13. David Halliday and Robert Mechanics of Particles and 4. Paul G. Hewitt, Conceptual Resnick, Physics for Students Continua (McGraw-Hill, Physics (Little, Brown, & of Science and Engineering 1980), p. 221. Company, 1985), p. 285. (Wiley, 1960), p. 417. 22. Reuben Benumof, Am. J. Phys. 5. J. Rekveld, Am. J. Phys. 26, 14. Richard B. Minnix and D. Rae 48, 387–392 (1980). 159–163, (1958). Carpenter, Jr., Phys. Teach. 21, 23. John A. Elliot, Am. J. Phys. 50, 6. PASCO scientific, PO Box 53–54 (1983). 1148–1150 (1982). 619011, 10101 Foothills Blvd., 15. Robert W. Leonard, Am. Phys. 24. For information on participat- Roseville, CA 95678-9011. Teach. 5, 175–176 (1937). ing in our summer Advanced 7. Central Scientific Company, 16. Sue Gray AlSalam and Ronald Placement Teachers Workshop, 3300 CENCO Parkway, D. Edge, Phys. Teach. 18, 518 contact Rebecca Pow at Franklin Park, IL 60131. (1980). rpow@ccs.ua.edu, 205-348- 8. F. P. Clay, Jr. and R. L. Kernell, 17. Salvatore Ganci, Am J. Phys. 3021 (University of Alabama, Am. J. Phys. 50, 910–912 52, 250–251 (1984) College of Continuing Studies, (1982). 18. A. S. McWilliams, Am. J. Phys. PO Box 870388, Tuscaloosa, 9. David D. Lockhart, Phys. 43, 1112 (1975). AL 35487) or A. K. Smith, Teach. 9, 283 (1971). 19. A B. Pippard, The Physics of Chapel Hill High School, 1709 10. Alan H. Cromer and David Vibration (Cambridge, 1978) High School Road, Chapel Hill, Chap. 10. NC 27516.Melde’s Experiment with an Aquarium Aerator Vol. 36, May 1998 THE PHYSICS TEACHER 279