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Standing Waves

Standing Waves. Review Harmonic Oscillator Review Traveling Waves Formation of Standing Waves Standing Waves and Resonance Resonance Variables String instruments Examples Standing Waves and Electron Orbitals. Brazilian guitarist Badi Assad. Summary - Simple Harmonic Oscillator.

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Standing Waves

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  1. Standing Waves • Review Harmonic Oscillator • Review Traveling Waves • Formation of Standing Waves • Standing Waves and Resonance • Resonance Variables • String instruments • Examples • Standing Waves and Electron Orbitals Brazilian guitarist Badi Assad

  2. Summary - Simple Harmonic Oscillator • Energy • Motion • Harmonic frequency • Frequency and period

  3. Summary – Traveling Waves • Behaves as coupled harmonic oscillator • Motion of individual oscillators • Motion of disturbance between oscillators • Wave equation for string • F = ma on individual segment • Sinusoidal traveling solutions (also pulse) • Velocity • Properties of sinusoidal waves • Amplitude • Wavelength • Frequency/Period • Velocity • Other topics • Transverse vs. longitudinal • Energy spreading for spherical wave

  4. Traveling wave animation http://www.animations.physics.unsw.edu.au/waves-sound/travelling-wavesII

  5. Standing wave animation • Animation http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/reflect.html • Note Peaks oscillate in place • Nodes • Antinodes • Only certain wavelengths “fit” • Basis for stringed instruments (guitar/violin/piano) • Electromagnetic resonances • Quantum mechanics

  6. Formation of Standing Wave • Add incident wave traveling to right • With reflected wave traveling to left • And use trig identify: • Produces standing wave – oscillates in place

  7. Another Standing wave animation • Can turn on/off reflection http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html • Settings • Oscillate • No end / fixed end • Small, but some damping • Turn amplitude down to 1 • Note: almost gets out of control!

  8. Standing wave with both fixed ends • Both ends (approximately) fixed http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta2fix.html • Only certain wavelengths “fit”

  9. Standing waves and resonance • If string fixed both ends, only certain wavelengths “fit”

  10. Standing waves and resonance • String anchored between 2 points • Allowed opening widths • In general • Allowed wavelengths • ,2,3…. • Allowed frequencies

  11. Stringed Instruments • 2 key equations • Factors effecting frequency • String length (guitar frets) • Tension (violin tuning) • Mass/length (guitar vs. bass)

  12. Example 11-14 • Wave velocity needed for length, frequency, and n • Mass/length • Tension needed for velocity and mass/length • Harmonics

  13. Problem 53 • Assume same v and n for both frequencies. • Write original frequency • Write shortened frequency • Tale the ratio

  14. Problem 56 • Write harmonic (n) as function of fundamental • Write harmonic (n+1) as function of fundamental • Subtract the difference

  15. Problem 58 • Combining frequency and velocity equations • After tuning • Ratio Decrease 4.8%

  16. Problem 59 • Find velocity needed for number of antinodes • Find tension needed for those velocities

  17. Problem 59 (2) • Hold frequency constant and vary velocity with n • Normally we do allowed frequencies in terms of allowed wavelengths • Now we do allowed velocities in terms of allowed wavelengths, with frequency constant • The we do allowed tensions, assuming frequency constant • Masses are thus

  18. Problem 59 (3) • For 1 loop • For 2 loops • For 5 loops

  19. Standing waves and electron orbitals http://www.upscale.utoronto.ca/PVB/Harrison/Flash/QuantumMechanics/CircularStandWaves/CircularStandWaves.html (sorry the downloadable .swf doesn’t seem to work)

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