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 Energy 𝐸= 1 2 𝑘 𝑥 2 + 1 2 𝑚 𝑣 2 = 1 2 𝑘 𝐴 2 = 1 2 𝑘 𝑣 𝑚𝑎𝑥 2 Motion 𝑥=𝐴𝑠𝑖𝑛 𝜔𝑡 𝑜𝑟 𝑥=𝐴𝑐𝑜𝑠 𝜔𝑡 Harmonic frequency 𝜔= 𝑘 𝑚 Frequency and period 𝜔=2𝜋𝑓 𝜔= 2𝜋 𝑇

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 𝑣= 𝑇 𝜇 Properties of sinusoidal waves Amplitude Wavelength Frequency/Period Velocity 𝑣=𝑓𝜆 Other topics Transverse vs. longitudinal Energy spreading for spherical wave

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

Standing wave 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

Formation of Standing Wave Add incident wave traveling to right 𝑌 𝑖𝑛𝑐 =𝐴𝑠𝑖𝑛(𝑘𝑥−𝜔𝑡) With reflected wave traveling to left 𝑌 𝑟𝑒𝑓 =𝐴𝑠𝑖𝑛(𝑘𝑥+𝜔𝑡) And use trig identify: sin 𝑎±𝑏 = sin 𝑎 cos⁡(𝑏)± cos 𝑎 sin⁡(𝑏) 𝑌 𝑡𝑜𝑡 = 𝑌 𝑖𝑛𝑐 + 𝑌 𝑟𝑒𝑓 =2𝐴𝑠𝑖𝑛 𝑘𝑥 cos⁡(𝜔𝑡) Produces standing wave – oscillates in place

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!

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”

Standing waves and resonance If string fixed both ends, only certain wavelengths “fit” 𝑳= 𝝀 𝟐 𝑳=𝝀 𝑳= 𝟑 𝟐 𝑳

Standing waves and resonance String anchored between 2 points Allowed opening widths 𝐿= 𝜆 2 𝐿=𝜆 𝐿= 3𝜆 2 In general 𝐿= 𝑛𝜆 2 𝑛=1,2,3… Allowed wavelengths 𝜆 𝑛 = 2𝐿 𝑛 𝑛=1,2,3…. Allowed frequencies 𝑓 𝑛 = 𝑣 𝜆 𝑛 = 𝑛𝑣 2𝐿 𝑛=1,2,3….

Stringed Instruments 2 key equations Factors effecting frequency 𝑓 𝑛 = 𝑣 𝜆 𝑛 = 𝑛𝑣 2𝐿 𝑛=1,2,3…. 𝑣= 𝑇 𝜇 Factors effecting frequency String length (guitar frets) Tension (violin tuning) Mass/length (guitar vs. bass)

Example 11-14 Wave velocity needed for length, frequency, and n 𝑓 1 = 𝑣 𝜆 1 = 1𝑣 2𝐿 𝑣=2𝐿 𝑓 1 = 2.2 𝑚 131 𝑠 =288 𝑚 𝑠 Mass/length 𝜇= .009 𝑘𝑔 1.1 𝑚 =0.0082 𝑘𝑔 𝑚 Tension needed for velocity and mass/length 𝑣= 𝑇 𝜇 𝑇= 𝑣 2 𝜇= 288 𝑚 𝑠 2 .0082 𝑘𝑔 𝑚 =679 𝑁 Harmonics 𝑓 𝑛 =𝑛 𝑓 1 =131, 261, 393, 524 𝐻𝑧

Problem 53 𝑓 𝑛 = 𝑛𝑣 2𝐿 𝑓 𝑛 ′= 𝑛𝑣 2 2 3 𝐿 Assume same v and n for both frequencies. Write original frequency 𝑓 𝑛 = 𝑛𝑣 2𝐿 Write shortened frequency 𝑓 𝑛 ′= 𝑛𝑣 2 2 3 𝐿 Tale the ratio 𝑓′ 𝑓 = 𝑛𝑣 2 2 3 𝐿 𝑛𝑣 2𝐿 = 3 2 𝑓 ′ = 3 2 294 𝐻𝑧=441 𝐻𝑧

Problem 56 Write harmonic (n) as function of fundamental 𝑓 𝑛 = 𝑛𝑣 2𝐿 =𝑛 𝑓 1 Write harmonic (n+1) as function of fundamental 𝑓 𝑛+1 = (𝑛+1)𝑣 2𝐿 =(𝑛+1) 𝑓 1 Subtract the difference 𝑓 𝑛+1 − 𝑓 𝑛 = 𝑛+1 𝑓 1 −𝑛 𝑓 1 = 𝑓 1 𝑓 𝑛+1 − 𝑓 𝑛 =350 𝐻𝑧 −280 𝐻𝑧=70 𝐻𝑧= 𝑓 1

Problem 58 Combining frequency and velocity equations After tuning 𝑓 𝑛 = 𝑛𝑣 2𝐿 = 𝑛 𝑇 𝜇 2𝐿 After tuning 𝑓 𝑛 ′= 𝑛𝑣 2𝐿 = 𝑛 𝑇 ′ 𝜇 2𝐿 Ratio 𝑓′ 𝑓 = 𝑇′ 𝑇 𝑇′ 𝑇 = 𝑓′ 2 𝑓 2 = 200 2 205 2 =95.2 % Decrease 4.8%

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

Problem 59 (2) Hold frequency constant and vary velocity with n Allowed wavelengths 𝜆 𝑛 = 2𝐿 𝑛 𝑛=1,2,3…. Normally we do allowed frequencies in terms of allowed wavelengths 𝑓 𝑛 = 𝑣 𝜆 𝑛 = 𝑛𝑣 2𝐿 Now we do allowed velocities in terms of allowed wavelengths, with frequency constant 𝑣 𝑛 =𝑓 𝜆 𝑛 =𝑓 2𝐿 𝑛 The we do allowed tensions, assuming frequency constant 𝑣= 𝑇 𝜇 𝑇 𝑛 =𝜇 𝑣 𝑛 2 = 4𝜇 𝑓 2 𝐿 2 𝑛 2 Masses are thus 𝑚 𝑛 = 𝑇 𝑛 𝑔 = 4𝜇 𝑓 2 𝐿 2 𝑔 𝑛 2

Problem 59 (3) For 1 loop 𝑚 𝑛 = 4𝜇 𝑓 2 𝐿 2 𝑔 𝑛 2 = 4 3.9∙ 10 −4 𝑘𝑔 𝑚 60 𝑠 2 1.5 𝑚 2 9.8 𝑚 𝑠 2 1 2 =1.29 𝑘𝑔 For 2 loops 𝑚 𝑛 = 4𝜇 𝑓 2 𝐿 2 𝑔 𝑛 2 = 4 3.9∙ 10 −4 𝑘𝑔 𝑚 60 𝑠 2 1.5 𝑚 2 9.8 𝑚 𝑠 2 2 2 =0.32 𝑘𝑔 For 5 loops 𝑚 𝑛 = 4𝜇 𝑓 2 𝐿 2 𝑔 𝑛 2 = 4 3.9∙ 10 −4 𝑘𝑔 𝑚 60 𝑠 2 1.5 𝑚 2 9.8 𝑚 𝑠 2 5 2 =0.052 𝑘𝑔

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)