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Traveling and Standing Waves

Traveling and Standing Waves. Review Excel Animation Superposition of + / - waves Standing wave animation Resonance conditions Stringed instruments Examples Power in 3d traveling wave. Review. Wave – coupled harmonic oscillator http :// www.youtube.com/watch?v=buNchmyile8

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Traveling and Standing Waves

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  1. Traveling and Standing Waves • Review • Excel Animation • Superposition of + / - waves • Standing wave animation • Resonance conditions • Stringed instruments • Examples • Power in 3d traveling wave

  2. Review • Wave – coupled harmonic oscillator http://www.youtube.com/watch?v=buNchmyile8 • Wave equation Derivation • Newton’s 2nd law on segment • Traveling wave solution • v = sqrt(T/ μ) • Pulse wave vs. periodic wave (slinky) • Transverse vs. longitudinal wave (animation) • Properties of periodic waves • Amplitude • Wavelength • Frequency, Period • Velocity

  3. Traveling periodic wave • Excel spreadsheet • Y = A sin (kx – ωt) to right • Y = A sin (kx + ωt) to left • Snapshot in time • Motion at fixed position • v = f λ • What are k and ω?

  4. Superposition of Waves • Standing wave Animation • Superimpose wave in + with wave in – Y = A sin (kx – ωt) + A sin (kx + ωt) • Use trig identity Y = A sin(kx) cos(ωt) + A cos(kx) sin(ωt) + A sin(kx) cos(ωt) - A cos(kx) sin(ωt) • Result Y = 2A sin(kx) cos(ωt) • Standing wave, not traveling wave!

  5. Standing Wave Animation • Animation • Y = A sin(kx) cos(ωt) • “Nodes” and “Antinodes” remain fixed • Basis for stringed instruments • Violin • Guitar • Piano • Electromagnetic resonances • Quantum mechanics

  6. Standing waves and resonance • String anchored between 2 points • Allowed wavelength • L = λ/2 • L = λ • L = 3λ/2 • In general L = nλ/2 , n = 1,2,3 … • Or λ = 2L/n, n = 1,2,3 …. • Or f = n v/2L = nf1 n = 1,2,3, …..

  7. Stringed instruments • f = n v/2L and v = sqrt(T/μ) • Frequency goes up as: • String gets shorted (examples) • Tension goes up (examples) • Mass/unit length goes down (examples) • Examples • Guitar • Piano

  8. Examples • Problem 57 • Example 11-14 • Problem 53 • Problem 56 • Problem 57 • Problem 59

  9. Power in 3-dimensional Traveling Wave • Energy harmonic oscillator ½ k A2 • Intensity = power/area • Intensity = power/4π r2 • I2/I1 = 4π r12 / 4π r22 • Examples • Example 11-13 • Problem 46

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