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

• Motion

• Harmonic frequency

• Frequency and period

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

Traveling wave animation

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

clock_waveforms_1.swf

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

reflect.swf

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

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!

wave-on-a-string.swf

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”

sta2fix.swf

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

Standing waves and resonance• String anchored between 2 points

• Allowed opening widths

• In general

• Allowed wavelengths– ,2,3….

• Allowed frequencies

Stringed Instruments

• 2 key equations

• 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

• Mass/length

• Tension needed for velocity and mass/length

• Harmonics

Problem 53

• Assume same v and n for both frequencies.

• Write original frequency

• Write shortened frequency

• Tale the ratio

Problem 56

• Write harmonic (n) as function of fundamental

• Write harmonic (n+1) as function of fundamental

• Subtract the difference

Problem 58

• Combining frequency and velocity equations

• After tuning

• Ratio

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

• 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

Problem 59 (3)

• For 1 loop

• For 2 loops

• For 5 loops

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)