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
𝐸= 1 2 𝑘 𝑥 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

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

4. Traveling wave animation

5. Standing wave animation
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
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

7. Another Standing wave animation
Can turn on/off reflection
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
Only certain wavelengths “fit”

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

10. Standing waves and resonance
String anchored between 2 points
Allowed opening widths
𝐿= 𝜆 2
𝐿=𝜆
𝐿= 3𝜆 2
In general
𝐿= 𝑛𝜆 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….

11. 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)

12. 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 𝑠 =288 𝑚 𝑠
Mass/length
𝜇= .009 𝑘𝑔 1.1 𝑚 = 𝑘𝑔 𝑚
Tension needed for velocity and mass/length
𝑣= 𝑇 𝜇 𝑇= 𝑣 2 𝜇= 𝑚 𝑠 𝑘𝑔 𝑚 =679 𝑁
Harmonics 𝑓 𝑛 =𝑛 𝑓 =131, 261, 393, 524 𝐻𝑧

13. 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 𝐿
Tale the ratio
𝑓′ 𝑓 = 𝑛𝑣 𝐿 𝑛𝑣 2𝐿 = 𝑓 ′ = 𝐻𝑧=441 𝐻𝑧

14. 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

15. 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 = =95.2 %
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
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

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

19. Standing waves and electron orbitals
(sorry the downloadable .swf doesn’t seem to work)