1. Standing Waves: Modes of Oscillation and Harmonic Series Phys/Mus 102 Spring 2015

2. Resonance Every oscillating system has a set of natural frequency, the frequencies at which the system “wants” to vibrate.

3. Key terms Resonance Reflection, fixed end, free end Superposition Standing wave Node/anti-node Harmonic series Mersenne’s Law Chladni pattern

4. Reflection Hard Boundary/Fixed End  Phase change! Soft boundary/Free End  No phase change

5. Standing Waves and Modes Result from superposition of incident waves and reflected waves Only certain combinations of wavelengths and frequencies allowed! = Mode

6. Harmonic Series f 1, 1 st harmonic (fundamental) f 2, 2 nd harmonic f 3, 3 rd harmonic

7. A few more in the series… Question: How can you excite mode or prevent certain ones?

8. Overtone series

9. At which frequencies do these modes occur? What physical variables matter?

10. Mersenne’s Law f 1 = fundamental frequency L = length of string T = tension in string  = mass per unit length of string Marin Mersenne

11. Mersenne’s Law Generalized Nth harmonic occurs at frequency: N = 1 = first harmonic N = 2 = second harmonic N = 3 = third harmonic …

12. Superposition of Modes

13. Modes on Guitar String

14. What happens with more complex objects: Pianos, Drums, Guitars, Violins, etc? Chladni Patterns

15. More Chladni Patterns (Guitar!) Instruments with complex structures have harmonic series/ resonance frequencies too!

16. Guitar Bracing

17. Chladni Patterns (Drums!) Animations courtesy of Dan Russel, Kettering University.

18. Piano soundboard modes

19. Chladni Patterns (Violin!)

20. Resources and Citations http://newt.phys.unsw.edu.au/jw/strings.html http://www.foreverparenting.com/1336/summer-fun-now-with-more-learning/ http://www.maestromindsguitarschool.com/guitar_website/Tools%20and%20Articles/guide _to_buying_your_first_guitar.html http://www.maestromindsguitarschool.com/guitar_website/Tools%20and%20Articles/guide _to_buying_your_first_guitar.html http://hep.physics.indiana.edu/~rickv/Standing_Waves_on_String.html http://www.animations.physics.unsw.edu.au/waves-sound/standing-waves/index.html Berg and Stork, Physics of Sound, 3 rd ed. http://www.phys.unsw.edu.au/~jw/guitar/guitarchladni_engl.html http://vanadium.rollins.edu/~tmoore/piano_soundboard.htm http://www.speech.kth.se/music/5_lectures/conklin/howdoes.html http://thecraftsmanspath.com/2008/04/06/dreadnought-guitar-bracing-the-top/