Standing Waves: Modes of Oscillation and Harmonic Series

Phys/Mus 102

Spring 2015

Resonance

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

Key terms

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

Reflection

Hard Boundary/Fixed EndPhase change!

Soft boundary/Free End No phase change

Standing Waves and Modes

Result from superposition of incident waves and reflected waves Only certain combinations of wavelengths and frequencies allowed!

= Mode

Harmonic Series

f1, 1st harmonic (fundamental)

f2, 2nd harmonic

f3, 3rd harmonic

A few more in the series…

Question: How can you excite mode or prevent certain ones?

Overtone series

At which frequencies do these modes occur?

What physical variables matter?

Mersenne’s Law

f1 = fundamental frequency

L = length of stringT = tension in stringm = mass per unit length of stringMarin Mersenne

Mersenne’s Law Generalized

Nth harmonic occurs at frequency:

N = 1 = first harmonicN = 2 = second harmonicN = 3 = third harmonic…

Superposition of Modes

Modes on Guitar String

What happens with more complex objects: Pianos, Drums, Guitars, Violins, etc?

Chladni Patterns

More Chladni Patterns (Guitar!)

Instruments with complex structures have harmonic series/ resonance frequencies too!

Guitar Bracing

Chladni Patterns (Drums!)

Animations courtesy of Dan Russel, Kettering University.

Piano soundboard modes

Chladni Patterns (Violin!)

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://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, 3rd 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/