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

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Presentation transcript:

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 End  Phase 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 f 1, 1 st harmonic (fundamental) f 2, 2 nd harmonic f 3, 3 rd 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 f 1 = fundamental frequency L = length of string T = tension in string  = mass per unit length of string Marin Mersenne

Mersenne’s Law Generalized Nth harmonic occurs at frequency: N = 1 = first harmonic N = 2 = second harmonic N = 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 _to_buying_your_first_guitar.html _to_buying_your_first_guitar.html Berg and Stork, Physics of Sound, 3 rd ed