Standing Waves: Modes of Oscillation and Overtone Series

The Week Ahead…
Mon: Waves on stringed instruments (lab 4)
Tues: Oscillations in wind instruments (lab 5)
Wed: No class; work on your labs and hw assignment
Thur: Harmonic Series Analysis (lab 6)
Fri: Fun demos!
Labs 4 and 5 due Friday at beginning of class.
HW 1 due Friday, 5pm
Standing Waves: Modes of
Oscillation and Harmonic Series
Phys/Mus 102
Spring 2014
Every oscillating system has a natural frequency.
Force on object occurs exactly in phase with the displacement,
leading to a “build up” of maximal displacement.
Key terms
Reflection, fixed end, free end
Standing wave
Harmonic series
Mersenne’s Law
Chladni pattern
Hard Boundary/Fixed End
Phase change!
Soft boundary/Free End
 No phase change
Standing Waves
Result from superposition of incident waves and reflected
Only certain wavelengths/frequencies are allowed!
Harmonic Series
f1, 1st harmonic
f2, 2nd harmonic
f3, 3rd harmonic
A few more in the series…
Question: What will you hear if you touch your finger to the middle of the string?
The overtone series notes
Mersenne’s Law
f1 = fundamental frequency
L = length of string
T = tension in string
m = mass per unit length of string
Mersenne’s Law Generalized
Nth harmonic occurs at
N = 1 = first harmonic
N = 2 = second harmonic
N = 3 = third harmonic
Chladni Patterns
What happens with
more complex objects:
Pianos, Drums, Guitars,
Violins, etc?
Modes on Guitar String
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!)
Image/Animation Sources
Berg and Stork, Physics of Sound, 3rd ed.

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