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15-8. Normal Modes

A string with both ends fixed, length L, for standing wave to exist:

2λnL = n=1, 2, 3, …

nL

n2

=λ

Standing wave wavelength

Lvvf

211 ==

λ

Fundamental frequency

12nf

Lvnvf

nn ===

λ

Harmonics, or overtones if n>1

Wave functiontxkAtxy nnSWn ωsinsin ),( =

nth harmonic is (n-1)th overtone

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Harmonics

µF

LLvf

21

21 ==

Since v= µ/F

Normal mode: a motion in which all particles of the system movesinusoidally with the same frequency.

Resonant frequency: frequency at which standing waves are producedMultiple resonant frequencies / normal modes for a stringSingle resonant frequency for a spring / pendulam

for string fixed at both ends