Section 26.8
The Vibrating String
Applying the Wave Equation to music and harmonics.
1
Introduction
Problem:
BCs: (Fixed ends)
ICs: (Initial shape) and (Initial velocity).
2
Normal Modes
Superposition
are the natural frequencies.
3
Worked Examples
Example 1: Plucked String
Initial velocity .
Then (no sine terms in time).
.
are regular Fourier Sine coeffs of the initial shape .
Example 2: Struck String
Initial shape , velocity (Piano/Hammer).
Then .
at is .
We expand into sine series, but coeff is .
Example 3: Harmonics
Press play (hypothetically) to see the complex vibration.
4
Practice Quiz
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