Section 25.4
Fourier Sine Series
If we odd-extend a function defined on , we can use just sines.
1
Introduction
This series is natural for problems with Dirichlet conditions (like the string).
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The Coefficients
Formula
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Worked Examples
Example 1: Square Wave
Expand on .
.
.
If n even, 0. If n odd, .
.
Example 2: Visualizing
Gibbs Phenomenon: The overshoot at jump discontinuities.
Example 3: Interpretation
What does this mean physically?
If a string is plucked in a flat shape (impossible, but theoretical), it creates these harmonics.
The "sound" component has amplitude .
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Practice Quiz
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