Acoustics and Vibration Animations
Daniel A. Russell
Graduate Program in Acoustics, The Pennsylvania State University

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Fourier Synthesis of Periodic Signals

Fourier composition of a square wave

Animation showing a square wave is built up from sine waves with integer multiple frequency components.

A square wave is built up from sine functions having odd-integer multiples of a fundamental frequency ( \( f_n = n f_1 \) with n being an odd integer) and amplitudes that decrease as \( 1/n \).

Fourier composition of a sawtooth wave

Animation showing a sawtooth wave is built up from sine waves with integer multiple frequency components.

A sawtooth wave is built up from sine functions having integer multiples of a fundamental frequency ( \( f_n = n f_1 \) with n being all integers) and amplitudes that decrease as \( 1/n \).

Fourier composition of a triangle wave

Animation showing a triangle wave is built up from sine waves with integer multiple frequency components.

A triangle wave is built up from sine functions having odd-integer multiples of a fundamental frequency ( \( f_n = n f_1 \) with n being an odd integer) and amplitudes that decrease as \( \frac{(-1)^n}{n^2} \).

Fourier composition of a pulse train

Animation showing a pulse wave is built up from sine waves with integer multiple frequency components.

An impulse (large amplitude but very short time duration) is created by adding together functions having integer multiples of a fundamental frequency ( \( f_n = n f_1 \) with n being all integers) and identical amplitudes.