A Fourier series represents a periodic function as a sum of sine and cosine waves. If a waveform repeats with fundamental frequency \(f\), then it can be written as
\[
f(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}\left(a_n\cos(2\pi nft)+b_n\sin(2\pi nft)\right).
\]
Each harmonic \(n\) oscillates at an integer multiple of the fundamental frequency, and the coefficients determine how strongly that harmonic contributes.
This idea is central in physics, engineering, acoustics, and signal processing. Complex periodic signals can be synthesized from simple sinusoidal components. In sound, for example, the harmonic content determines the waveform shape and strongly affects timbre.
A standard square wave of amplitude \(A\) has the Fourier series
\[
f(t)=\frac{4A}{\pi}\left(\sin(2\pi ft)+\frac{1}{3}\sin(2\pi 3ft)+\frac{1}{5}\sin(2\pi 5ft)+\cdots\right).
\]
Only odd harmonics appear, and all cosine coefficients are zero. In coefficient form,
\[
a_0=0,\qquad a_n=0,\qquad
b_n=
\begin{cases}
\dfrac{4A}{\pi n}, & n \text{ odd},\\[4pt]
0, & n \text{ even}.
\end{cases}
\]
A sawtooth wave uses all harmonics, with amplitudes that decay like \(1/n\). A triangle wave uses only odd harmonics again, but its coefficients decay faster, like \(1/n^2\), which makes the approximation smoother and the waveform less sharp.
In practice, a synthesizer or numerical tool uses a finite number of harmonics,
\[
f_N(t)=\frac{a_0}{2}+\sum_{n=1}^{N}\left(a_n\cos(2\pi nft)+b_n\sin(2\pi nft)\right).
\]
This is called a partial sum. As \(N\) increases, the approximation gets better over most of the interval. However, near jump discontinuities, something special happens: the approximation overshoots the target waveform. This is the Gibbs phenomenon.
For a square wave, even with many harmonics, the oscillations near the jump do not disappear completely. They become narrower, but the overshoot remains at a fixed fraction of the jump height. This is an important feature of Fourier approximations and is not a numerical bug.
For the sample setup “square wave, harmonics = 10,” the calculator uses
\[
b_n=\frac{4}{\pi n}
\]
for odd \(n\) and zero for even \(n\). So the first few nonzero terms are
\[
\frac{4}{\pi}\sin(2\pi ft)
+\frac{4}{3\pi}\sin(2\pi 3ft)
+\frac{4}{5\pi}\sin(2\pi 5ft)
+\cdots
\]
up to the chosen harmonic count. The resulting graph shows how these harmonics combine to approximate the ideal square wave.
At more advanced level, Fourier ideas extend to complex exponentials, transforms, and discrete FFT methods. But the finite Fourier series used here is the essential starting point for understanding how harmonic addition creates familiar periodic waveforms.
This calculator lets you inspect the coefficients, visualize the synthesized waveform, compare it with the ideal target, and watch the harmonic build-up directly in the contained animation.
