vibrating string
Pin a string down at both ends, pluck it, and let go: it springs into motion as a shimmering blend of arches — one big arch, plus a faster two-arch wiggle, plus a three-arch one, all sounding together. This is the problem Fourier series were practically invented to solve, the place where 'a signal is a sum of harmonics' meets real physics. The string's possible shapes are exactly sine curves that vanish at the two fixed ends, and its motion is a sum of those shapes each oscillating in time.
The motion u(x,t) of a string of length L fixed at both ends obeys the wave equation, u_tt = c^2 u_xx, with u(0,t) = u(L,t) = 0. Separation of variables says look for products X(x) T(t); the boundary conditions force X(x) = sin(n pi x / L), the standing-wave modes, and each mode oscillates in time at its own frequency, c n pi / L. The general solution is a Fourier sine series in space whose coefficients ride in time: u(x,t) = sum over n of sin(n pi x / L) [ A_n cos(c n pi t / L) + B_n sin(c n pi t / L) ]. To match the initial pluck shape and initial velocity, you expand them as half-range sine series — and the Euler-Fourier formulas hand you A_n and B_n directly. Each n is a normal mode; the n = 1 mode is the fundamental tone, the higher n are the overtones.
This is the canonical bridge from Fourier series to partial differential equations, and the physical reason musical strings make harmonious sound: the overtone frequencies are exact integer multiples of the fundamental, c n pi / L, because the mode shapes are evenly indexed sines. The honest qualifier: this clean harmonic ladder relies on idealizations — perfect flexibility, small amplitude, uniform tension and density. Real strings have stiffness that pushes the overtones slightly sharp (inharmonicity), which is why a real piano is tuned by ear, not by the formula alone.
Pluck a string into a triangle shape centered at L/2 and release it from rest: only odd modes are excited, and u(x,t) = sum over odd n of (8h/(n^2 pi^2)) sin(n pi/2) sin(n pi x/L) cos(c n pi t/L) — the spatial profile is a half-range sine series, the time factor is a pure cosine because the string starts at rest.
Where you pluck decides which harmonics sing — plucking at the center silences every even mode.
The neat result that overtones are exact integer multiples of the fundamental is a property of the idealized flexible string; real strings have bending stiffness that makes the overtones progressively sharp, so the perfect harmonic series is a model, not the literal physics.