the normal modes
Strike a bell, a wine glass, or a guitar string and it does not move at random — it settles into a handful of pure, persistent patterns of vibration, each with its own fixed pitch. These special patterns are the normal modes: the natural shapes in which a system likes to vibrate, every part of it moving up and down together at one shared frequency.
For a string of length L pinned at both ends, separation of variables finds them exactly. Guess u(x,t) = X(x)T(t), substitute into u_tt = c^2 u_xx, and divide by X T: the x-part must equal the t-part, so both equal a constant, giving two ordinary differential equations. The fixed ends force X(x) = sin(n pi x / L) for n = 1, 2, 3, ... — these are the spatial shapes, the modes — and the matching T(t) oscillates at frequency f_n = n c / (2 L). So the n-th normal mode is a sine shape with n humps, vibrating at n times the base frequency. The number n counts the half-wavelengths that fit on the string, and the points that stay still are the nodes.
Normal modes are the building blocks of all vibration: because the wave equation is linear, any motion of the string is a superposition (a sum) of normal modes with various strengths, and that sum is precisely a Fourier sine series. The strength of each mode is its Fourier coefficient. This is why the same idea reappears everywhere — the eigenfunctions of a Sturm-Liouville problem, the standing waves of a drum (Bessel-function modes), the orbitals of an atom. Finding a system's normal modes is finding the natural alphabet in which its behaviour is written.
On a guitar string, the fundamental (n = 1) is a single arch with no interior nodes. Lightly touch the exact midpoint and pluck: you silence n = 1 and force a node there, so the string rings on the n = 2 mode (two arches) one octave higher — a guitar harmonic.
Mode n has n humps and n - 1 interior nodes; their sum is a Fourier sine series.
Normal modes in this clean, evenly spaced form require an ideal one-dimensional string. Real drums, bells, and stiff strings have modes whose frequencies are NOT simple integer multiples, which is exactly why they sound less 'tuneful' than a string.