an eigenfunction
Think of the natural shapes a vibrating string falls into: a single smooth hump, then two humps with a still point in the middle, then three, and so on. Each is a self-consistent pattern the string can hold and oscillate in without changing form. An eigenfunction is exactly such a privileged shape — the standing-wave pattern that belongs to one permitted frequency of a boundary value problem.
Precisely, an eigenfunction is a non-trivial solution of a homogeneous boundary value problem at one of its special parameter values. If the eigenvalue problem is L[y] = lambda y with given boundary conditions — where L is a differential operator like taking minus the second derivative — then for each eigenvalue lambda_n there is an eigenfunction y_n satisfying L[y_n] = lambda_n y_n and the boundary conditions. The key feature is that the operator does not distort the function into something of a new shape; it merely scales it by the number lambda_n. For y'' + lambda y = 0 on [0, L] with zero ends, the eigenfunctions are y_n = sin(n pi x / L), each with its own number of arches.
Eigenfunctions are powerful because they form a complete, mutually orthogonal set: nearly any reasonable function on the interval can be written as a sum of them, just as ordinary Fourier series rebuild a signal from sines and cosines. That is why they are the natural building blocks for solving heat, wave, and Schrodinger equations. One caution: an eigenfunction is fixed only up to a constant multiple — if y_n works, so does 5 y_n — so people usually normalise it (rescale so its weighted length is 1) to pick a canonical representative.
For y'' + lambda y = 0 on [0, 1] with y(0) = y(1) = 0, the eigenfunction for lambda = 4 pi^2 is y = sin(2 pi x): it rises, returns to zero at x = 1/2, dips, and returns to zero at x = 1 — two arches. Differentiating twice gives y'' = -4 pi^2 sin(2 pi x), so the operator just multiplies y by -4 pi^2.
The n-th eigenfunction has exactly n arches and n-1 interior zeros — a clean fingerprint of which eigenvalue it belongs to.
An eigenfunction must be non-trivial: the zero function y = 0 technically satisfies L[0] = lambda·0 for every lambda, but it is excluded by definition, otherwise every number would be an eigenvalue and the idea would collapse.