Wikipedia article on nodes offers this 1D illustration: a node is an interior point at which a standing wave does not move.
(At the endpoints the wave is forced to stay put, so I would not count them as nodes despite being marked on the plot.)
A standing wave in one dimension is described by the equation , where is its (angular) frequency. The function solves the wave equation : the wave vibrates without moving, hence the name. In mathematics, these are the (Dirichlet) eigenfunctions of the Laplacian.
Subject to boundary conditions (fixed ends), all standing waves on the interval are of the form for . Their eigenvalues are exactly the perfect squares, and the nodes are equally spaced on the interval.
Things get more interesting in two dimensions. For simplicity consider the square . Eigenfunctions with zero value on the boundary are of the form for positive integers . The set of eigenvalues has richer structure, it consists of the integers that can be expressed as the sum of two positive squares: 2, 5, 8, 10, 13, 17,…
The zero sets of eigenfunctions in two dimensions are called nodal lines. At a first glance it may appear that we have nothing interesting: the zero set of is a union of equally spaced horizontal lines, and equally spaced vertical lines:
But there is much more, because a sum of two eigenfunctions with the same eigenvalue is also an eigenfunction. To begin with, we can form linear combinations of and . Here are two examples from Partial Differential Equations by Walter Strauss:
When , the square is divided by nodal lines into 12 nodal domains:
After slight perturbation there is a single nodal line dividing the square into two regions of intricate geometry:
And then there are numbers that can be written as sums of squares in two different ways. The smallest is , with eigenfunctions such as
This is too good not to replicate: the eigenfunctions naturally extend as doubly periodic functions with anti-period .