## Nodal lines

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 ${f''+\omega^2 f=0}$, where ${\omega}$ is its (angular) frequency. The function ${u(x,t) = f(x)\cos \omega t}$ solves the wave equation ${u_{tt}=u_{xx}}$: the wave vibrates without moving, hence the name. In mathematics, these are the (Dirichlet) eigenfunctions of the Laplacian.

Subject to boundary conditions ${f(0)=0 = f(\pi)}$ (fixed ends), all standing waves on the interval ${(0,\pi)}$ are of the form ${\sin nx}$ for ${n=1,2,3,\dots}$. 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 ${Q=(0,\pi)\times (0,\pi)}$. Eigenfunctions with zero value on the boundary are of the form ${f(x,y) = \sin mx \sin ny}$ for positive integers ${m,n}$. 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 ${\sin mx \sin ny}$ is a union of ${n-1}$ equally spaced horizontal lines, and ${m-1}$ 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 ${\sin mx \sin ny}$ and ${\sin nx \sin my}$. Here are two examples from Partial Differential Equations by Walter Strauss:

When ${f(x,y) = \sin 12x \sin y+\sin x \sin 12y }$, the square is divided by nodal lines into 12 nodal domains:

After slight perturbation ${f(x,y) = \sin 12x \sin y+0.9\sin x \sin 12y }$ 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 ${50=1^2+7^2 = 5^2+5^2}$, with eigenfunctions such as

$\displaystyle f(x,y) = \sin x\sin 7y +2\sin 5x \sin 5y+\sin 7x\sin y$

pictured below.

This is too good not to replicate: the eigenfunctions naturally extend as doubly periodic functions with anti-period ${\pi}$.