Nodal lines

Wikipedia article on nodes offers this 1D illustration: a node is an interior point at which a standing wave does not move.

Standing wave and its nodes
Standing wave and its nodes

(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:

Boring nodal lines
This is a square, not a tall rectangle

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:

Frequency 145, twelve nodal domains
Eigenvalue 145, twelve 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:

Also frequency 145, but two  nodal domains
Also eigenvalue 145, but two nodal domains

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.

Frequency 50
Frequency 50

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

Periodic extension
Periodic extension

1 thought on “Nodal lines”

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