|VW - CVM 1.1|
In the previous example , a traditional separation of variables was possible because the translations occurred in the direction of the coordinate axes. Here, we construct eigenfunctions of the Laplacian that are periodic with respect to any lattice.
This picture shows a wallpaper function with translations in two directions that are not the simple coordinate directions. The idea here is to develop coordinates adapted to the lattice.
Suppose we wish to construct eigenfunctions periodic with respect to translations along two linearly independent vectors t1 and t2, and therefore along every vector in the lattice generated by these two.
Consider complex exponential functions of the form
Since we require
The most general eigenfunction periodic with respect to the lattice generated by t1 and t2 is
where n and m are integers, not necessarily positive.
For computational purposes, it will be convenient to expand this formula in
terms of the components of k1 and
Since, at this point, we are interested in real-valued functions, we now have cos(nX+mY) and sin(nX+mY), the real and imaginary parts of the complex wave above, as the only eigenfunctions periodic with respect to the given lattice. We call these functions lattice waves.
The basic result of Fourier analysis (see [Ga]), is that any function (say a continuous one) periodic with respect to the given lattice can be constructed as a superposition of these lattice waves. Thus, every recipe for wallpaper functions starts with these ingredients. To see what is involved in imposing other symmetries, continue with the example in the hexagonal lattice .