The simple physical situations involving waves on an L-shaped region include a vibrating L-shaped membrane, or tambourine, and a beach towel blowing in the wind, constrained by a picnic basket on onefourth of the towel. A more practical example involves microwave waveguides. One such device is a waveguide-to-coax adapter. The active region is the channel with the H-shaped cross section visible at the end of the adapter. The ridges increase the bandwidth of the guide at the expense of higher attenuation and lower power-handling capability. Symmetry about the dotted lines in the contour plot of the electric field in the channel implies that only one-quarter of the cross section needs to be modeled and that the resulting geometry is our L-shaped region. The boundary conditions are not the same as the membrane problem, but the differential equation and the solution techniques are the same.



You can use classic finite difference methods to compute the eigenvalues and eigenfunctions of the L-shaped membrane in MATLAB with

n = 200 h = 1/n A = delsq(numgrid('L',2*n+1))/h^2 lambda = eigs(A,12,0)

The resulting sparse matrix A has order 119201 and 594409 nonzero entries. The eigs function uses Arnoldi’s method from the MATLAB implementation of ARPACK to compute the first 12 eigenvalues. This takes only a little over a minute on a 1.4 GHz Pentium laptop. However, the corner singularity causes the computed eigenvalues to be accurate to only three or four significant digits. If you try for more accuracy with a finer mesh and a larger matrix, you soon exceed half a gigabyte of main memory.



For the L-shaped membrane and similar problems, a technique using analytic solutions to the underlying differential equation is much more efficient and accurate than finite difference methods. The building blocks are the fractional order Bessel functions and trig functions that yield eigenfunctions of circular sectors. Remember Pac-Man? How would Pac-Man vibrate? This simple graphics character from one of the earliest video games provides a two-dimensional test domain. When he was not chomping ghosts, Pac-Man was three-quarters of the unit disc.With polar coordinates \(r\) and \(\theta\), and parameters \(\alpha\) and \(\lambda\), the eigenfunctions of a circular sector are

\[v(r,\theta) = J\left(\alpha, \sqrt{\lambda} r\right) \sin(\alpha \theta)\]