The MathWorks Logo is an Eigenfunction of the Wave Equation
By Cleve Moler, MathWorks
We hope you’ve seen it many times. It’s on the covers of our books. It’s on our business cards and stationery. It’s even on a “sponsor a highway” sign on Route 9 in Natick,Massachusetts. But, do you really know what the logo is?
I’m talking about the L-shaped membrane.We’ve used various pictures of it ever since The MathWorks was founded almost twenty years ago, but it only recently became the official company logo. I’d like to tell you about its mathematical background.
The wave equation is a fundamental model in mathematical physics that describes how a disturbance travels through matter. If t is time and x and y are spatial coordinates with the units chosen so that the wave propagation speed is equal to one, then the amplitude of a wave satisfies the partial differential equation
Periodic time behavior gives solutions of the form
The quantities
The L-shaped region formed from three unit squares is interesting for several reasons. It is one of the simplest geometries for which solutions to the wave equation cannot be expressed analytically, so numerical computation is necessary. The 270º nonconvex corner causes a singularity in the solution.Mathematically, the gradient of the first eigenfunction is unbounded near the corner. Physically, a membrane stretched over such a region would rip at the corner. This singularity limits the accuracy of finite difference methods with uniform grids.
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
The eigenvalues are determined by the requirement that
The circular portion of the boundary has
Contour plots of the first six eigenfunctions show that two are symmetric about the center line,
These functions are exact solutions to the eigenvalue differential equation. They also satisfy the boundary conditions on the two edges that meet at the origin. All that remains is to pick
A least squares approach employing the matrix singular value decomposition is used to determine
Then, for any vector
Each
The MATLAB function membrane
uses an older version of this algorithm to compute eigenfunctions of the L-shaped membrane. Contour plots of the first six eigenfunctions show that the first, fifth, and sixth are symmetric about the center line; the second and fourth are antisymmetric about the center line; and the third is actually the first eigenfunction of a unit square, reflected into the other two squares.
The MATLAB function logo
generates our company logo, a lighted, reflective, surface plot of a variant of the first eigenfunction. After being so careful to satisfy the boundary conditions, the logo uses only the first two terms in the sum. This artistic license gives the edge of the logo a more interesting, curved shape.
Published 2003