Analysis of eigenfunctions and eigenvalues of the Laplacian is important in the understanding of distributed parameter vibration systems and quantum mechanics. The elliptical domain is advantageous for such work because it allows the explicit representation of eigenfunctions as products of the Mathieu functions. In [Ann. Phys., 9 (1960), pp. 24-75], Keller and Rubinow used wave propagation, geometrical optics, and WKB methods to show that two special types of eigenmodes, whispering gallery and bouncing ball modes, exist on a general convex domain, and they illustrated the case of an elliptical domain. In this paper, we develop a numerical package of the Mathieu and modified Mathieu functions to actually construct profiles of a sequence of eigenfunctions in order to visualize such special eigenmodes and others. In the process, we have also been able to observe a new type of transition state named 'focusing modes' herein, which seem to have a complementary behavior to the bouncing ball modes. Numerical eigenvalues are tabulated for comparison and some discussions on the 'focusing modes' are presented.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computational Mathematics
- Applied Mathematics