| n this project, we examine the interior solutions of the Helmholtz equation within several polygonal domains formed by piecing together identical 30;First, the bound-states of four three-body systems are examined. It is shown that the Schrodinger problem reduces to the two-dimensional Helmholtz problem--with Dirichlet boundary conditions--within either a 60;Numerical solutions of the hexagon problem are determined using the imaginary-time-step method with finite differences. Contour plots of the lowest 69 eigenfunctions and the corresponding eigenvalues--accurate to at least five digits--are reported.;Next, the Helmholtz equation is solved within the context of the waveguide problem. The uniform waveguide cross-sections in which many of the lowest TE (Neumann) and TM (Dirichlet) modes are examined include the 30;Accurate, high-order numerical solutions using the point-matching method are reported for the lowest 26 non-closed-form modes. In addition, a new eigenvalue bounding method is presented. To illustrate, the lowest TM cutoff wavenumbers for the unit-edged hexagon and rhombus are bounded by 2.674946522... |
