The Modified Galerkin Method for the Modified Wave Equation for the Shape of Superellipsoid

The objective of this work was to find the numerical solution of the Impendence problem for the Helmholtz equation for a smooth superellipsoid. The superellipsoid is a shape that is controlled by two parameters. There are some numerical issues in this type of an analysis; any integration method is affected by the wave number k, because of the oscillatory behavior of the fundamental solution. The Helmholtz equation, which is the modified wave equation, is used in many scattering problems. This project was funded by NASA RI Space Grant for testing of the Robin boundary condition for the shape of the superellipsoid. One practical value of all these computations can be getting a shape for the engine nacelles in a ray tracing the space shuttle. We significantly reduced the number of terms in the infinite series needed to modify the original integral equation and used the Green's theorem to solve the integral equation for the boundary of the surface.