David Oliphant and Dr. R. Steven Turley, Physics and Astronomy

Scattering problems are common in physics. A scattered (reflected) wave or particle from an object can give us hints to the nature of the object’s shape, size, color, or composition. This is the basic principle behind sonar, radar, ultrasound, NMR, and even vision. One technique in determining the nature of a scattered wave is called the Method of Moments. It involves solving the Helmholtz equation by representing the object’s response or current as a sum of basis functions and weights. The number of basis functions used is important. If there are too many the computational time becomes inordinate. However, there must be enough basis functions to accurately represent the current. The most difficult place to meet both criteria is when the wavelength and the length of the object are comparable.

I have been learning and improving a computer program that models, using the Method of Moments, how scalar waves scatter off a soft cylinder (i.e. sound waves bouncing off a cylindrical bubble). We use Legendre polynomials as our basis functions in the z direction. With these we are able to accurately represent the current with reasonable computational time, even when the cylinder and the wavelength are comparable.

As I began changing the computer code to fix what I thought was another singularity problem I found a more serious bug. The basis being used for the azimuthal direction was not a complete set. The basis exp(imx) is only complete if m is allowed to vary from the negative value of its maximum to its maximum value. I found that m was only being varied from 0 to its maximum. After fixing this problem I started testing. I found that the individual integrals in the code are accurate to at least five significant figures. I also worked on testing the final answer the scattering program produced. Not enough numbers have been calculated to conclude that it is all working properly, but initial results are promising. We may yet find that the singularity problem with touching patches is significant and must be fixed. However, I have modified the code such that this will be a simple change.

The program has a singularity problem similar to one I have already solved. After fixing this problem I will modify the program to model a bent cylinder. This involves a coordinate transformation from the bent space to a simpler straight one. Later I will modify the code to model hard rather than soft cylinders (Neumann rather than Dirichlet boundary conditions). For this I will need to choose different high-order basis functions because Neumann boundary conditions require the derivative of the wave to be zero at the boundary. This boundary condition is more common in acoustical problems. In addition it will make the model more easy to modify for electromagnetic wave scattering problems.

I have already made significant contributions to this project by solving a singularity problem and simplifying the code. This has given me a competent understanding of how the program works and a great desire to see it completed. I am currently enrolled in a graduate level mathematical physics course (Physics 517). We are learning about mapping, vector spaces, operator algebra, and complex analysis. These have given me a deeper understanding of the fundamentals of this project, and will help me implement the changes spoken of. I am a physics senior with only some GE and one physics course to complete. All this combined with the encouragement and help from Dr. Steven Turley will facilitate my completion of this project. The ORCA Scholarship will financially allow me to spend about six hours a week next semester finishing this project.