Geometric properties for reflected fields from a general surface using finite differences

Abstract

It is well known that geometrical acoustics (GA) approximations to reflected fields depend on the surface shape incident to the field in a neighborhood. However, because of the general complexity of these solutions certain canonical problems pertaining to regular shapes, such as cones and spheres, have been solved, but little attention has been given to the problem of approximating the essential geometric properties for a general surface in a neighborhood. Approximations to the reflected field (ka≫1) usually assume (although this is not a limitation of GA as pointed out by Fock [Electro Magnetic Diffraction and Propagation Problem (Perganom, New York, 1965), 1st ed. ] that the surface is a well known geometric shape, i.e., ellipse, cone, etc. This is done to make calculations of curvature and other geometric properties straightforward. This paper will discuss the problem of obtaining the needed geometric properties for more general geometric shapes. It will consider only a small neighborhood of the surface. Using differential geometry and finite differences, it will then show that all geometric properties in a neighborhood may be approximated by a small number of points from that neighborhood. This technique can with a few practical limitations geometrically describe any neighborhood on a surface. These neighborhoods form ’’areas’’ called patches. When the entire surface meets certain conditions, these patches when ’’tied’’ together actually describe the entire surface and all its geometric properties.