# Nonreflective boundary conditions for Schr"odinger's equation

#### Abstract

The propagation of sound under water is modeled by the wave equation. Under certain conditions, this equation can be approximated by the parabolic Schroedinger-type equation 2 i k$\sb0$ u$\sb{\rm r}$ + u$\sb{\rm zz}$ + k$\sb0\sp2$ (n$\sp2$(r,z) $-$ 1) u = 0 where u represents the acoustic pressure at range r and depth z in the ocean. This parabolic equation has been solved numerically in the past in the semi-infinite domain r $>$ 0, z $>$ 0, usually by introducing an artificial absorbing ocean bottom at a depth far below that of the physical ocean bottom.^ In this dissertation, global non-reflective boundary conditions along the physical ocean bottom are derived, and the new boundary problem (in a bounded domain) is shown to be equivalent to the original problem. A numerical algorithm is derived to solve this new problem, and conditions under which this scheme is stable, convergent, and consistent are exhibited. A computer code which implements the algorithm is developed, and numerical results for test cases are obtained. Finally, the boundary conditions are generalized, by using the Fast Fourier Transform, to solve problems representing acoustic propagation under different physical conditions. ^

#### Subject Area

Mathematics

#### Recommended Citation

Marie Elizabeth Mayfield,
"Nonreflective boundary conditions for Schr"odinger's equation"
(1989).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI9007242.

http://digitalcommons.uri.edu/dissertations/AAI9007242