Exact and operator rational approximate solutions of the Helmholtz, Weyl composition equation in underwater acoustics−The quadratic profile

Abstract

The application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one-way Helmholtz equation is briefly reviewed. The role of operator symbols is stressed and their properties briefly discussed. The construction of the operator symbol requires the exact or approximate solution of the (Helmholtz) Weyl composition equation in the Weyl pseudodifferential operator calculus. The exact symbols for several quadratic profiles are presented and briefly analyzed. The symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case. These results are used to illustrate several points pertinent to wide-angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics.