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.

Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations in Underwater Acoustics

“…It is therefore appropriate to define the coupled evolution equation on the symbol level of the two involved operators (cf. [20,25]). Without attenuation both the SPE and the WAPE conserve the L 2 -norm and the discrete analogue of this conservation is the main ingredient for showing unconditional stability of the finite difference scheme in Section 3.…”

“…Therefore we postulate that the coupled model also has to conserve the L 2 -norm. This can be achieved if the operator A on the right-hand side of (B.3) is interpreted as the Weyl operator (see [20] Due to the pole of the symbol (ξ ) it would be quite difficult to appropriately discretize (B.3), (B.5), and it is beyond our scope here. We remark that finite difference schemes of pseudo-differential equations with smooth symbols have recently been studied in [31].…”

“…For a detailed description and motivation of this procedure we refer to [12,20,21,28,43,44]. The linear approximation of √ 1 − λ by 1 − λ/2 gives the narrow angle or standard "parabolic" equation (SPE) of Tappert [43],…”

The Bremmer series for a multi-dimensional acoustic scattering problem

“…The vertical-propagation operator Γ can be represented in a variety of ways, e.g., left-, right-, Weyl-symbols and spectral theory, see [5,7,9]. Here, we use a leftsymbol representation, i.e., the operator is defined as the action of the integral and an upper bound on the inverse…”

“…This gives a non-local one-way wave operator asymptotically correct in both the high frequency and high wave number limit. A more careful decomposition is proposed in [6][7][8][9].…”