Abstract
The Bremmer series solution of the acoustical wave equation in generally inhomogeneous media, requires the introduction of pseudodifferential operators. Such operators describe the (de)composition, propagation and interaction of the up- and downgoing waves. The exact solutions of the corresponding symbols can be derived for several medium profiles. Of particular interest is the symbol associated with the vertical slowness or square root operator, involved in the propagation. This symbol constitutes the generalized slowness surface. These symbols serve as benchmarks for numerical one-way wave schemes involving waveguiding structures. Starting from the kernel representation of the transverse Helmholtz operator’s resolvent, a normal mode expansion is set up. The Schwartz kernels of negative fractional powers of the transverse Helmholtz operator and their left symbols can then be expressed in these normal modes as well. Using the composition relation for the original operator and its inverse square, a recursive relation for the left symbols is obtained. Results of the quadratic waveguiding (focusing) profile and a localized profile will be presented. The connection with the eigenfunctions and eigenvalues (normal mode expansion) will be highlighted.
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