Environmentally adaptive wedge modes
The notion of environmentally adaptive, curvilinear, uncoupled normal modes is developed by writing the Helmholtz equation in curvilinear coordinates. The depth-dependent vertical scaling function provides some coordinate freedom missing from the Cartesian formulation. This freedom permits adaptation of the normal mode approach to variable depth and variable sound-speed problems for environments that change slowly in range. A W.K.B. approximation to the eigenfunctions shows modal decoupling can be realized when the vertical scaling function is chosen equal to the reciprocal of the depth-dependent vertical wave number of the lowest mode. Numerical simulations demonstrate that a requirement for constant inter-modal ratios of the vertical wave number is valid throughout several wedges with different bottom boundary conditions. The horizontal refraction of such modes agrees closely with analytic solutions. For a wide-angle wedge, the acoustic field agrees qualitatively well with the benchmark solution of Dean and Buckingham [J. Acoust. Soc. Am. 93, 1319-1328 (1993)].