Evolution of local spectra in smoothly varying nonhomogeneous environments−Local canonization and marching algorithms

By applying the windowed Fourier transform directly to the Helmholtz wave equation a new formulation that governs the evolution of the local spectra of wave fields in a general nonhomogeneous environment is derived. By further invoking the so‐called locally homogeneous approximation, a simplified evolution equation, termed as the locally homogeneous wave equation is developed, together with an upper bound on the error associated with the approximation. It is shown how simple analytical solutions of the new wave equation in a general smoothly varying nonhomogeneous environment can be obtained using well‐known analytical techniques, and how the marching methodology connects these new solutions to the original problem described by the Helmholtz equation.