Abstract This article presents a multimodal formulation for parabolic equations (multimodal PE) that can be applied to the modelling of directional sound propagation in stratified, range-dependent fluid environments. Multimodal PE projects sound pressure onto a space spanned by a set of basis modes, which are defined in a bounded domain with the water surface and a false bottom truncating the sediment layer. This projection results in a one-way equation for the projection coefficients, with the depth operator implemented in the projection space. The proposed multimodal PE offers three advantages: (1) it provides stable solutions that analytically account for discontinuities in sound speed and density at stratified interfaces, making it well-suited to multi-layer sediment environments; (2) its computational complexity remains essentially unchanged even as the number of sediment layers increases; and (3) it exhibits strong convergence behaviour proportional to 1/M2, where M is the number of basis modes, enabling efficient solutions when high accuracy is required. Moreover, multimodal PE supports very-wide-angle directional sound propagation within a single PE simulation, without added numerical complexity compared with omnidirectional propagation. Numerical simulations validate the accuracy, efficiency and source-directivity modelling capability of the proposed approach.
He et al. (2026) studied this question.