Spectral and modal formulations for the Doppler-shifted field scattered by an object moving in a stratified medium

Spectral and normal mode formulations for the three-dimensional field scattered by an object moving in a stratified medium are derived using full-field wave theory. The derivations are based on Green's theorem for the time-domain scalar wave equation and account for Doppler effects induced by target motion as well as source and receiver motion. The formulations are valid when multiple scattering between the object and waveguide boundaries can be neglected, and the scattered field can be expressed as a linear function of the object's plane wave scattering function. The advantage of the spectral formulation is that it incorporates the entire wave number spectrum, including evanescent waves, and therefore can potentially be used at much closer ranges to the target than the modal formulation. The normal mode formulation is more computationally efficient but is limited to longer ranges. For a monochromatic source that excites N incident modes in the waveguide, there will be roughly N2 distinct harmonic components in the scattered field. The Doppler shifts in the scattered field are highly dependent upon the waveguide environment, target shape, and measurement geometry. The Doppler effects are illustrated through a number of canonical examples.