Extrapolating Green's functions using the waveguide invariant theory.

The broadband interference structure of sound propagation in a waveguide can be described by the waveguide invariant, β, that manifests itself as striations in the frequency-range plane. At any given range (r), there is a striation pattern in frequency (ω), which is the Fourier transform of the multipath impulse response (or Green's function). Moving to a different range (r+Δr), the same pattern is retained but is either stretched or shrunken in ω in proportion to Δr, according to Δω/ω=β(Δr/r). The waveguide invariant property allows a time-domain Green's function observed at one location, g(r,t), to be extrapolated to adjacent ranges with a simple analytic relation: g(r+Δr,t)≃g(r,α(t-Δr/c)), where α=1+β(Δr/r) and c is the nominal sound speed of 1500 m/s. The relationship is verified in terms of range variation of the eigenray arrival times via simulations and by using real data from a ship of opportunity radiating broadband noise (200-900 Hz) in a shallow-water environment, where the steep-angle arrivals contributing to the acoustic field have β≈0.92.