Transmutation of non-local boundary conditions in ocean acoustics

We employ a transmutation technique to construct a new non-local boundary condition for the paraxial approximation in ocean acoustics. The transmutation operator introduced by De Santo and Polyanskii, when applied to the Helmholtz equation governing the acoustic pressure in the water column, leads to the so-called parabolic equation of Fock and Tappert. This transmutation operator acting on the N-D map at the water–bottom interface yields an intermediate non-local boundary condition for the parabolic equation which eliminates the backscattering terms in the Schwartz kernel of the N-D map. The kernel of the intermediate condition is approximated by a uniform stationary phase formula taking account of the possible coalescence of the brach points of the integrand with the stationary points of the phase, and it leads to a non-local boundary condition of Volterra-type for the parabolic equation. This condition is quite different than similar conditions derived by other approximations, in that the kernel of the Volterra operator is smooth, the smoothing effect coming from the fact that the horizontal range coordinate is scaled with the relative refraction index between the water column and the bottom.