Time-harmonic acoustic wave scattering in an ocean with depth-dependent sound speed

Abstract

Time-harmonic acoustic wave propagation in an inhomogeneous ocean with depth-dependent sound speed can be modeled by the Helmholtz equation in an infinite, three-dimensional (3D) waveguide of finite height. Using variational theory in Sobolev spaces we prove well posedness of the corresponding scattering problem from a bounded inhomogeneity inside such an ocean. To this end, we introduce an exterior Dirichlet-to-Neumann operator for depth-dependent sound speed and prove boundedness, coercivity, and holomorphic dependence of this operator in function spaces adapted to our weak solution theory. Analytic Fredholm theory then yields existence and uniqueness of solution for the scattering problem for all but a countable sequence of frequencies. The latter result generalizes corresponding theory for waveguide scattering with constant sound speed and easily extends to various related scattering problems, e.g. to scattering from impenetrable obstacles.

Keywords:

• scattering in waveguides
• depth-dependent background sound speed
• variational solution theory

AMS Subject Classifications:

• 35P25
• 35J05
• 35J20
• 34B24

Acknowledgements

The authors would like to thank Prof. Houssem Haddar for various fruitful discussions on the subject of the paper.

Notes

No potential conflict of interest was reported by the authors.