Abstract
We study the scattering of acoustic waves by an obstacle embedded in a uniform waveguide with planar boundaries, which is a fundamental model in shallow ocean acoustics. Under certain geometric assumptions on the shape of the obstacle, we prove a Rellich-type uniqueness theorem in the case of soft obstacles. The solvability of the scattering problem is proved via boundary integral equations and the limiting absorption principle is established. An eigenfunction expansion in terms of the scattering solutions reveals the appropriate (partial) scattering amplitudes for the waveguide. We show that these scattering amplitudes for the propagating modes, at a fixed frequency, define uniquely the shape of a soft obstacle.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. S. Ahluwalia and J. B. Keller, Exact and asymptotic representation of the sound field in a stratified ocean, in Wave Propagation and Underwater Acoustics, eds. J. B. Keller and J. S. Papadakis, Springer-Verlag, pp (1977), 14-85.
M. Callan, C. M. Linton and D. V. Evans, Trapped modes in two-dimensional waveguides, J. Fluid Mech. 229 51–64 (1991).
D. V. Evans, M. Levitin, D. Vassiliev, Existence theorems for trapped modes, J. Fluid Mech. 261 21–31 (1994).
D. V. Evans, C. M. Linton and F. Ursell, Trapped mode frequencies embedded in the continuous spectrum, Q. J. Mech. appl. Math. 46(2) 253–274 (1993).
C. I. Goldstein, Eigenfunction expansion associated with the Laplacian for certain domains with infinite boundaries I, Trans. Amer. Math. Soc. 135 1–31 (1969).
R. P. Gilbert and Y. Xu, Starting fields and far fields in ocean acoustics, Wave Motion 11 507–54 (1989).
R. P. Gilbert and Y. Xu, Dense sets and the projection theorem for acoustic waves in a homogeneous finite depth ocean, Math. Meth. in the Appl. Sci. 12 67–76 (1989).
R. P. Gilbert and Y. Xu, The propagation problem and far-field patterns in a stratified finite-depth ocean, Math. Meth. in the Appl. Sci. 12 199–208 (1990).
K. Morgenröther and P. Werner, Resonances and standing waves, Math. Meth. in the Appl. Sci. 9 105–126 (1987).
K. Morgenröther and P. Werner, On the principles of limiting absorption and limiting amplitude for a class of locally perturbed waveguides. Part 1: Time-independent theory, Math. Meth. in the Appl. Sci. 10 125–144 (1988).
A. G. Ramm, Scattering by obstacles (Reidel, 1986).
A. G. Ramm, Multidimensional inverse scattering: solved and unsolved problems, Proc. Intern. Confi on Dynamical Syst. and Applic, eds. G. Ladde and M. Sabandham (Atlanta, Georgia, 1994), pp.287-296.
A. G. Ramm, Scattering amplitude as a function of the obstacle, Appl. Math. Lett. 6 (5) 85–87 (1993).
A. G. Ramm, New method for proving uniqueness theorems for obstacle inverse scattering problem, Appl. Math. Lett. 6 (6) 89–92 (1993).
A. G. Ramm, The scattering problem analyzed by means of an integral equation of the first kind, J. Math. Anal Appl. 201 324–327 (1996).
A. G. Ramm and A. Ruiz, Existence and uniqueness of scattering solutions in non-smooth domains, J. Math. Anal Appl. 201 329–338 (1996).
A. G. Ramm, Investigation of the scattering problem in some domains with infinite boundaries, I, II, Vestnik (1963) 45-66, 67-767 (19).
A. G. Ramm, Uniqueness theorems for inverse obstacle scattering problems in Lipschitz domains, Applic. Analysis 59 377–383 (1995).
A. G. Ramm, Continuous dependence of the scattering amplitude on the surface of an obstacle, Math. Methods in the Appl. Sci. 18 121–126 (1995).
A. G. Ramm, Stability of the solution to inverse obstacle scattering problem, J. Inverse and Ill-Posed Problems 2(3) 269–275 (1994).
A.G. Ramm, Stability estimates forobstacle scattering, J. Math. Anal. Appl. 188(3) 743–751 (1994).
A. G. Ramm, Examples of nonuniqueness for an inverse problems of geophysics, Appl. Math. Lett. 8(4) 87–90 (1995).
A. G. Ramm, Stability of the solution to 3D inverse scattering problems with fixed-energy data, Inverse Problems in Mechanics, (Proc. ASME, AMD-Vol. 186 1994), pp. 99-102.
A. G. Ramm, Remark on recent papers by R. Kleinman, T. Angeli and coauthors
A. G. Ramm and P. Werner, On limit amplitude principle, Jour. fuer die reine und angew. Math. 360 19–46 (1985).
F. Rellich, Das Eigenwertproblem von Δu+ℝu = 0 in unendlichen Gebieten, Jber. d. deutsch. Math.-Verein. 53 47–65 (1943).
K. J. Witsch, Examples of embedded eigenvalues for the Dirichlet—Laplacian in domains with infinite boundaries, Math. Meth. in the Appl. Sci. 12 177–182 (1990).
Y. Xu, The propagating solution and far field patterns for acoustic harmonic waves in a finite depth ocean, Appl. Anal 35 129–151 (1990).
Y. Xu, An injective far-field pattern operator and inverse scattering problem in a finite depth ocean, Proc. Edinburgh Math. Soc. 43 295–311 (1991).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Ramm, A.G., Makrakis, G.N. (1998). Scattering by Obstacles in Acoustic Waveguides. In: Ramm, A.G. (eds) Spectral and Scattering Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1552-8_7
Download citation
DOI: https://doi.org/10.1007/978-1-4899-1552-8_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-1554-2
Online ISBN: 978-1-4899-1552-8
eBook Packages: Springer Book Archive