Abstract
This chapter is concerned with the basic mathematical and physical aspects of solving the scattering problem of a plane sound wave on spherical objects. Important properties of the eigenfunctions of the scalar Helmholtz equation in spherical coordinates are discussed. These eigenfunctions are used as expansion functions for all the fields that are involved in solving the scattering problem. Especially their transformation behavior with respect to a rotation and a translation of the laboratory frame is considered in detail. Explicit expressions are given for the computation of the total and differential scattering cross-sections, and the so-called “optical theorem” for the computation of the former quantity is introduced. A section is moreover included that provides some hints of how to estimate the accuracy and physical reliability of an obtained scattering solution. A short outlook on the usage of Debye potentials for solving the corresponding electromagnetic scattering problem is given, and the Python programs, which are related to the subjects of this chapter, are described. Appendix A provides a complete listing of these programs.
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References
Sommerfeld, A.: Partial Differential Equations in Physics. Academic Press, New York (1949)
Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley, New York (1983)
Rother, T., Kahnert, M.: Electromagnetic Wave Scattering on Nonspherical Particles: Basic Methodology and Applications. Springer, Heidelberg (2014)
van de Hulst, H.C.: Light Scattering by Small Particles. Dover, New York (1981)
Mishchenko, M.I., Travis, L.D., Lacis, A.A.: Scattering, Absorption, and Emission of Light by Small Particles. Cambridge University Press, Cambridge (UK) (2002)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Harri Deutsch, Frankfurt/Main (1984)
Martin, P.A.: Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles. Cambridge University Press, Cambridge (UK) (2006)
Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)
Eyges, L.: Some nonseparable boundary value problems and the many-body problem. Ann. Phys. 2, 101–128 (1957)
Newton, R.G.: Optical theorem and beyond. Am. J. Phys. 44, 639–642 (1976)
Saxon, D.S.: Tensor scattering matrix for the electromagnetic field. Phys. Rev. 100, 1771–1775 (1955)
Mishchenko, M.I., Hovenier, J.W., Travis, L.D. (eds.): Light Scattering by Nonspherical Particles: Theory, Measurement, and Applications. Academic Press, London (2000)
Rother, T.: Green’s Functions in Classical Physics. Springer International Publishing AG, Cham, Switzerland (2017)
Debye, P.: Der Lichtdruck auf Kugeln von beliebigem Material. Ann. Phys. 30, 57–136 (1909)
Born, M., Wolf, E.: Principles of Optics. Pergamon Press, Oxford (1980)
Borghese, F., Denti, P., Toscano, G., Sindoni, O.I.: Electromagnetic scattering by a cluster of spheres. Appl. Opt. 18, 116–120 (1979)
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Rother, T. (2020). Basics of Plane Wave Scattering. In: Sound Scattering on Spherical Objects. Springer, Cham. https://doi.org/10.1007/978-3-030-36448-9_1
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DOI: https://doi.org/10.1007/978-3-030-36448-9_1
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