Abstract
As we mentioned earlier we interested in the manner in which a signal emitted by a transmitter evolves through a medium and in the form that it assumes at a receiver. In this chapter we illustrate how this information can be obtained. In the approach adopted here a given wave problem, defined in R n × R, is reduced to a first order system which is defined in a suitably chosen abstract space. From a practical point of view there are two main advantages in adopting this approach. First, by carefully selecting the abstract space setting energy aspects can be very simply accommodated. A first order system structure can allow a ready application of results from the theory of semigroups given in Chapter 5. In particular, we have seen that conditions can be given which ensure that the first order system, and hence the given wave problem, is well-posed. We shall assume in this chapter that these conditions are satisfied.
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References and Further Reading
H. Amann: Ordinary Differential Equations, An Introduction to Nonlinear Analysis, W. de Gruyter, Berlin, 1990.
T. Kato: Perturbation Theory for Linear Operators, Springer, New York, 1966.
G.F. Roach: Greens Functions (2nd Edn). Cambridge Univ. Press, London, 1970/1982.
G.F. Roach: An Introduction to Linear and Nonlinear Scattering Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics 78, Longman, Essex, 1995.
E.J.P. Schmidt: On scatteringby time dependent potentials, Indiana Univ. Math. Jour. 24(10), 1975, 925–934.
N.A. Shenk: Eigenfunction expansions and scattering theory for the wave equation in an exterior domain, Arch. Rational Mech. Anal. 21, 1966, 120–150.
P.E. Sobolevski: Equationsof parabolic typeina Banach space. Amer. Math. Soc. Transl. 49, 1996, 1–62.
H. Tanabe: Evolution Equations, Pitman Monographs and Studies in Mathematics 6, Pitman, London, 1979.
C.H. Wilcox: Scattering Theory for the d’Alembert Equation in Exterior Domains, Lecture Notes in Mathematics, No. 442, Springer, Berlin, 1975.
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(2008). An Approach to Echo Analysis. In: An Introduction to Echo Analysis. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84628-852-4_7
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