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Abstract The mathematical analysis of scattering theory has been a major area of interest in mathematics and physics research since the latter half of the twentieth century. The aim of this work is to examine, in a functional analytic setting, properties of the differential operator L and solutions involved for scattering on the line −∞ < x < ∞. The characterisation of the spectrum of L will provide insight into the physical interpretation of the problem. The study of scattering theory will proceed with the major results in the field being presented with particular focus on reflectionless scattering. Attention is then directed to the inverse reflectionless case. We look at scattering on the line with a matrix transfer condition at the origin in addition an overview of the inverse case is presented.

Theoretical and experimental work on sound propagation in suspensions and emulsions is reviewed. Three theoretical approaches are identified: scattering theory, coupled phase theory and porous media theory. Coupled phase theory is extended and compared analytically and numerically to scattering and porous media theory. Important regimes of scattering theory - the lossless and viscothermal scattering, single and multiple scattering and incoherent scattering regimes - are examined. Experimental data is used to corroborate lossless multiple scattering theory in the short wavelength, high volume fraction region. Previous coupled phase theories have modelled sound propagation in two phase media (i) with heat transfer assuming incompressible particles and (ii) with a compressible particulate phase neglecting heat transfer. Type (i) models are examined analytically and compared to scattering theory. Types (i) and (ii) are compared and brought together in a more general coupled phase theory. The new theory provides an alternative model to scattering theory for sound propagation in emulsions. Predictions of the new theory are compared to experimental data and predictions of scattering theory. Conditions for the equivalence of the frameless Biot porous media theory and coupled phase theory are identified. Predictions of the two approaches are compared to experimental data. New measurements of pore size distribution are used to predict measured acoustical properties of air saturated glass beads. Other extensions to coupled phase theory are reviewed and developed. Predictions including the effect of high volume fraction on the drag and the induced mass force are compared to experimental data and predictions of porous media theory. Coupled phase theory including heat transfer is extended to include particle size distributions; predictions of tl-ds are compared to measurements. The effect of non-spherical particles is investigated. Using the theory of Culick, frequency shifts for modes in an enclosure into which a suspension has been introduced are calculated. These are compared to the predictions of an intuitive approach. The method of Margulies and Schwartz for modelling particle diffusion is discussed. Areas where further work is required are identified.

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