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Application to the Wave Equation

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Wave Propagation

Part of the book series: Mathematics and Its Applications ((MAIA,volume 17))

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Abstract

In this chapter we will take up the study of the wave equations in one dimension and study the propagation of the wave in a region with inhomogeneous properties of refractive index by analyzing the reflection and transmission functions for the region. In the previous chapter we studied these functions in the context of particle transport. Similar studies carry over in the case of wave propagation. An order of scattering analysis of the emergent and internal solutions leads to Bremmer series [1] solutions under certain conditions.

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References

  1. Bremmer, H., ‘The WKB Approximation as the First Term of a Geometric Optical Series’, Theory of Electromagnetic Waves Symposium, Interscience Publishers, Inc., New York, 1951, 169.

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  2. Bellman, R., and R. Kalaba, ‘Functional Equations, Wave Propagation and Invariant Imbedding’, Journal of Mathematics and Mechanics 8 (1959), 683.

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  3. Stokes, G., Mathematical and Physical Papers of Sir George Stokes, Vol. 14, Cambridge, (1904), 145.

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  4. Bellman, R. and G.M. Wing, Introduction to Invariant Imbedding, John Wiley & Sons, Inc., New York, 1975.

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  5. ‘Wave Propagation and the Bremmer Series’, Journal of Mathematical Analysis and Applications 1 (1960), 255.

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  6. Taylor, A.E., Introduction to Functional Analysis, John Wiley & Sons, Inc., New York, 1958.

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  7. Bellman, R., Stability Theory of Differential Equations, McGraw Hill, New York, 1953. References in this volume may be useful.

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  8. Scott, M., Invariant Imbedding and Its Applications to Ordinary Differential Equatins, Addison-Wesley Publishing Company, London, 1973.

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  9. Bellman, R., R. Vadusevan, and S. Ueno, ‘On the Matrix Riccati Equation of Transport Processes’. Journal of Mathematical Analysis and Applications 44 (1973), 472.

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  10. Bellman, R., and R. Vasudevan, ‘Wave Equations with Sources, Invariant Imbedding and Bremmer Series Solutions’, Journal of Mathematical Analysis and Applications 48 (1974) 17.

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  11. Wing, G.M., ‘Invariant Imbedding and Generalization of JWKB Method and the Bremmer Series’. Journal of Mathematical Analysis and Applications 48 (1974), 400.

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© 1986 D. Reidel Publishing Company

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Bellman, R., Vasudevan, R. (1986). Application to the Wave Equation. In: Wave Propagation. Mathematics and Its Applications, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5227-0_4

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  • DOI: https://doi.org/10.1007/978-94-009-5227-0_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8811-4

  • Online ISBN: 978-94-009-5227-0

  • eBook Packages: Springer Book Archive

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