*ANDRÉ AUTHIER*

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198528920
- eISBN:
- 9780191713125
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528920.003.0011
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter describes Takagi's dynamical theory of the diffraction of incident spherical waves. It considers the crystal wave to be developed as a sum of modulated waves. The fundamental equations ...
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This chapter describes Takagi's dynamical theory of the diffraction of incident spherical waves. It considers the crystal wave to be developed as a sum of modulated waves. The fundamental equations are generalized as a set of partial differential equations (Takagi's equations). Their solutions for an incident spherical wave are first obtained by the method of integral equations for both the transmission and reflection geometries. The hyperbolic nature of Takagi's equations is shown and their solution derived using the method of Riemann functions for a point source located on the entrance surface or away from the incident surface. An appendix describes the properties of hyperbolic partial differential equations.Less

This chapter describes Takagi's dynamical theory of the diffraction of incident spherical waves. It considers the crystal wave to be developed as a sum of modulated waves. The fundamental equations are generalized as a set of partial differential equations (Takagi's equations). Their solutions for an incident spherical wave are first obtained by the method of integral equations for both the transmission and reflection geometries. The hyperbolic nature of Takagi's equations is shown and their solution derived using the method of Riemann functions for a point source located on the entrance surface or away from the incident surface. An appendix describes the properties of hyperbolic partial differential equations.

*Luciano Rezzolla and Olindo Zanotti*

- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780198528906
- eISBN:
- 9780191746505
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528906.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter is devoted to the conditions under which nonlinear hydrodynamical waves are produced and to the study of the flow properties across such waves. Special emphasis is given to the ...
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This chapter is devoted to the conditions under which nonlinear hydrodynamical waves are produced and to the study of the flow properties across such waves. Special emphasis is given to the mathematics of hyperbolic systems of partial differential equations, showing that the relativistic-hydrodynamics equations can be cast in both quasi-linear hyperbolic form and in conservative form. Attention is focused to the discussion of rarefaction and shock waves, which are treated to highlight the similarities and also the differences with Newtonian physics. Within this framework, the Riemann problem for the relativistic-hydrodynamics equations in flat spacetime is studied in great detail, both for one-dimensional and multidimensional flows. The chapter is completed by two more advanced topics, namely the stability of nonlinear waves and the properties of discontinuous solutions in full general relativity.Less

This chapter is devoted to the conditions under which nonlinear hydrodynamical waves are produced and to the study of the flow properties across such waves. Special emphasis is given to the mathematics of hyperbolic systems of partial differential equations, showing that the relativistic-hydrodynamics equations can be cast in both quasi-linear hyperbolic form and in conservative form. Attention is focused to the discussion of rarefaction and shock waves, which are treated to highlight the similarities and also the differences with Newtonian physics. Within this framework, the Riemann problem for the relativistic-hydrodynamics equations in flat spacetime is studied in great detail, both for one-dimensional and multidimensional flows. The chapter is completed by two more advanced topics, namely the stability of nonlinear waves and the properties of discontinuous solutions in full general relativity.

*George Jaroszkiewicz*

- Published in print:
- 2016
- Published Online:
- January 2016
- ISBN:
- 9780198718062
- eISBN:
- 9780191787553
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198718062.003.0012
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses the mathematical reasons why time has one dimension rather than two or more dimensions. It examines the flow of information, through a spacetime with a given signature, in ...
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This chapter discusses the mathematical reasons why time has one dimension rather than two or more dimensions. It examines the flow of information, through a spacetime with a given signature, in terms of partial differential equations. These are classified as elliptic, parabolic or hyperbolic. The role of boundary conditions in ruling out different temporal dimensions is explained. The chapter discusses the stability of solutions in spacetimes with various signatures and give a mathematical explanation for the observation that the world we live in has four spacetime dimensions best described by hyperbolic partial differential equations. The chapter ends with commentary on empirical studies in cosmology and biology that have used multi-component time concepts to fit their data.Less

This chapter discusses the mathematical reasons why time has one dimension rather than two or more dimensions. It examines the flow of information, through a spacetime with a given signature, in terms of partial differential equations. These are classified as elliptic, parabolic or hyperbolic. The role of boundary conditions in ruling out different temporal dimensions is explained. The chapter discusses the stability of solutions in spacetimes with various signatures and give a mathematical explanation for the observation that the world we live in has four spacetime dimensions best described by hyperbolic partial differential equations. The chapter ends with commentary on empirical studies in cosmology and biology that have used multi-component time concepts to fit their data.