## Waves and Compressible FlowMathematics is playing an ever more important role in the physical and biol- ical sciences, provoking a blurring of boundaries between scienti?c disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and tea- ing, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Contents The starred sections are self-contained and may be omitted at a ?rst reading. |

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### Contents

The starred sections are selfcontained and may be omitted at a first reading | 1 |

Exercises | 18 |

Theories for Linear Waves | 41 |

Nonlinear Waves in Fluids | 99 |

Shock Waves 135 | 134 |

Epilogue | 181 |

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acoustic waves amplitude angle approximation assume axis axisymmetric body boundary conditions Chapter characteristics compressible flow conservation of mass consider constant curve Deduce deﬁned density derive dimensions dispersion relation downstream eikonal equation energy entropy Exercise expansion fan ﬁnd ﬁow ﬁuid ﬂow function gasdynamics given gravity waves group velocity Helmholtz Hence homentropic hyperbolic incompressible initial integral inviscid jump condition leads linear Mach number mathematical momentum non-zero nonlinear nozzle Ockendon one-dimensional partial differential equations past a thin plane pressure Rankine–Hugoniot rays region satisﬁes satisfy Section shallow water shallow water equations shock relations shock wave Show shown in Figure solution solve steady Stokes wave subsonic supersonic Supersonic flow past Suppose surface theory thin wing tion two-dimensional unsteady variables velocity potential vorticity wave equation wave propagation wavenumber weak shock write