The Application of Differential Geometry to Ray Acoustics in Inhomogeneous and Moving Media

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Abstract:

Starting from the sound wave equation, we show first that the sound rays in any stationary, subsonically moving medium are the geodesic lines of a Finslerian metric in which the arc length has the physical interpretation of propagation time; thus, it is shown that the sound rays satisfy Fermat's principle of least propagation time. In supersonic currents, the front and back sides of the wave surface must be distinguished between; only the rays corresponding to the front side are the lines of the least propagation time; in contradistinction, those corresponding to the back side are the lines of the greatest propagation time.

Zusammenfassung

Ausgehend von der Schallwellengleichung wird zuerst gezeigt, daß die Schallstrahlen in einem beliebigen stationären, mit Unterschallgeschwindigkeit bewegten Medium geodätische Linien einer Finslerschen Metrik sind, wobei die Bogenlänge physikalisch als Ausbreitungszeit zu inter pretieren ist. Damit ist gezeigt, daß Schallstrahlen dem Fermatschen Prinzip der kürzssten Lauf- zeit gehorchen. In Überschallströmungen muß man zwischen der Vorder- und der Rückseite der Wellenfläche unterscheiden; lediglich die der Vorderseite entsprechenden Strahlen sind Linien kürzester Laufzeit; im Gegensatz dazu sind die der Rückseite entsprechenden Strahlen Linien längster Laufzeit.

Sommaire

A partir de l'équation des ondes acoustiques on montre que les rayons sonores dans un milieu stationnaire en mouvement subsonique sont les lignes géeodésiques d'une métrique de Finsler où la longueur d un arc s'interprète physiquement comme une durée de propagation. On en déduit que les rayons acoustiques obéeissent au principe de Fermat du temps de propagation minimal. Dans le cas des écoulements supersoniques, il faut opéerer une distinction entre le front avant et le front arrière de la surface d'onde: seuls les rayons correspondant au front antérieur sont des lignes de durée de propagation minimale, tandis que les rayons relatifs au front postérieur sont des lignes de durée de propagation maximale.

Document Type: Research Article

Publication date: January 1, 1981

More about this publication?
  • Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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