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Modal-Analyse im Bogen- oder Ring-Kanal. Teil II: Ring-Kanal

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Sound fields in ducts with ring-shaped cross-sections or in flat ducts with a circular bend of their axis can be represented as mode sums of modes in the cylindrical co-ordinate system in which the duct wall surfaces are co-ordinate surfcaces. The modal analysis of the sound fields in both types of ducts proceeds in parallel over wide parts; the numerical solutions, however, reveal significant differences. The case of the duct bend was treated in Part I of this paper; this part will be concerned with the ring-shaped duct. Several forms of developments of the characteristic equation for the radial wave numbers as mode solutions will be described; all of them intend to approximate the characteristic equation by a polynomial equation so that its solutions can be used as starters for a numerical solution of the complete charateristic equation, thus leading to sets of mode solutions. Finally, a set of modes from a flat duct with un-symmetrical lining, which are easily and safely determined by a continued-fraction expansion, are used as starters for modes of a ring-shaped channel with same width and lining. This works quite well under certain parameter conditions.

Schallfelder in Kanälen mit kreisringförmigem Querschnitt oder mit rechteckigem Querschnitt und kreisbogenförmiger Biegung der Längsachse können mit Moden beschrieben werden in dem zugrunde liegenden kreiszylindrischen Koordinatensystem, in welchem die Kanalwände Koordinatenflächen sind. Die analytische Behandlung der Felder in beiden Kanalformen läuft über weite Strecken parallel zueinander; bei der numerischen Lösung der Aufgabe gibt es jedoch signifikante Unterschiede. Der Fall des Bogen-Kanals wurde in Teil I dieses Artikels beschrieben; dieser Teil wendet sich dem Ring-Kanal zu. Es werden verschiedene Formen der Entwicklung der charakteristischen Gleichung für die radialen Wellenzahlen als Moden-Eigenwerte (Moden-Lösungen) beschrieben, welche alle zum Ziel haben, diese Gleichung auf Polynom-Form zu bringen, deren Lösungen zumindest geeignet sind als Startwerte für eine Nachbesserung durch numerische Lösung der vollständigen charakteristischen Gleichung, so daß sich ein Satz von Moden- Lösungen berechnen läßt. Schließlich werden auch noch Moden-Eigenwerte aus einem unsymmetrisch ausgekleideten Flach-Kanal, welche mit Kettenbruch-Entwicklung einfach, sicher und schnell zu berechnen sind, als Startlösungen für einen ebenso ausgekleideten und gleich weiten Ring-Kanal benutzt. Unter bestimmten Parameter-Bedingungen funktioniert das sehr gut.
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Document Type: Research Article

Publication date: July 1, 2008

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  • 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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