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Generalized formulation for acoustics

Jean-Dominique Polack 1
1 IJLRDA-LAM - Lutheries - Acoustique - Musique
DALEMBERT - Institut Jean Le Rond d'Alembert
Abstract : In 1937, Janowski and Spandöck (Akust. Z., 2:322–331) experimentally demonstrated the curvature of sound rays travelling at grazing incidence above absorbing materials. 30 years later, Cremer and Müller (Die wissenschaftlichen Grundlagen der Raumakustik, Band II, Hirzel Verlag, 1978) showed that this curvature was created by the surface admittance of the material; however, they were not able to demonstrate the curvature of the rays. By making use of the stress-energy tensor, introduced in Acoustics by Morse and Ingard (Theoretical Acoustics. Mc Graw-Hill, 1968), we show that this formalism, borrowed from the general relativity theory, makes it possible to derive the curvature of grazing rays, provided that general coordinates are adopted so that the metric tensor adapts itself to the admittance at the boundary. We show that the metric tensor can be arbitrarily defined, with the only condition that the normal derivative of the velocity potential be null. Absorption is then taken into account by the residual Christoffel symbols that define normal derivation of the stress-energy tensor at the boundary. A generalized formulation of energy conservation is then obtained, which generalizes earlier work on energy propagation in corridors and flat rooms by Dujourdy et al. (Acta Acustica, 103:480-491, 2019; Wave Motion 87:193-212, 2019). An unexpected result from the formalism is that the imaginary, or reactive, part of the stress-energy tensor satisfies Maxwell equations.
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Submitted on : Monday, November 18, 2019 - 3:25:08 PM
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Jean-Dominique Polack. Generalized formulation for acoustics. 2019. ⟨hal-02148492v2⟩

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