Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields

Nolan, M and Davy, J 2019, 'Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields', Journal of the Acoustical Society of America, vol. 145, no. 6, pp. 3330-3340.


Document type: Journal Article
Collection: Journal Articles

Title Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields
Author(s) Nolan, M
Davy, J
Year 2019
Journal name Journal of the Acoustical Society of America
Volume number 145
Issue number 6
Start page 3330
End page 3340
Total pages 11
Publisher AIP Publishing LLC
Abstract There are two definitions of the inner product of modal spatial functions used in the literature. Both definitions integrate the product of the modal spatial functions over a line, area, or volume. The only difference is that one of the definitions takes the complex conjugate of one of the modal spatial functions before multiplying the modes together. The definitions are the same if the modal spatial functions are real. If the modal spatial functions are complex, only the definition which takes the complex conjugate is an inner product. If the specific acoustic impedance of the boundaries has a real part, then the modes are only orthogonal with the definition which does not take the complex conjugate, although this definition is not strictly an inner product because the modal spatial functions are complex in this situation. However, this definition of "inner product" can be used to calculate the coefficients in the modal expansion of the system response. On the other hand, when it comes to calculating the mean pressure squared and the mean sound intensity, the modal spatial functions cross-products cannot be ignored because the modes are not orthogonal for the definition which takes the complex conjugate.
Subject Architectural Science and Technology (incl. Acoustics, Lighting, Structure and Ecologically Sustainable Design)
Acoustics and Noise Control (excl. Architectural Acoustics)
Acoustics and Acoustical Devices; Waves
Keyword(s) Modes
Inner product
Reverberant sound fields
Non-diffuse
DOI - identifier 10.1121/1.5109662
Copyright notice © 2019 Acoustical Society of America
ISSN 0001-4966
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