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

Mélanie Nolan, John L. Davy

Research output: Contribution to journalJournal articleResearchpeer-review

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.

Original languageEnglish
JournalJournal of the Acoustical Society of America
Volume145
Issue number6
Pages (from-to)3330-3340
ISSN0001-4966
DOIs
Publication statusPublished - 1 Jun 2019

Cite this

@article{73a933c511a3497092f47f750c1437ea,
title = "Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields",
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.",
author = "M{\'e}lanie Nolan and Davy, {John L.}",
year = "2019",
month = "6",
day = "1",
doi = "10.1121/1.5109662",
language = "English",
volume = "145",
pages = "3330--3340",
journal = "Acoustical Society of America. Journal",
issn = "0001-4966",
publisher = "A I P Publishing LLC",
number = "6",

}

Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields. / Nolan, Mélanie; Davy, John L.

In: Journal of the Acoustical Society of America, Vol. 145, No. 6, 01.06.2019, p. 3330-3340.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Two definitions of the inner product of modes and their use in calculating non-diffuse reverberant sound fields

AU - Nolan, Mélanie

AU - Davy, John L.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

U2 - 10.1121/1.5109662

DO - 10.1121/1.5109662

M3 - Journal article

VL - 145

SP - 3330

EP - 3340

JO - Acoustical Society of America. Journal

JF - Acoustical Society of America. Journal

SN - 0001-4966

IS - 6

ER -