Purdue University Purdue e-pubs Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering 8-216 Experimental Investigation of the Use of Equivalent Sources Model in Room Acoustics Simulations Yangfan Liu Purdue University, liu278@purdue.edu J Stuart Bolton Purdue University, bolton@purdue.edu Follow this and additional works at: http://docs.lib.purdue.edu/herrick Liu, Yangfan and Bolton, J Stuart, "Experimental Investigation of the Use of Equivalent Sources Model in Room Acoustics Simulations" (216). Publications of the Ray W. Herrick Laboratories. Paper 135. http://docs.lib.purdue.edu/herrick/135 This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.
Yangfan Liu Advisor: J. Stuart Bolton
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Room Acoustics with Source of Finite Size Sound prediction of a flat screen TV: Not point sources Arbitrary room shapes Fast Traditional Methods: Ray Tracing Image Source Model (hybrid) BEM/FEM Available techniques are not suitable for a fast and accurate room acoustics simulations of finite-size sources The Use of Equivalent Source Models (ESM) The actual sound field Governing Equation Boundary Conditions Sound field from sources Boundary Conditions Estimate ESM parameters by matching the BCs 4
Sound Field Components in Room Acoustics Room surface Source surface Free-space component Incoming component Scattering component Pressure: P f, Particle velocity: u f Room component: (P r, u r ) Total sound field: p p p ut u f ur t f r Free-space ESM vs. Room Acoustics ESM Free-space ESM Outgoing waves only Pressure of the total sound field (sampled at a measurement surface) Source type Boundary condition Room Acoustics ESM Both incoming and outgoing waves Impedances of room component (sampled at both source and room surfaces) 5
BC of the Room Components in Room Acoustics o BC on source surface, Γ 1, (admittance β 1 ): Total: Free-space: β 1 (x)p t (x) + u (x) = u nt (x) β 1 (x)p f (x) + u (x) = u nf (x Room surface (Γ 2 ) Source surface (Γ 1 ) Free-space component Room component o BC on room surface, Γ 2, (admittance β 2 ): Total: In-vacuo driving velocity on the source surface. (source characteristics) β 2 (x)p t (x) = u nt (x) Total sound field: p p p u u u t f r t f r Boundary conditions used in the room acoustics ESM: ( x) p ( x) u ( x) x ( x) p ( x) u ( x) u ( x) ( x) p ( x) x 1 r nr 1 2 r nr nf 2 f 2. Can be viewed as a weighted The problem of room acoustics: combination of the pressure BC Given: and the P f velocity, u f ; Calculate: BC P r, u r 6
General Procedure of Constructing Room Acoustics ESM N p( x) gi( x, yi ) Qi, i1 N 1 u( x) gi ( x, yi ) Qi, j i 1 Total sound pressure Total particle velocity Location in the sound field Location of each source g i (x, y i ) sound field of source with unit strength Q i - the source strength 7
General Procedure of Constructing Room Acoustics ESM Matrix form used to estimate the ESM parameters: (1) (1) B 1Ap A u n Q u B p (2) (2) B 2 A p A u nf n 2 f source surfaces room surfaces B diag( ( x ), ( x ),..., ( x )), 1 1 1 1 2 1 B diag( ( x ), ( x ),..., ( x )) 2 2 M 1 2 M 2 2 M 1 1 (1) (1) ( Ap ) ij g j ( xi, y j ), ( Au ) (, ), n ij ng j xi y j j (2) (2) ( Ap ) ij g j ( xm, ), ( ) (, ), 1i y j Au n ij ng j xm1i y j j M 1 1 1 Estimate the source strengths Least squares and regularization 8
Sound field expression for multipole sources (3D): o Source of order zero (monopole) with unit strength is expressed as: jkr out e in P ( x, y), P ( x, y) 4r jkr e 4r o Source of order n with unit strength include all the following: P n n i j xy z k P with i j k n time dependence e jωt N = 1: dipole (3) N = 2: quadrupole (6) N = 3: octupole (1) N = 4:? (15) Nn ( ) C 1, n n n31, n Liu, Y., & Bolton, J. S. (216). On the completeness and the linear dependence of the Cartesian multipole series in representing the solution to the Helmholtz equation. The Journal of the Acoustical Society of America, 14(2), EL149-EL153. 9
Comparison of ESM and BEM at 5 Hz Room surface: absorption coefficient.5 Source surface: impedance: 9 Rayls In-vacuo driving velocity: u ( ) 2 cos (2 ) /4 else - θ is the angle to the x axis o Free-space sound field is calculated using BEM 1
Sound Pressure (Imaginary Part) Sound Pressure (Real Part) Comparison of ESM and BEM at 5 Hz 1 5-5 Comparison at 5 Hz Bottom Plane Middle Plane Top Plane BEM ESM -1 5 1 15 2 25 3 Receiver Index 5 BEM ESM -5-1 5 1 15 2 25 3 Receiver Index BEM nodes: 4715 ESM sampling points: 332 ESM parameters: 7 - The ESM contains outgoing and incoming multipoles both up to order 4. 11
Measure the sound field from a loudspeaker in a small room Rectangular room (1.867 m 1.771 m 1.95 m) with additional small features as shown. Loudspeaker is at the room center. Hard Surfaces Most surfaces are made of plywood covered by sound absorbing materials. Some small surfaces (highlighted in the figure) are uncovered and are assumed to be acoustically hard. Sound Absorbing Materials on Hard Surfaces Room Geometry 12
Imaginary Part - rayl Real Part - rayl Impedance measurements o Impedance of the covered surfaces are measured using the twomicrophone measurement on impedance tubes. 1 Measured Impedance 5 1 2 3 4 5 6 Frequency - Hz -5-1 -15-2 -25 1 2 3 4 5 6 Frequency - Hz 13
z - m o Sound pressure measured on planes parallel to 4 faces of the loudspeaker. Field o 72 microphones for near field (.25 m away from the loudspeaker faces). o 72 microphones for far field. (.5 m away from the loudspeaker faces). o 487 sampling points on the room surface. o 166 sampling points on the source surface. -.2 -.6Field.6.6 x - m.8.8 o Room acoustics ESM includes outgoing and incoming multipoles y - m up to order 3 (4 model parameters.).6.4.2 -.4 -.2.2.4.4.2 -.2 -.4 o Source information is calculated by using the free-space Multipole ESM up to order 3 (2 model parameters). 14
Imaginary Part Sound Pressure (Pa) Real Part Sound Pressure (Pa) Model performance at 52 Hz o Use the impedance boundary condition to calculate the source strengths..15.1.5 Right Back Left Model Performance at 52 Hz Right Experiment ESM Prediction Free-Space Back -.5 -.1 5 1 15 Receiver Index 4 x 1-3 2 Back Left Back -2-4 Experiment -6 ESM Prediction Free-Space -8 5 1 15 Receiver Index 15
Imaginary Part Sound Pressure (Pa) Real Part Sound Pressure (Pa) Model performance at 124 Hz o Use the impedance boundary condition to calculate the source strengths. 4 x 1-3 2 Right Back Left Model Performance at 124 Hz Right Back Left Back -2 Experiment -4 ESM Prediction Free-Space -6 5 1 15 Receiver Index Back 6 x 1-3 4 Experiment ESM Prediction Free-Space 2-2 -4 5 1 15 Receiver Index 16
Imaginary Part Sound Pressure (Pa) Real Part Sound Pressure (Pa) Model performance at 2 Hz o Use the impedance boundary condition to calculate the source strengths. 1 x 1-3 5 Right Back Left Model Performance at 2 Hz Right Experiment ESM Prediction Free-Space Back -5 5 1 15 Back Left Receiver Index Back 6 x 1-3 4 2 Experiment ESM Prediction Free-Space -2-4 -6 5 1 15 Receiver Index 17
Model performance (Fixed location ESM) at 52 Hz o First calculate sound pressure and normal velocities on the boundaries using BEM (3438 nodes). o Compare ESM results using pressure BC and impedance BC. The Pressure Boundary Condition: (1) (1) A p p r Q (2) (2) A p p r Impedance Boundary Condition: (1) (1) B 1Ap A u n Q u B p (2) (2) B 2 A p A u nf n 2 f Can be viewed as a weighted combination of the pressure BC and the velocity BC 18
Model performance (Fixed location ESM) at 52 Hz o First calculate sound pressure and normal velocities on the boundaries using BEM (3438 nodes). o Compare ESM results using pressure BC and impedance BC. o Pressure BC gives reasonable prediction accuracy. o ESM can be used as a reduced order modeling to BEM. o Impedance BC prediction is a weighted average of the pressure and the velocity BC predictions. o Admittance (very small values) is the weighting factor of pressure BC, so the velocity BC is favored. 19
Equivalent source models were constructed to perform room acoustics simulations with finite-size sources, arbitrary geometry and nonuniform surface normal impedances. In room acoustics ESMs, both outgoing and incoming waves should be included, and the impedance boundary conditions for the room component sound field are used for parameter estimation. Calculating the model parameters using the impedance boundary condition produces a weighted combination of the results using the pressure boundary condition and the results obtained using velocity boundary condition. Prediction accuracy needs to be improved. In the case of a small rectangular room, the multipole ESM can be used in conjunction with a BEM to create a reduced order model for sound field prediction. Free field model build on velocity measurements may improve the accuracies. Alternative choices of basis functions may also improve the results. 2
Liu, Y., & Bolton, J. S. (216). On the completeness and the linear dependence of the Cartesian multipole series in representing the solution to the Helmholtz equation. The Journal of the Acoustical Society of America,14(2), EL149- EL153. Liu, Y., & Bolton, J. S. (213). The use of equivalent source models for reduced order simulation in room acoustics. Proceedings of the 21st International Congress of Acoustics, ICA 213, Montreal, Canada, June 2-7, 213. Liu, Y., & Bolton, J. S. (215). Simulation of sound fields radiated by finite-size sources in room environments by using equivalent source models: three-dimensional validation Proceedings of INTER-NOISE Conference 215, San Francisco, CA, USA, August 9-12, 215 (pp. 4246-4259). Institute of Noise Control Engineering. 21