Living room acoustics You can use the heading from the model documentation for the title. © 2013 COMSOL. All rights reserved.

Background and Motivation This is a model that solves for the Eigenfrequencies and for the acoustic field inside a living room where the sound sources are placed at the loudspeaker location. The model enables the user to get the Eigenfrequencies around 90 Hz and the response at any point in the living room up to 500 Hz. This is useful when optimizing for loudspeaker locations inside the living room.

Geometry The geometry corresponds to a living room with different furniture: Shelf Loudspeaker boxes TV Table Aluminum legs TV Table Couch Carpet

Absorbing and damping boundaries The living room has many surfaces that are not sound-hard but rather act as absorbers. The couch is for example made of a porous material like, for example, foam covered with tissue or leather. These surfaces can be modeled by means of an impedance boundary condition defined by the normal impedance Zn(f) (Pas/m). The normal impedance is a complex-valued function that is function of the frequency. The normal impedance is related to the reflection coefficient R (complex ratio of reflected to incident pressure) as: The absorption coefficient, , is a real-valued quantity describing the amount of absorbed energy at a surface. It is related to the reflection coefficient as: 𝑍 𝑛 =𝜌𝑐 1+𝑅 1−𝑅 𝛼=1− |𝑅| 2

Absorbing and damping boundaries The surface properties are best described by the reflection coefficient or the surface impedance, as they contain both amplitude and phase information. Absorbing surfaces are often only characterized by their absorption coefficient (f). Because it has no phase information ( is unknown), this impedance is only an approximate description of the acoustic surface properties: arbitrarily setting   0 will result in a real-valued impedance. This is NOT the fully correct description and may result in erroneous results. The importance of the phase decreases with increasing frequency. In the model, the absorption coefficient (f) of the different materials is defined by means of various interpolation functions under Global Definitions and we crudely assume   0. 𝑍 𝑛 =𝜌𝑐 1+|𝑅| 1−|𝑅| with 𝑅 = 𝑒 𝑖𝜃 1−𝛼

Modeling Interfaces The “Pressure Acoustics, Frequency Domain“ user interface is used to perform Eigenfrequency and Frequency Domain studies.

Modeling Interfaces A “Normal Acceleration“ boundary condition is used to define the sources.

Modeling Interfaces “Impedance“ boundary conditions are used to define the different materials.

Modeling Interfaces The reflection coefficients and the normal impedances are calculated according to these expressions:

Modeling Interfaces The absorption coefficients of the materials are defined by means of interpolation functions which contain the alpha of each material vs. Frequency from 125 Hz to 4000 kHz.

Modeling Interfaces A “Sound Hard Boundary (Wall) “ boundary condition is used for the remaining boundaries.

Mesh and Solver Settings In acoustics, the computational mesh must resolve the acoustic wavelength. The requirement is to have at least 5 second order elements per wave length. The maximal mesh size is given by: The mesh size is defined under: Global Definitions > Parameters ℎ 𝑚𝑎𝑥 = 𝜆 5 = 𝑐 5𝑓

Mesh and Solver Settings

Mesh and Solver Settings For large acoustic models it is efficient using an iterative solver with the geometric multigrid pre-conditioner. Decrease the relative tolerance (also lower than shown here) to get a more converged solution. This is important for higher frequencies with many standing modes.

Results – Eigenfrequency Acoustic Pressure (acpr) f = 77 Hz f = 72 Hz f = 80 Hz f = 83 Hz

Results – Eigenfrequency Sound Pressure Level (acpr) f = 77 Hz f = 72 Hz f = 80 Hz f = 83 Hz

Results – Eigenfrequency Acoustic Pressure, Isosurfaces (acpr) f = 77 Hz f = 72 Hz f = 80 Hz f = 83 Hz

Results – Frequency sweep Acoustic Pressure (acpr) f = 200 Hz f = 100 Hz f = 300 Hz f = 400 Hz

Results – Frequency sweep Sound Pressure Level (acpr) f = 200 Hz f = 100 Hz f = 300 Hz f = 400 Hz

Results – Frequency sweep Acoustic Pressure, Isosurfaces (acpr) f = 200 Hz f = 100 Hz f = 300 Hz f = 400 Hz

Results – Frequency sweep Slice f = 200 Hz f = 100 Hz f = 300 Hz f = 400 Hz

Results – Frequency sweep Intensity f = 200 Hz f = 100 Hz f = 300 Hz f = 400 Hz

Results – Frequency sweep Response Left Center Right A Cut Point 3D data set is used to plot and evaluate a value at these points which would correspond to the audience.

Results – Frequency sweep Response Resonance peaks are clear and due to the losses at the impedance boundaries.