Comparison of sound fields in rooms computed by finite-element method and by cone beam method

The eigenmode and free‐wave models of steady‐state reverberant sound field in rooms are sometimes mistakenly believed to be unrelated representations. These two models are shown to coincide for the case of rectangular rooms, for which the axial, tangential, and oblique modes can be decomposed into two, four and eight travelling plane waves, respectively. (This was pointed out by Morse and Bolt [Rev. Mod. Phys. 16 (2), 85 (1944)].) Above the frequency of modal overlap, therefore, the reverberant sound field in a rectangular room may be viewed as an ensemble of many plane waves arriving from many directions in space. This provides justification for use of the free‐wave model.

Prediction of sound fields in rooms using statistical energy analysis

Scattering coefficients (SCs) are crucial for modeling room sound fields in rooms. Due to the lack of SCs on many practical surfaces and room objects, this study aims to derive a relation between surface SCs and the angulardependent<br/>SCs obtained by a spherical receiver array in the room. The work is based on a computational approach. The sound field at a spherical receiver array is computed for a room with and without surface scattering at the room<br/>boundaries. A hybrid model of the Image Source Model and the Radiosity Model is used for this purpose. The sound field on the spherical receiver is then decomposed into plane waves, leading to ambisonic impulse responses<br/>(IRs). Then, the angular-dependent SCs are derived from these ambisonic IRs in both scenarios with a coherence method. Results show that angular-dependent SCs vary significantly depending on the apparent area of scattering<br/>surfaces as seen from the receiver, and a qualitative relationship between surface SCs and angular-dependent SCs is observed. This approach offers a promising framework for quantifying the scattering properties of room boundaries<br/>based on spherical array receivers.

Eigenfrequencies are found in sound fields in rooms by analyzing wave equations for a rectangular room surrounded by hard walls. The relationship between eigenfrequencies and eigenfunctions holds well for the wave equations as well as for linear equations. It might be quite rare to see a perfectly rectangular room in our daily life, but, room acoustics in rectangular rooms provide a good fundamental basis for better understanding of room acoustics in general. Representation of an impulse response and its frequency characteristics can be derived using the Green function in a rectangular room. Consequently, the wave theoretic representation of a sound field is derived from the expression in terms of geometric acoustics, such as the mirror image theory. Finally, a general solution form of the wave equation is presented using an integration formula in three-dimensional space.

By considering that surface irregularities of building facades are sufficient to produce diffusion in streets, the diffuse sound field modeling by the mathematical theory of diffusion is used to predict sound propagation and reverberation in rectangular streets. This modeling was first developed to predict the sound field in rooms with diffusely reflecting boundaries [J. Picaut, J.-D. Polack, and L. Simon, Acust. Acta Acust. 83(4), 614–621 (1997)]. In this paper, the modeling is applied to urban acoustics without significant modification. A 3-D diffusion equation is derived for the sound energy density and solved for time-varying sources and in steady state. A diffusion coefficient is introduced to characterize the degree of diffusion of the street, and an exchange coefficient depending on the absorption coefficient is used to take the wall absorption into account. When the diffusion coefficient is chosen correctly, predicted reverberation times and sound attenuation are in accordance with experimental data obtained in a scale model of a street. With new developments, by introducing in the model, for example, intersections, moving sources, meteorological effects..., this original modeling should be in the future, a simple solution to predict the noise impact in large urban areas.

The use of image and ray tracing methods (and variations thereof) for the computation of sound fields in rooms is relatively well developed. In their regime of validity, both methods work well for prediction in rooms with small amounts of diffraction and mostly specular reflection at the walls. While extensions to the method to include diffuse reflections and diffraction have been made, they are limited at best. In the fields of illumination and computer graphics the ray tracing and image methods are joined by another method called luminous radiative transfer or radiosity. In radiosity, an energy balance between surfaces is computed assuming diffuse reflection at the reflective surfaces. Because the interaction between surfaces is constant, much of the computation required for sound field prediction with multiple or moving source and receiver positions can be reduced. In acoustics the radiosity method has had little attention because of the problems of diffraction and specular reflection. The utility of radiosity in acoustics and an approach to a useful development of the method for acoustics will be presented. The method looks especially useful for sound level prediction in industrial and office environments. [Work supported by NSF.]

In 1939, Dr. Dah-You Maa presented a paper entitled "The Distribution of Eigentones in a Rectangular Chamber at Lower Frequency Ranges" [D. Y. Maa, J. Acoust. Soc. Am. 10, 258 (1939)]. Since then, his interest in room acoustics has not diminished. More than six decades later, Maa proposed an idea of adding a monopole solution to the normal modal expansion to improve the accuracy in simulating the near-field sound field in rooms [D. Y. Maa, Acta Acust. (Beijing) 27, 385-388 (2002)]. Following this idea, a hybrid model that combines the free field Green's function and a modal expansion has been proposed by the authors based on a rigorous mathematical derivation [B. Xu et al, J. Acoust. Soc. Am.128, 2857-2867 (2010)]. The hybrid modal expansion can be further extended for complex sound sources by introducing the multipole expansion to the solution. In this paper, the theoretical derivation of the hybrid modal expansion will be reviewed, followed by examples demonstrating the use of the hybrid modal expansion in real-world applications.

In the Bergeron method [Shimoda et al., Trans. Inst. Electron. Commun. Eng. Jpn., Part A 72, 1–11 (1989); English transl., 1989 Electron. Commun. Jpn., Vol. 72; Scripta Technica, Inc. (Wiley, New York, 1990)], sound reflection surfaces in rooms are modeled by the simple resistances corresponding to the rigid walls. And the low-frequency responses for a middle scale of auditorium are agreed preferably with the real fields [K. Nakagawa and H. Shimoda, ASVA 97]. However, for big absorption acoustic models such as recording studios, or mixing rooms, the surfaces in rooms must be modeled by the impedance reflections. In this paper, the basic treatments for the impedance reflection models in the Bergeron method are investigated, and the wideband time-stretched pulses are used to simulate one-dimensional sound fields. The feasibility of numerical simulations using the Bergeron method are confirmed by supposing a simple surface impedance reflection model.

The room acoustic design of the low frequency characteristics of control and recording rooms in studio facilities still poses a challenging task to acoustic designers and consultants. Due to the low density of eigenmodes at frequencies below the Schroeder frequency, the statistical assumptions underlying geometrical acoustics theory do not hold for this frequency range. Consequently simulation methods such as 'raytracing' or the 'mirror-image-method' cannot supply reliable results here. On the other hand analytical solutions for the wave equation of the sound field in enclosed spaces are generally restrained to rather simple room geometries and boundary conditions. Therefore numerical methods such as the finite element method (FEM) are promising contenders for the simulation of low frequency sound fields in arbitrarily shaped rooms with complex boundary conditions. In the course of this study the FEM was used to simulate the low frequency sound field in the control room of a recording studio in Munich, Germany. The results are compared to measurements in the real room in order to evaluate the performance of the simulation. Special attention is paid to a realistic representation of the acoustic characteristics of the room boundaries in the model and the limitations of an impedance approach are discussed. It is made clear, that the Achilles' heel of room acoustic FEM simulations appears to be the realistic boundary specification and that further work has to focus on the identification or development of robust and reliable measurement or modelling techniques to determine the acoustic characteristics of the various materials on the room boundaries.

Acoustical observations and measurements have been made in a number of large rooms in Italy where the size, architectural detail, and decorations, especially the sculpture, produce several interesting acoustical phenomena. The wide range of air volume studied permitted special attention to be given to the facility with which sound fields can be established in them. Large variations of room shapes and extremes in the degree of acoustical coupling between airspaces also made interesting observations possible. The buildings studied afforded great variations in architecture and styles of decoration so that the scatterers for sound diffusion covered a considerable range, effecting the character of the reverberation and the effective sound absorption of complex surfaces. In the largest spaces, the effect of air absorption produced significant effects not only at the higher frequencies but also at intermediate frequencies. Measurements made with large variations in relative humidity permitted this effect to be isolated.

A frequency domain finite element solver for acoustic simulations of 3D rooms with microperforated panel absorbers

Chapter 4 - Sound Fields in Rooms

Among frequently used approaches for evaluation of the sound field inside a room is the one based on using of image source method. This method has the advantage of a very accurate calculation of the reflections, but for higher order reflections, the calculation explodes due to an exponential increase of the number of possible image sources. Numerical method which makes the convergence of the infinite row in the sound level computation faster, will be presented in this paper. In this way the sound level at an arbitrary room point can be predicted only by means of the energy contribution of a relatively small number of image sources considerably shortening the computing time

The image source method (ISM) is often relied upon to simulate room acoustical fields due to its computational efficiency. While it is common to find implementations of the ISM for shoebox-shaped rooms, the ISM can be used to simulate more general room shapes. However, including the directivities of sources and receivers has not been well resolved for arbitrary room geometry in the literature. In this work, we extend the diffraction-enhanced image source method, which uses spherical harmonic descriptions to include directional sources and receivers, to arbitrary room geometries. The proposed extension is evaluated using finite element method simulations for different room configurations and is shown to capture many of the relevant features of the room transfer functions. In addition, discrepancies in the evaluation results indicate possible missing acoustic effects that are difficult to capture with ISM-based methods and need to be addressed.

International Symposium : Computer Modelling and Prediction of Objective and Subjective Properties of Sound Field in Rooms