A hybrid modal analysis for enclosed sound fields and its applications

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.

Sound fields in realistic rooms were simulated by different kinds of computational methods, and comparisons were conducted. One is based on the wave acoustics and the other is based on the geometrical acoustics, i.e., the finite-element method (FEM) and the conical beam method (CBM). As for the software applied here, the authors have developed their own FEM code, namely LsFE-SFA, to conduct FEM computations, and they employed commercial software, i.e. raynoise, to conduct CBM computations. First, theoretical basis of LsFE-SFA was given briefly. Next, sound-pressure distributions in a reverberation room with the volume of 165 cubic meters were computed by both methods to compare the agreement with measured sound-pressure levels. The boundary conditions due to the sound absorption were changed to examine the relation between the absorbent condition and the agreements. The agreement between FEM and measurement were found to be better throughout the examination. Next, both FEM and CBM were applied onto the analysis of large-scale sound fields in a multipurpose hall with the volume of 37<th>000 cubic meters to discuss the agreement and difference in detail.

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.

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

The method of broadband Green's functions with low wavenumber extractions (BBGFL) is used to calculate Green's function for inhomogeneous waveguides filled with different dielectrics and with irregular boundaries. To construct the BBGFL modal solutions, we derive governing equations of the linear eigen-matrix problem and orthonormalization condition. In BBGFL, the Green's function is represented in modal expansions with convergence accelerated by higher order low wavenumber extractions. To obtain a linear eigenvalue problem for the modes, we use two BBGFLs of rectangular waveguides with two dielectric wavenumbers. The orthonormalized mode functions are used to construct the Green's function. Current wavenumber derivatives and Green's function wavenumber derivatives are computed by a single low wavenumber MoM impedance matrix. The wavenumber derivatives are used to accelerate the convergence of modal summations to 6th order. Numerical results are illustrated and compared with the direct MoM method of using free space Green's function. Results show accuracies and computation efficiencies for broadband simulations of Green's functions.

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 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 dyadic Green's functions of the cylindrical chirowaveguide are derived by modal expansion. Bohren's decomposition of the electromagnetic field is used to obtain the vector wave functions. The magnetic dyadic Green's function that is purely solenoidal is first derived and the electric dyadic Green's function is then obtained by manipulating the magnetic dyadic Green's function. The singular term in the expression of the electric dyadic Green's function is reinstated through the manipulation procedure.

It is useful to compute sound field of a source in a virtual environment which is different from the measurement environment. For example, some properties of sound source, such as directivity index and frequency response curve, are required to be measured in an anechoic room or free space, but both of them cannot be always accessible. Consequently, it will be useful to compute sound field of a source in free space when sound field of the source is not measured in the free space. In the aforementioned example, the free space is a virtual environment. Based on reciprocity theorem and modal expansion, a method to predict sound field of a source in a virtual environment is given in this paper when the scattering effect of the source can be neglected. Reciprocity theorem builds the relationship between measured sound field and predicted sound field, which plays an important role in the method. Green’s function in the virtual environment is needed in the method. To restrict measurement points on an enclosed surface, the Green’s function is expanded by a set of modes. A simulation is given to examine the validity of the method.

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.

Motivated by the problem of spatial sampling of a sound field by a spherical array, Chap. 3 presented methods for sampling functions on a sphere, followed by methods for reconstructing a function from its samples. These could form the basis for computing the sound pressure on the surface of a sphere, given measurements by an array of microphones. However, in spherical microphone array processing one may also be interested in computing the sound field around the array by decomposing the sound field into plane-wave components, for example. In this case, placing pressure or omni-directional microphones on the surface of a single sphere in free-field may not allow accurate plane-wave decomposition , due to zeros of the spherical Bessel function . This problem is presented at the beginning of the chapter. One possible solution is to place microphones on the surface of a rigid sphere . This configuration offers a practical advantage—the rigid sphere provides an ideal housing for all microphone wiring and conditioning electronics. However, one drawback of the rigid sphere is that sound scattered from the sphere can be reflected back by surrounding objects, thereby modifying the sound field it measures. This is particularly important for arrays used for sound field analysis in room acoustics, for example, in which case placing microphones in a free field, in an open-sphere configuration, may be preferable. Open spherical array configurations that avoid the problem of the zeros of the spherical Bessel function are therefore presented next. The array configuration may also affect other aspects of array performance related to the frequency range of operation and to the sensitivity to sensor noise and to other uncertainties. A general framework for array design that considers a range of objectives is introduced, followed by example designs. The chapter concludes with a description of an open spherical array configuration in which the microphones are placed within the volume of a shell. Other array configurations, including the hemispherical array , another array comprised of concentric rigid and open spheres , and an array incorporating non-spherical sampling surfaces, are also discussed.

We study light propagation in biological tissue using the radiative transport equation. The Green's function is the fundamental solution to the radiative transport equation from which all other solutions can be computed. We compute the Green's function as an expansion in plane-wave modes. We calculate these plane-wave modes numerically using the discrete-ordinate method. When scattering is sharply peaked, calculating the plane-wave modes for the transport equation is difficult. For that case we replace it with the Fokker-Planck equation since the latter gives a good approximation to the transport equation and requires less work to solve. We calculate the plane-wave modes for the Fokker-Planck equation numerically using a finite-difference approximation. The method of computing the Green's function for it is the same as for the transport equation. We demonstrate the use of the Green's function for the transport and Fokker-Planck equations by computing the point-spread function in a half-space composed of a uniform scattering and absorbing medium.