Institute of Fundamental Technological Research

The paper examines the low-frequency reverberation sound field in coupled-room systems. In theoretical model, the modal expansion of sound pressure was applied, while in numerical procedure, the discrete Hilbert transform was used to determine the amplitude of decaying sound. Computer simulations were performed for a room system consisting of two connected rectangular rooms. Eigenfunctions and eigenfrequencies of this system were determined by the finite element method. Simulation results showed that for the hard-walled room system the sound decay is almost exponential for frequencies of modes localized in one of the subrooms. Acoustical treatment of the ceiling significantly reduced reverberation. However, due to beating effects and modal overlap, a large irregularity of sound decay curves has occurred. This makes it difficult to correctly qualify the sound decay, because in this case it is practically impossible to characterize the reverberation process with only one or at most two decay times.

room acoustics, coupled rooms, reverberant sound field, mode localization, sound decay

Enclosures with a mixed rectangular-cylindrical geometry are common in sacred, historic and modern buildings. Therefore, obtaining efficient analytical formulas for describing the sound radiation inside such structures is of practical importance. These formulas can be implemented in open-source software and can aid in the analysis of acoustic fields. In this study, the problem of sound radiation by a point source located inside a rectangular enclosure with a sound-absorbing quarter-cylindrical ceiling was analytically solved. The formulas presented in this paper can also be used to obtain the eigenfrequencies and eigenfunctions. To obtain the analytical solution, two connected subregions, namely rectangular space, and quarter-cylinder, were considered, and the continuity conditions were imposed on the region’s coupling interface. The acoustic field was described using the solution for a rigid-walled room and the additional components containing the Fourier and Dini series. These components allow the solution to satisfy the impedance boundary condition and the continuity conditions. The formulas and their computer implementation were validated using the finite element method (FEM). A good agreement was achieved between results given by both methods. The numerical analysis demonstrated that the use of the formulas is less time-consuming than using FEM when the frequency is greater than 150 Hz. Moreover, compared to FEM, the analytical solution can be applied to higher frequencies for which the use of FEM requires hardware with high computing capability.

Room acoustics,Irregular enclosure,Efficient analytical solution,Acoustic pressure field,Acoustic intensity vector field,Complex eigenfrequencies

The paper investigates the sound field excited by a boundary pure-tone source in rigid-walled rectangular rooms. This approach is applicable in the low-frequency range, where sound absorption by wall surfaces can be considered negligible. The sound pressure was theoretically determined by applying the Green's function based on the cut-on and evanescent modes expansion instead of the usual normal mode expansion. The theoretical model was used to predict the spatial distribution of the sound pressure level at different source frequencies. The calculation results have shown that for audible frequencies below the cut-off frequency, the plane wave mode and evanescent modes strongly interfere which results in an interference pattern with large dips in the pressure level forming a continuous curve. A shape of this curve is highly dependent on the excitation frequency. These dips have been found to occur when the sound associated with the plane wave mode is cancelled by the sound produced by evanescent modes.

rectangular room, evanescent modes, plane wave mode, sound cancellation, sound intensity

Accurate definition of boundary conditions is of crucial importance for room acoustic predictions because the wall impedance phase angle can affect the sound field in rooms and acoustic parameters applied to assess a room reverberation. In this paper, the issue was investigated theoretically using the convolution integral and a modal representation of the room impulse response for complex-valued boundary conditions. Theoretical considerations have been accompanied with numerical simulations carried out for a rectangular room. The case of zero phase angle, which is often assumed in room acoustic simulations, was taken as a reference, and differences in the sound pressure level and decay times were determined in relation to this case. Calculation results have shown that a slight deviation of the phase angle with respect to the phase equal to zero can cause a perceptual difference in the sound pressure level. This effect was found to be due to a change in modal frequencies as a result of an increase or decrease in the phase angle. Simulations have demonstrated that surface distributions of decay times are highly irregular, while a much greater range of the early decay time compared to the reverberation time range indicates that a decay curve is nonlinear. It was also found that a difference between the decay times predicted for the complex impedance and real impedance is especially clearly audible for the largest impedance phase angles because it corresponds approximately to 4 just noticeable differences for the reverberation metrics.

room acoustics, complex wall impedance, indoor sound field, room impulse response, reverberation parameters

Absorbing properties of multi-layer acoustic absorbers were modeled using the impedance translation theorem and the Garai and Pompoli empirical model, which enables a determination of the characteristic impedance and propagation constant of fibrous sound-absorbing materials. The theoretical model was applied to the computational study of performance of single-layer acoustic absorber backed by a hard wall and the absorber consisting of one layer of absorbing material and an air gap between the rear of the material and a hard back wall. Simulation results have shown that a high thickness of absorbing material may cause wavy changes in the frequency relationship of the normal and random incidence absorption coefficients. It was also found that this effect is particularly noticeable for acoustic absorbers with a large thickness of air gap between the absorbing material and a hard back wall.

sound absorption, multi-layer absorber, surface impedance, fibrous materials, air gap

In this paper, the steady-state and transient sound fields inside 3D enclosed spaces were examined theoretically using the modal expansion method. An analytic formula for the Green's function was derived allowing to predict the interior sound field for both the harmonic and impulse boundary excitations. A theoretical model was tested numerically for a 3D car-like cavity with a sound absorbing material placed on walls. For a harmonic excitation, calculation results revealed high impact of a frequency and a sound damping on a distribution of the steady-state pressure amplitude. They also indicated irregularly located energy vortices in the active intensity vector field. Simulations of a transient sound field were carried out for a boundary excitation producing a sine wave pulse. Temporal changes in this field were studied using the pressure amplitude determined via the discrete Hilbert transform. A large temporal variability of a distribution of this amplitude was noted due to a strong dependence of a transient sound on a position of the observation point. Simulation results also shown that for a small sound damping on cavity walls, there was an intense sound reverberation inside the cavity.

enclosed spaces, sound field prediction, modal expansion method, Green's function, sound intensity, reverberation time

A room impulse response obtained for complex-valued boundary conditions on wall surfaces was used to determine the frequency response of arbitrary shaped room. Based on theoretical findings, a numerical procedure was developed to test the effectiveness of a high-density sound absorbing material in improving low-frequency spectral flatness. The impedance of absorbing material was determined using the two-parameter Komatsu model. The simulation results have shown that the smoothing effect of the frequency response becomes apparent when the thickness of absorbing material is large enough. This is because as the material thickness increases, the sound absorption tends to increase at lower frequencies.

frequency response of room, spectral flatness, room impulse response, wall impedance

In the low-frequency range, the acoustical behaviour of enclosed spaces is strongly influenced by excited acoustic modes resulting in a spatial irregularity of a steady-state sound field. In the paper, this problem has been examined theoretically and numerically for a system of coupled spaces with complex-valued conditions on boundary surfaces. Using a modal expansion method, an analytic formula for the Green's function was derived allowing to predict the interior sound field for a pure-tone excitation. To quantify the spatial irregularity of steady-state sound field, the parameter referred to as the mean spatial deviation was introduced. A numerical simulation was carried out for the system consisting of two coupled rectangular subspaces. Eigenfunctions and eigenfrequencies for this system were determined using the high-accuracy eigenvalue solver. As was evidenced by computational data, for small sound damping on absorptive walls the mean spatial deviation peaks at frequencies corresponding to eigenfrequencies of strongly localized modes. However, if the sound damping is much higher, the main cause of spatial irregularity of the interior sound field is the appearance of sharp valleys in a spatial distribution of a sound pressure level.

interior acoustics, coupled spaces, steady-state sound field, modal expansion method, sound damping, Green's function

In the low-frequency range, a sound reproduction in enclosures is strongly influenced by excited room modes. While the spectral impact of acoustic modes on a room response is well recognized, there is no sufficient knowledge on how these modes affects transients. In the paper this issue has been examined theoretically and numerically for a room excited by a tone burst by using a modal expansion method supported by a computer implementation. To quantify a temporal accuracy of a sound reproduction, the new metrics referred to as the tone burst reproduction error was introduced. The basis for determining this quantity was a deviation between the tone burst amplitude and the amplitude of a sound pressure computed via the Hilbert transform. A numerical simulation was performed for an irregularly shaped enclosure having a form of two-room coupled system. Calculation results have proved that a high inaccuracy of a tone burst reproduction occurs at receiving points corresponding to sharp dips in a distribution of the steady-state sound pressure level. This is because in these points an amplitude of transient sound is much bigger than a tone burst amplitude. It was discovered that strong narrow peaks in the tone burst reproduction error are located at centers of vortices in the active sound intensity vector field. An influence of a sound damping in a room on a reproduction of a tone burst was also examined and it was found that a substantial increase in a wall sound absorption does not significantly improves a tone burst reproduction because it does not eliminate sharp dips in a distribution of the steady-state sound pressure level.

room acoustics, room modes, transients, tone burst, discrete Hilbert transform, sound intensity vector field

A modal representation of a room impulse response has been used to formulate expressions for low-frequency sound field in rooms of arbitrary shape. Based on theoretical results, a simulation program has been developed to predict a sound pressure distribution and a room transfer function for rectangular enclosure having walls covered by a material of complex impedance. Calculation results have shown that changes in the wall reactance entail a substantial modification of a sound pressure distribution. Furthermore, an influence of wall reactance on the room transfer function was investigated and it was discovered that a change in a reactance sign causes a shift in frequencies of modal vibrations excited in the room.

room acoustics, modal vibrations, room impulse response, complex wall impedance

A new method for determining optimum dimension ratios for small rectangular rooms has been presented. In a theoretical model, an exact description of the room impulse response was used. Based on the impulse response, a frequency response of a room was calculated to find changes in the sound pressure level over the frequency range 20-200 Hz. These changes depend on the source and receiver positions, thus, a new metric equivalent to an average frequency response was introduced to quantify the overall sound pressure variation within the room for a selected source position. A numerical procedure was employed to seek a minimum value of the deviation of the sound pressure level response from a smooth fitted response determined by the quadratic polynomial regression. The most smooth frequency responses were obtained when the source was located at one of the eight corners of a room. Thus, to find the best possible dimension ratios, in the numerical procedure the optimal source position was assumed. Calculation results have shown that optimum dimension ratios depend on the room volume and the sound damping inside a room, and for small and medium volumes these ratios are roughly 1:1.48:2.12, 1:1.4:1.89 and 1:1.2:1.45. When the room volume was suitably large, the ratio 1:1.2:1.44 was found to be the best one.

room acoustics, small rooms, optimum dimension ratios, room impulse response, frequency room response

A small rectangular room with hard walls has a number of acoustic flaws and the most serious drawback is a long reverberation time. A technique commonly used for improving room acoustics consists in increasing a sound absorption on a ceiling. In this study, the impact of acoustical treatment of a ceiling on reverberant properties of a small rectangular room was examined. Changes in the modal reverberation time due to this treatment were investigated by the analytical method. As was evidenced by calculations, the initial increase in a sound absorption on a ceiling causes a substantial decrease in the modal reverberation time and a treatment efficiency decreases with a further absorption increase. It was found also that for a room with hard walls statistical and wave theories give the same result as the modal reverberation time for oblique modes and the Sabine's reverberation time are identical. A more detailed information about reverberant properties of a room was provided by the numerical method employing a backward integration of the squared room impulse response. Using this method, global and local reverberation parameters were determined. Numerical simulations discovered a quite good agreement between global and local reverberation time and high differences between global and local early decay time resulting from a nonlinear shape of a decay curve. Therefore, one can conclude that the global decay times characterize reasonably well a reverberation process in a late stage of sound decay but they are not correctly describe this process in an initial stage.

small room acoustics, modal expansion method, room impulse response, reverberation time, early decay time

A theoretical method has been presented to describe sound decay in enclosures and simulate the room impulse response (RIR) employed for prediction of the indoor reverberation characteristics. The method was based on a solution of wave equation with the form of a series whose time-decaying components represent responses of acoustic modes to an impulse sound source. For small sound absorption on room walls this solution was found by means of the method of variation of parameters. A decay function was computed via the time-reverse integration of the squared RIR. Computer simulations carried out for a rectangular enclosure have proved that the RIR function reproduces the structure of a sound field in the initial stage of sound decay sufficiently well. They have also shown that band-limitedness of the RIR has evident influence on the shape of the decay function and predicted decay times.

room acoustics, indoor reverberation, room impulse response, modal expansion method, decay times

The paper focuses on a theoretical description and numerical evaluation of active and reactive components of sound intensity in enclosed spaces. As the study was dedicated to low-frequency room responses, a modal expansion of the sound pressure was used. Numerical simulations have shown that the presence of energy vortices whose size and distribution depend on the character of the room response is a distinctive feature of the active intensity field. When several modes with frequencies close to a source frequency are excited, the vortices within the room are positioned irregularly. However, if the response is determined by one or two dominant modes, a regular distribution of vortices in the room can be observed. The irrotational component of the active intensity was found using the Helmholtz decomposition theorem. As was evidenced by numerical simulations, the suppression of the vortical flow of sound energy in the nearfield permits obtaining a clear image of the sound source.

room acoustics, modal analysis, sound intensity, energy vortex, stagnation point

The main aim of room acoustics is to predict reverberant properties of enclosures from measured or numerically simulated room responses. In this paper, this issue was examined in low-frequency range where acoustic characteristics of rooms are strongly frequency dependent due to differences in a modal damping. A theoretical description of a decaying sound field was based on a modal expansion of the sound pressure, and a reverberant response of the room was initiated by emission of Dirac delta time impulse or by switching off a time-harmonic source. Theoretical findings were employed to determine reverberant characteristics of a coupled-room system containing two connected rectangular subrooms. Simulation results have shown that a sound decay after steady-state harmonic excitation is strongly influenced by the sound frequency and due to large fluctuations in a decaying pressure; numerical techniques for smoothing decay curves are needed. Calculations also revealed that in some one-third octave bands, the decay function found via backward integration of squared room impulse response (RIR) changes very irregularly impeding a proper qualifying of nature of sound decay.

room acoustics, modal analysis, decay times, coupled rooms

The paper presents a theoretical basis of calculations of the sound intensity in enclosed spaces and shows results of numerical visualization of the active intensity in a room with absorptive walls formed by two coupled rectangular subrooms. The study was focused on the low-frequency range, therefore to describe the active and reactive intensities, the modal theory of room acoustics was applied. Space distribution of eigenfunctions, modal frequencies and modal damping coefficients were calculated numerically using the forced oscillator method (FOM) and the nite difference time-domain (FDTD) method. Based on theoretical and numerical results, the computer program has been developed to simulate the active intensity vector eld when the room is excited by a harmonic point source. Calculation data have shown that the active intensity was extremely sensitive to position of the source since at a fixed source frequency, different source locations always generate different distributions of characteristic objects of the active sound eld such as energy vortices and stagnation points. Because of complex room shape, the vortex centers are in most cases positioned irregularly inside the room. Almost regular arrangement of vortices was found only in the case when the source frequency was tuned to the frequencies of modes which were strongly localized in one of the subrooms.

room acoustics, coupled rooms, sound intensity, energy vortex, stagnation point

The modal expansion method has been used to formulate expressions for real and imaginary parts of the complex sound intensity inside enclosures. Based on theoretical results, the computer program has been developed to simulate the acoustic intensity vector field inside the irregular room whose shape resembles the capital letter L. Calculation results have shown that a low-frequency distribution of the acoustic intensity is strongly influenced by the modal localization and the characteristic objects in the active intensity field are energy vortices and saddle points positioned irregularly inside the room. It was found that for small sound damping the vortex centers lie exactly on the lines corresponding to zeros of the eigenfunction for a dominant mode. An increase in a sound attenuation results in the change of vortex positions and can cause the formation of new vortices. Finally, an influence of the wall impedance on the quantitative relation between the acoustic and reactive intensities was studied and it was concluded that for very small sound damping the behavior of the sound intensity is basically only oscillatory.

Room acoustics, Irregular room, Sound intensity, Energy vortex, Modal localization

The analytical method has been developed to determine the acoustic behaviour of a lightly damped room excited by a sound source. Using the classical modal analysis, a solution of the wave equation was found as a sum of an infinite number of components representing individual responses of acoustic modes to the sound source. For weak sound absorption a general form of this solution was determined by means of the method of variation of parameters. From this solution the equations for the Green's function and the room impulse response were found. By employing the Green's function, the expressions describing a sound build-up and a sound decay for a harmonic excitation were derived.

room acoustics, wave equation, room impulse response, decay times

Reverberant responses are widely used to characterize acoustic properties of rooms, such as the early decay time (EDT) and the reverberation times T20 and T30. However, in real conditions a sound decay is often deformed by background noise, thus a precise evaluation of decay times from noisy room responses is the main problem. In this paper this issue is examined by means of numerical method where the decay times are estimated from the decay function that has been determined by nonlinear polynomial regression from a pressure envelope obtained via the discrete Hilbert transform. In numerical experiment the room responses were obtained from simulations of a sound decay for two-room coupled system. Calculation results have shown that background noise slightly affects the evaluation of reverberation times T20 and T30 as long as the signal-to-noise ratio (SNR) is not smaller than about 25 and 35 dB, respectively. However, when the SNR is close to about 20 and 30 dB, high overestimation of these times may occur as a result of bending up of the decay curve during the late decay.

room acoustics, reverberation, decay times, room response, background noise, coupled rooms

The paper describes briefly some main aspects of the active feedback control system that has been developed and constructed for reduction of vibroacustic emission of vibrating plate structures with arbitrary boundary conditions. Relations between the forms and frequency of the vibrations induced by an external harmonic excitation and the distribution of the generated acoustic pressure field are investigated using the developed numerical model based on the Indirect Variational Boundary Element Method. The aim of the control is to minimize the sound pressure level in a given point of the ambient space. The system uses small, rectangle-shaped piezoelectric transducers as both sensors and actuators. The transducers are connected in a number of independent feedback loops, and the feedback gains are the control parameters which are optimized using the developed optimal control algorithm. The constructed active system has been tested for the stability and control performance during experimental research performed in an anechoic chamber. Results of experiments are presented in the paper, proving a high level of noise reduction and a good agreement with numerical predictions.

Vibrating plate structures, Active feedback control system, Vibrational modes

In this paper, the modal expansion method supported by a computer implementation has been used to predict steady-state distributions of the potential and kinetic energy densities, and the active and reactive sound intensities inside two coupled enclosures. The numerical study was dedicated to low-frequency room responses. Calculation results have shown that the distribution of energetic quantities in coupled spaces is strongly influenced by the modal localization. Appropriate descriptors of the localization effect were introduced to identify localized modes. As was evidenced by numerical data, the characteristic objects in the active intensity field are vortices positioned irregularly inside the room. It was found that vortex centers lie exactly on the lines corresponding to zeros of the eigenfunction for a dominant mode. Finally, an impact of the wall impedance on the quantitative relationship between the active and reactive intensities was analyzed and it was concluded that for very small sound damping the behavior of the sound intensity inside the room space is essentially only oscillatory.

room acoustics, coupled rooms, acoustic energy density, active sound intensity, reactive sound intensity

The Hilbert transform is an important tool for a signal analysis, and it has been widely used in many areas of science and technology. An application of this method to an analysis of reverberant decay of modal vibrations in enclosures enables a detection of a pressure envelope resulting in a more accurate prediction of a reverberation time. It is shown that the discrete Hilbert transform is not accurate in an envelope detection even for the moderately damped harmonic signals, and a generated error strongly depends on the ratio of a modal damping coefficient to a mode frequency. This inexactness is due to the fact that for exponentially decaying harmonic signals the Bedrosian identity is not satisfied. In order to increase the accuracy of the envelope prediction, appropriate modifications of discrete signals corresponding to different kinds of modes are proposed. This method is successfully tested in the multi-mode case using discrete signals simulating a reverberant response of two coupled rooms.

Hilbert transform, modal vibrations, reverberation time, room acoustics

In the paper, an accuracy of the discrete Hilbert transform method in a detection of amplitude and frequency of vibration signals has been examined. This issue was studied for sinusoidal damped vibrations often encountered in practical applications. It has been shown that an inexactness of the Hilbert transform generates fictitious oscillations of amplitude and frequency. For small damping of vibrations these oscillations were significant only at both ends of a discrete signal. An error induced by a numerical algorithm was especially severe for strongly damped vibrations because it manifested itself by a large overestimation of amplitude and erroneous jumps of frequency. To reduce the error generated by the Hilbert transform method, the appropriate modification of discrete signal was proposed.

vibration signals, discrete Hilbert transform, Bedrosian theorem

In the paper, a usefulness of the Hilbert transform in a computer modelling of reverberant behaviour of rooms is demonstrated for the enclosure consisting of two coupled subrooms. In numerical simulations a decay of the sound pressure is computed and the Hilbert transform methodology is used to determine an envelope of this decay. Calculation results have shown that, because of the mode localization, a distribution of absorbing material has a great influence on decay times evaluated from changes in the pressure envelope. For example, when a difference between the sound damping in subrooms is large, the localization effect is responsible for a creation of a nonlinear decay of the sound pressure level characterized by rapid early and slow late sound decays.

room acoustics, reverberant behaviour of rooms, coupled rooms, Hilbert transform

In this paper, the computer modelling application based on the modal expansion method is developed to study the influence of a sound source location on a steady-state response of coupled rooms. In the research, an eigenvalue problem is solved numerically for a room system consisting of two rectangular spaces connected to one another. A numerical procedure enables the computation of shape and frequency of eigenmodes, and allows one to predict the potential and kinetic energy densities in a steady-state. In the first stage, a frequency room response for several source positions is investigated, demonstrating large deformations of this response for strong and weak modal excitations. Next, a particular attention is given to studying how the changes in a source position influence the room response when a source frequency is tuned to a resonant frequency of a strongly localized mode.

coupled room system, steady-state room response, potential and kinetic energy densities, localization of modes

In the present study, the acoustic characteristics of coupled rooms were investigated using the computer modelling application based on the modal expansion method. A numerical procedure enables the computation of shape and frequency of eigenmodes, and allows one to predict sound decay times at any point of a room space. Results of numerical simulation have shown that along with a change in a coupling area the effect of a mode degeneration appeared and there are two main reasons for this. First, a mode degeneration is associated with a convergence of frequencies of neighbouring modes with a decrease of coupling area. Another reason for the mode degeneration are variations in a sequence of modes with a change in the coupling area. It was demonstrated that in the coupled rooms several modes exhibit a considerable localization. A degree of mode localization was described theoretically by the “existence surface” of mode and it was found that the localization phenomenon has a great effect on the reverberation process because for some absorbing material distributions it caused the double–slope decay characterized by steep initial and slow late sound decays.

A simple analytical method is developed to estimate frequencies of longitudinal modes in closed hard-walled ducts with discontinuities in a cross-sectional area. The approach adopted is based on a general expression for the acoustic impedance for a plane wave motion in a duct and conditions of impedance continuity at duct discontinuities. Formulae for mode frequencies in a form of transcendental equations were found for one, two and three discontinuities in a duct cross-section. An accuracy of the method was checked by a comparison of analytic predictions with calculation data obtained by use of numerical implementation based on the forced oscillator method with a finite difference algorithm.

longitudinal duct modes, discontinuity in duct cross-section, contraction ratio

In this paper spectral characteristics and a mode localization in acoustically coupled enclosures were studied numerically. In a theoretical model a low-frequency limit was considered where eigenmodes are weakly damped, thus they were approximated by normal acoustic modes for rigid room walls. The effect of acoustic coupling on the mode frequency, the integrated density of states (IDOS) and the existence surface of modes were investigated for several coupling areas. Numerical results indicate that a decrease in coupling area causes a decrease of IDOS and results in a stronger localization of modes.

coupled rooms, spectral characteristics, localization of modes

In the paper, the simulation program predicting the sound level has been applied to one of buildings neighbouring a road. In the simulation are considered: geometry of road surroundings, the shape of the investigated building façade including, road geometry (number of lanes and their positions) and traffic structure (vehicle flow rates and their average speeds). Adjustable accuracy in an urban system modelling allows finding the model which is adequate to the real system. For the model of the accuracy limited by the simulation error due to vehicle speed estimation, by comparison with the field measurement, the model validation range has been established. Based on the simulation adjustable accuracy, possibilities of the validation range extension have been analysed.

simulation, traffic noise, noise protecting means

Acoustically coupled spaces have recently been drawing more and more attention in the architectural acoustics community, thus a determination of shapes and frequencies of eigenmodes in these room systems from computer-based models has become increasingly significant. In this investigation, an eigenvalue problem was solved numerically for a simple room system consisting of two connected rectangular spaces. In a numerical procedure, the forced oscillator method with a finite difference algorithm was applied. In order to determine the influence of irregularity of system shape on eigenmodes frequency, a modal behaviour in the coupled spaces was studied for several sizes of coupling area. Calculation results have shown that with the exception of a fundamental mode, the changes in resonant frequencies were relatively small. However, an increase in a system irregularity led to a coincidence of frequencies of neighbouring modes and variations in a sequence of modes on a frequency axis, which both contributed to a degeneration of eigenmodes.

coupled spaces, modal analysis, degenerate modes, mode localization

A study into the application of the Hilbert transform in numerical simulations of reverberant sound decay in irregularly shaped enclosures is presented. It is shown that there are some limitations in the use of this integral transform to exponentially decaying harmonic signals because the integration result consists not only of a signal, which differs from the original one by a phase shift of pi/2, but also of a decaying nonoscillating signal occurring due to the fact that a spectrum of exponential function is unbounded. An initial amplitude of this signal is directly proportional to the ratio between a damping coefficient and a mode frequency, thus for lightly damped rooms, where this ratio is much smaller than unity, it can be neglected. Results of numerical simulation carried out for L-shaped enclosure indicate that the Hilbert transform is a useful tool in calculating instantaneous properties of reverberant sound, especially an envelope of the sound pressure level. It is of special importance in the case of irregularly shaped rooms, where a deviation from the exponential sound decay often occurs because of differences between reverberant responses for particular modes.

Hilbert transform, analytic signal, irregularly shaped enclosures, reverberant room response, reverberation time, early and late decay times

In this paper, a modal analysis was used to describe a reverberation phenomenon in a room of complex shape. A theoretical model was limited to low sound frequencies, when eigenmodes are lightly damped, thus they may be approximated by uncoupled normal acoustic modes of a hard-walled room. A utility of this method was demonstrated in a numerical example where the enclosure in a form of two coupled rooms was considered. A reverberation time was evaluated from a time decay of spatial root mean square pressure, the overall measure of room pressure. The results of calculations, performed for three different distributions of absorbing materials on room walls, showed how various location of the material can effect a dependence of the reverberation time on a frequency of sound source.

coupled enclosures, modal analysis, reverberation time

The sound decay process in L-shaped room was the subject of this investigation. The room with walls covered by the low absorbent material constituted the initial configuration for the study. To decrease a reverberation time, alterations of acoustic room properties were carried out by adding a high absorbent material on lateral room walls. Two different material configurations were considered and their effects on a double slope sound decay were analysed. In order to predict the behaviour of decay curve, a numerical implementation based on the wave theory was applied. A space distribution of eigenfunctions was computed using the forced oscillator method with a finite difference algorithm. Numerical results have demonstrated that an introduction of high absorbent material on selected room walls may cause an intense double slope effect. It is characterized by rapid initial and slow late decays, thus the decay curve exhibits so-called a “sagging” appearance. It was found also that a phenomenon of mode localization is an important factor in modifying the double slope sound decay.

coupled rooms, modal analysis, reverberation, double slope effect, localization of eigenmodes

The acoustical properties of an irregularly shaped room consisting of two connected rectangular subrooms were studied. An eigenmode method supported by a numerical implementation has been used to predict acoustic characteristics of the coupled system, such as the distribution of the sound pressure in steady-state and the reverberation time. In the theoretical model a low-frequency limit was considered. In this case the eigenmodes are lightly damped, thus they were approximated by normal acoustic modes of a hard-walled room. The eigenfunctions and eigenfrequencies were computed numerically via application of a forced oscillator method with a finite difference algorithm. The influence of coupling between subrooms on acoustic parameters of the enclosure was demonstrated in numerical simulations where different distributions of absorbing materials on the walls of the subrooms and various positions of the sound source were assumed. Calculation results have shown that for large differences in the absorption coefficient in the subrooms the effect of modal localization contributes to peaks of RMS pressure in steady-state and a large increase in the reverberation time.

coupled rooms, modal analysis, reverberation time, modes localization

Acoustically coupled rooms have recently been drawing more and more attention in the architectural acoustics community. Determination of decay times in these coupled rooms from computer-based models has become increasingly significant. In computational modelling of coupled enclosures the geometrical acoustics methods do not apply at lower frequencies, and models must account for the wave-nature of sound fields, especially modal effects. In this investigation a combination of modal analysis with finite-difference algorithm was used to predict a decay in the sound field energy in an enclosure composed of two connected rectangular subrooms. Random incidence absorption coefficients inside subrooms have been modified in this study to create different configurations of absorbing materials on subrooms walls. When one of these subrooms was much more absorbent than the other one, calculation results have shown that, at any frequencies, there existed both the initial sudden and the late slow decrease in the sound field energy. This effect was characterized by two metrics: early decay time (EDT) and late decay time (LDT) determined from a sound level decay from –50 to –60 dB. The analysis of numerical data confirmed that the most influential factor in realizing the non-exponential sound decay is a phenomenon of eigenmodes localization.

coupled rooms, reverberation, double-sloped sound decay, localization of modes

In the paper, predictions of relative sound level distribution on building facades in city-centres obtained by using the simulation PROP11 program are compared with measurements. The simulation involves the geometry of surrounding buildings, road geometry (number of lanes and their positions) and traffic structure (vehicle flow rates and their average speeds). The agreement between measurement and simulation results is tested for different directivity characteristics of an equivalent point source representing the vehicles.

predicted sound level in built-up area, vehicle noise emission directivity

In the paper a modal analysis was used to describe a reverberation phenomenon in an irregularly shaped room. A theoretical model was limited to low sound frequencies, when eigenmodes are lightly damped, thus they may be approximated by normal acoustic modes of a hard-walled room. A utility of this method was demonstrated in a numerical example where the room in a form of two acoustically coupled rectangular subrooms was considered. A reverberation time was evaluated individually for each subroom from time decay of acoustic pressure amplitude for different distributions of absorbing materials of room walls and various positions of sound source under the condition that a total room absorption remained constant. Calculation results have shown a great influence of modes localization on a reverberant energy decay for a large difference between the absorption coefficients of walls in subrooms, because in this case for frequencies of some localized modes a substantial increase in the reverberation time was observed.

room acoustics, modal analysis, reverberation time, modes localization

In the paper a phenomenon of sound production in a resonator driven by an air jet has been investigated. Measurements of sound spectra at the closed end of the resonator cavity were performed to determine the influence of the jet velocity on the frequency and amplitude of excited acoustic oscillations. It was found that in the sound generation process two ranges of the jet velocity could be distinguished where different variations of the oscillation frequency with the growing jet velocity were observed. After the onset of oscillation the frequency increased fast with growing jet velocity and the frequency increment was directly proportional to a value of jet velocity. At higher flow velocities an increase in the frequency was still observed, but the frequency growth was much smaller. The experiment has shown also that due to an excitation of mechanical vibrations of the resonator elements a rapid change in the oscillation frequency occurred. Finally, the experimental data were compared to calculation results to examine the accuracy of the theoretical model, in which a force driving the resonator was predicted from the vortex sound theory and the resonator was modelled by an equivalent impedance circuit.

sound generation, flow-induced resonance, acoustic resonators

Acoustical properties of two rooms of irregular shapes have been investigated theoretically by a system of eigenmodes, which for low frequencies and weak losses on walls can be well approximated by a system of normal acoustic modes of hard wall room. The eigenfunctions and eigenfrequencies were determined numerically with an application of the forced oscillator method with a finite difference algorithm. It was found that a geometrical irregularity has three main effects. First, the low-frequency modal density is changed. Second, some of eigenmodes are localized, thus for particular frequencies a vibration amplitude will be enhanced locally inside the room. Third, the acoustic energy losses inside the room are increased with the amplitude of vibration on walls.

room acoustics, modal analysis, eigenfunctions, modal density, modes localization

The first of two companion papers theoretical and experimental results are presented for a Helmholtz resonator subject to external excitation by an acoustic plane wave. The response of the resonator was analysed in terms of the relation between the pressure induced in the interior of the resonator and the driving pressure. Equations for the resonance frequency and the quality factor were developed for the cavity that was a rectangular parallelepiped with a centered circular orifice. The flow from the cavity had a constant velocity profile. The resonator was modelled by an equivalent impedance circuit, predicted from the classic theory of sound radiation, with an additional resistive term connected with the viscous action inside the orifice. The theoretical results were compared with experimental data for a frequency chosen so as to tune the system to the lowest resonance mode. The experiment has shown that the acoustic response of resonator was changed considerably when mechanical vibrations of the resonator elements were excited. A phenomenon of flow-induced resonance in the considered system geometry will be presented in the companion paper.

Helmholtz resonator, flow-induced resonanse, acoustic response of resonator

cavity resonator, flow-excited sound, Strouhal number

Experimental and theoretical results are presented for a cavity resonator driven by an air jet. Cavity sound spectra have been investigated to found a dependence of sound frequency and amplitude versus jet speed. Two ranges of the jet speed were distinguished. In the stage I, identified with low speed range, the frequency increased fast with jet speed and a frequency increment was proportional to jet speed as in the edge–tones phenomenon. In a stage II, corresponding to higher jet speeds, an increase in the frequency was still observed, but a frequency growth was much smaller. A correlation of experimental data has shown that the observed oscillation amplitudes are within moderate–range. A theoretical model based on moving compact vortices was presented to describe flow–acoustic interaction in a resonator opening and find an aerodynamic force driving a resonator. A response of resonator was modelled by equivalent impedance circuit with added nonlinear term. From final equations a frequency and an amplitude of oscillation were calculated. A support for the proposed theory is provided by favourable comparison between theoretical predictions and experimental data.

cavity resonator, air jet, sound generation, vortex street

An air flow in a pipe with two closed side-branches can induce high pressure pulsations in a pipe system. This phenomenon has been investigated and the results are reported in this paper. A simple theoretical model based on a wave transmission was used to determine a resonance condition associated with an acoustic coupling between branches. In the model a plane wave approximation and an impedance representation of a branch were applied. The experiment was carried out in a pipe system with a relatively large distance between branches compared to branch lengths. A frequency and a pressure of pulsations were measured in a wide range of length of downstream branch. A support for the theory is provided by a favorable comparison between experimental data and calculated resonant frequencies of the system.

pipe system, closed side-branches, acoustic resonance

The sound attenuation in a Helmholtz resonator subjected to a plane incident wave is examined. The resonator closing a tube is considered. The average acoustic power absorbed by vortical field has been calculated in the low frequency range with the aid of the appropriate conservation equations. Considering this power loss and losses due to viscous damping inside the resonator orifice, the change of absorption properties of the resonator is analyzed. Results of this study indicate that the peak of the absorption coefficient occurs when the loss resistance is equal to the radiation resistance. Plots of the theoretically predicted absorption coefficient and normalized loss resistance as a function of incident pressure amplitude are in good agreement with experimental data obtained for different orifice diameters.

Helmholtz resonator, sound absorption, nonlinear resistance

An experimental study of an effect of the acoustic nonlinearity on absorption properties of Helmholtz resonators is presented in this work. By use of the classical standing wave method the changes in the absorption coefficient and the resonators impedance were investigated at moderate and high amplitudes of incident wave. As a result of nonlinearity a high absorption at resonance frequencies was observed and then a decrease in this absorption with increasing amplitude. Measurements of the total loss resistance of resonators have indicated that a change in the resistance at high amplitudes depends strongly on resonator orifice area, the smaller area - the higher increase in the resistance. The experimental results have also shown a growth in resonators reactance which causes an increase in resonance frequency. Quite a good agreement between experimental data and the theory presented in Part I was found.

Helmholtz resonator, acoustic nonlinearity, absorption coefficient

This is the first of two companion papers concerned with the nonlinear absorption of Helmholtz resonators at a high amplitude incident wave. The phenomenon has been examined theoretically by use of a model of the acoustic field in the neighborhood of the resonator placed at the end of cylindrical tube. The calculation results have shown that the peak of the absorption coeficient occurs when the nonlinear resistance is equal to the radiation resistance of the resonator. The full experimental investigations of this phenomenon and a comparison between theoretical and experimental data will be presented in the companion paper (Part II).

Helmholtz resonator, nonlinear sound absorption, high amplitude incident sound

In flow ducts with closed side branches strong acoustic pulsations are often induced. This was shown by test results performed for systems with a single side branch and co-axial branches with the same lengths. With growing the flow velocity an excitation of successive resonant modes was observed. Their frequencies were increased together with the flow velocity but at peaks of sound pressure there was an excellent agreement between measurements results and theoretical predictions. A conversion of fluctuating flow energy to energy of resonant acoustic field was included in theoretical consideration by means of negative resistance in impedance model of branches. Hence, it was possible to predict a stronger nonlinearity in the case of a duct with co-axial branches. It was found that a characteristic parameter of the analyzed phenomenon is Strouhal number. Its values for all modes are within the same range approximately and, in addition, it determines a change of acoustic inertance at the branch opening.

closed side branches, Strouhal number, resonant modes

An experimental study of flow-excited acoustic oscillations in piping systems with closed branches is presented in this paper. An influence of acoustic interactions between branches on a generated sound was investigated in arrangements with two and three branches. Significant changes of frequency and level of oscillation in terms of branch lengths were observed. In the system with three branches, at the specified lengths of one branch, a cancellation of oscillations was noticed.

piping system, closed side branches, flow-induced oscillations

acoustic emission, plate vibrations, frequency spectrum

This paper presents the model of the discrete sound induction effect due to a flow over deep cavity in the wall of a rectangular duct. Theoretical analysis applies shear layer approximation with a vortex sheet, with deflection satisfying the Kutta-Żukowski condition, and an equivalent impedance system of a deep cavity with cavity impedance change in the presence of the flow included. The applied theoreticl method makes it possible to determine the effect of resonanse modes of the cavity on disturbances of the vortex sheet and also to determine the frequency and relative value of pressure amplitude in the case of a discrete sound.

discrete sound generation, vortex sheet, deep cavity

This paper is concerned with a model of sound generation in a case of the deep cavity exposed to a grazing mean air flow. The problem was analyzed theoretically by means of infinitely thin shear layer assumption and the description of acoustic response of the cavity. The response was determined with the aid of the impedance model of flow excited quarter-wave resonator, including the nonlinear orifice damping. The parameters of unstable motion in the slot were found, basing on continuity conditions across the orifice plane. Calculated values were used in order to determine the acoustic pressure in far-field area. The theoretical predictions were compared to experimental results.

deep cavity, discrete sound generation, grazing flow

This paper presents an analysis of the mechano-acoustic feedback between a sound source and a resonance system. In the theoretical part, the change in the mechanical impedance of the sound source which occurs when the source is affected by the external acoustic field is determined; and subsequently, using the image source method, the distribution of the standing pressure wave between the source and the resonator is given. The combination of the relations thus derived permits the distribution of the standing pressure wave between the source and the resonator to be determined, with consideration given to changes in the radiation impedance of the source and the mechanical parameters of the resonator. The experimental part gives the results of measurements of the dependence of the mean value of the modulus of the pressure amplitude of the standing wave on frequency for different resonance frequencies of the resonator and a comparison of the experimental and theoretically determined results in the case of an interaction between a sound source and a rigid baffle.

resonance system, mechano-acoustic feedback, radiation impedance, absorption coefficient

A technique commonly used for improving room acoustics consists in increasing a total sound damping in a room. This objective can be achieved by using different configurations of a porous material for acoustical treatment of a room. In this work, that problem is analyzed theoretically by exploiting a modal representation of the impulse response (IR) function for steady-state sound field predictions. A formula for the IR function was obtained by solving a wave equation for an enclosure with complex-valued boundary conditions of walls. On the walls where the acoustic treatment is applied, these boundary conditions are related to the characteristic impedance, effective speed of sound and thickness of the porous material used for padding. Two different porous materials were considered in the analyses of the room with acoustic treatment, and to this end, the required effective properties were calculated for a rigid foam with a designed periodic microstructure, as well as for a poroelastic foam with specific visco-elastic properties of the skeleton.

wibroakustyka, analiza sygnałów, dyskretna transformata Hilberta

The modal expansion method has been used to formulate expressions for real and imaginary parts of the complex sound intensity inside enclosures. Based on theoretical results, the computer program has been developed to simulate the active intensity vector field inside L-shaped enclosure. Calculation results have shown that a distribution of the active intensity is strongly influenced by the modal localization and the typical objects in the active intensity field are energy vortices and saddle points positioned irregularly inside the room. It was found that an increase in a sound attenuation results in the change of vortex positions and can cause the formation of new vortices. An influence of the wall impedance on the quantitative relation between the active and reactive intensities was also studied and it was concluded that for very small sound damping the behavior of the sound intensity is basically only oscillatory.

room acoustics, active intensity vector field

In the paper, the computer modelling application based on the modal expansion method is developed to study an influence of a sound source location on a steady-state response of coupled rooms. In the research, an eigenvalue problem is solved numerically for a room system consisting of two rectangular spaces connected to one another. A numerical procedure enables the computation of shape and frequency of eigenmodes, and allows one to predict the potential and kinetic energy densities in a steady-state. In the first stage, a frequency room response for several source positions is investigated, demonstrating large deformations of this response for strong and weak modal excitations. Next, particular attention is given to studying how the changes in a source position influence the room response when a source frequency is tuned to a resonant frequency of a strongly localized mode.

Modelowanie pola dźwiękowego w pomieszczeniach sprzężonych akustycznie stanowiło przedmiot wielu prac badawczych, ponieważ przykładami tego rodzaju obiektów są teatry, gdzie szereg loż połączonych jest z salą główną, oraz kościoły, w których występuje wiele naw oraz kaplic. W pracy, przy zastosowaniu analizy modalnej i implementacji numerycznej przeanalizowano dla małych częstotliwości rozkład ciśnienia akustycznego w stanie ustalonym dla sali złożonej z dwóch połączonych ze sobą pomieszczeń prostopadłościennych. Wyniki symulacji numerycznej uzyskane dla różnego usytuowania materiału pochłaniającego wykazały silną zależność rozkładu ciśnienia od częstotliwości dźwięku. Zaobserwowano to szczególnie dla dużych różnic we własnościach absorpcyjnych obu pomieszczeń, gdyż w tych przypadkach dla szeregu częstotliwości pojawiły się silne maksima ciśnienia związane z efektem lokalizacji modów własnych.

Absorbing properties of multi-layer acoustic absorbers were modeled using the impedance translation theorem and the Garai and Pompoli empirical model, which enables a determination of the characteristic impedance and propagation constant of fibrous sound-absorbing materials. The theoretical model was applied to the computational study of performance of single-layer acoustic absorber backed by a hard wall and the absorber consisting of one layer of absorbing material and an air gap between the rear of the material and a hard back wall. Simulation results have shown that a high thickness of absorbing material may cause wavy changes in the frequency relationship of the normal and random incidence absorption coefficients. It was also found that this effect is particularly noticeable for acoustic absorbers with a large thickness of air gap between the absorbing material and a hard back wall.

W zakresie małych częstotliwości mody akustyczne mają duży wpływ na pole dźwiękowe w przestrzeniach zamkniętych, ponieważ ich wzbudzenie skutkuje nieregularnością rozkładu ciśnienia akustycznego oraz deformacją odpowiedzi częstotliwościowej, zaś w warunkach pogłosowych prowadzi do nieliniowości krzywej zaniku poziomu dźwięku. Do modelowania akustyki przestrzeni zamkniętych zastosowano teorię falową bazującą na analizie modalnej, w której podstawowe znaczenie ma wyprowadzona zależność na odpowiedź impulsową dla przestrzeni zamkniętej o dowolnym kształcie. Wykorzystując opracowany model teoretyczny przeanalizowano takie zagadnienia akustyczne jak: wpływ kształtu przestrzeni zamkniętej na częstotliwość modów własnych, efekt lokalizacji modów oraz zjawisko degeneracji modów. Zbadano też wpływ modyfikacji parametrów dźwiękochłonnych powierzchni brzegowych na nieliniowość zaniku poziomu ciśnienia akustycznego oraz rozkład przestrzenny składowej aktywnej i reaktywnej wektora natężenia dźwięku. Dla pomieszczenia o ścianach pokrytych materiałem dźwiękochłonnym o zespolonej impedancji powierzchniowej przeprowadzono natomiast symulacje odpowiedzi częstotliwościowej i porównano ich wyniki z rezultatami obliczeń uzyskanymi metodą elementów skończonych.

A modal representation of a room impulse response has been used to formulate expressions for low-frequency sound field in rooms of arbitrary shape. Based on theoretical results, a simulation program has been developed to predict a sound pressure distribution and a room transfer function inside rectangular enclosure having walls covered by a material of complex impedance. Damping properties of the material have been described by the random-incident absorption coefficient. Calculation results have shown that a wall reactance strongly influences a sound absorption inside a room and an increase in the absolute value of the reactance leads to a drop of a sound attenuation. Furthermore, it was found that changes in the wall reactance entail a substantial modification of a sound pressure distribution. Finally, an influence of wall reactance on the room transfer function was investigated and it was discovered that a change in a reactance sign causes a shift in frequencies of modal vibrations excited in the room.

In the low-frequency range, a sound reproduction in small rectangular rooms is strongly influenced by acoustic modes of a room. While the spectral impact of acoustic modes on a room response is well understood, there is less information on how modes alter the temporal response of a room. In the work this issue has been examined theoretically using the convolution theorem and a modal description of the room impulse response. To determine the effect of a sound frequency on the temporal room response, the usual tone burst was used as an acoustic excitation of a room. Consequently, the tone burst was modeled by the n-cycle continuous sine wave multiplied by the rectangular window. To evaluate an accuracy of the tone burst reproduction, a new metric was introduced. It compares the energy envelope of a source signal and the envelope of a pressure signal at different receiver positions. The pressure envelope was predicted by applying the discrete Hilbert transform. An accuracy of the tone burst reproduction was compared to the spatial distribution of a sound pressure at steady-state to determine a connection between the transient and steady-state room responses.