Schroeder Frequency Research Articles

The crossover frequency of acoustic simulation for small-scale enclosures by wave and statistic based methods

In the field of room acoustic simulation, wave-based and statistical numerical methods are pivotal for acquiring sound field characteristics. Although the Schroeder Frequency has been utilized to deduce the suitable frequency range of these methods in large enclosures, the crossover frequency warrants further investigation for the small-scale enclosures with complicated inner structures, such as car cabins. This work aims to explore the applicable frequency range of these acoustic simulation methods for a small-scale model, which means identifying the crossover frequency between wave-based Finite Element Method (FEM) and statistical Ray Tracing Method (RTM). First, the crossover frequency was analyzed by the Schroder Frequency equation and modal density estimated from the room transfer functions (RTFs) in a simplified car model, which considering the interior with and without seat occlusion. Further, the crossover frequency was validated by comparing the magnitude spectra of RTFs simulated by FEM and RTM. This work provides a foundational frequency reference for applying statistical acoustic methods in small-scale enclosures.

Reverberation room design optimisation for low-frequency diffuse sound absorption testing

The reproducibility of sound absorption testing with the reverberation room method is a long-standing concern. Absorptive samples induce directionality in the nearfield, while the farfield depends on the room geometry below the Schroeder frequency. Nevertheless, when properly accounting for nearfield effects, the theoretical diffuse absorption coefficient of a sample still represents its average performance across an ensemble of different rooms, even at very low frequencies. Recent research found that particular reverberation room designs allow for an accurate measurement of the diffuse sound absorption coefficient of highly absorptive samples at low frequencies. Pinpointing such designs hence opens up a possibility to sustainably improve the low-frequency reproducibility of sound absorption testing in reverberation rooms. The present paper introduces a numerical optimisation framework that serves this purpose. Specific room shapes are parametrised and the geometrical room parameters are optimised so as to minimise the difference between the measured and the diffuse absorption coefficient under appropriate constraints. The sound absorption testing of a sample in a particular reverberation room is numerically simulated using a method that is both accurate and computationally efficient at low frequencies. The diffuse absorption is computed with a hybrid deterministic-statistical energy analysis approach that accounts for the detailed absorber properties, geometry, and boundary conditions, as well as the nearfield effects. The methodology is applied to both cuboidal and hexahedral room shapes. Certain optimised designs are found not only to provide an excellent match for the absorber that was used during the optimisation, but they also maintain their performance across a range of absorptive samples. Additionally, potential geometrical deviations are found to be well tolerated by these reverberation room designs.

Estimation of sound absorption coefficient at modal frequencies using decay time measurements and eigenvalue model in a small room

Measuring the sound absorption coefficient in a small reverberation room below the Schroeder frequency is a challenging task and the results are subject to a high degree of uncertainty. Recent research shows that the low-frequency acoustic properties of the enclosure and the test sample should be characterized by quantities that describe the modal sound field. The aim of this paper is to present a method for estimating the absorption coefficient of a test sample at modal frequencies in the low frequency range and to demonstrate its applicability in a small room. The hybrid inverse approach is based on eigenvalue measurement data coupled with a finite element model to estimate the sound-absorbing properties of a small room and the test sample. The method was tested experimentally and the results were consistent with those of the long impedance tube method. The applicability of the proposed method was further demonstrated by the in-situ characterization of different acoustic samples in the low frequency range.

Active design of diffuse acoustic fields in enclosures.

This paper presents a numerical framework for designing diffuse fields in rooms of any shape and size, driven at arbitrary frequencies. That is, we aim at overcoming the Schroeder frequency limit for generating diffuse fields in an enclosed space. We formulate the problem as a Tikhonov regularized inverse problem and propose a low-rank approximation of the spatial correlation that results in significant computational gains. Our approximation is applicable to arbitrary sets of target points and allows us to produce an optimal design at a computational cost that grows only linearly with the (potentially large) number of target points. We demonstrate the feasibility of our approach through numerical examples where we approximate diffuse fields at frequencies well below the Schroeder limit.

Performance optimisation of small reverberant room with hanging diffusers

Small reverberant room can be an efficient and economic tool to provide fast sound absorption measurements in diffuse field of homogeneous sound absorber or more complex structure and useful for development, testing or evaluating specification requirements in many industrial fields. A 5.7 m3 reverberant room with a Schroeder frequency around 1275 Hz can have some diffusivity issues in lower and middle frequencies leading to an unsuitable level of consistency, reliability, and repeatability. This paper presents the results of a study to improve the measurement performance of this small reverberant room by adding hanging diffusers. Using Ray-Tracing method, a numerical parametric study was done to estimate sound absorption of various material by varying the number, the position, and the orientation of these diffusers based on ASTM C423 and E795 standard. Moreover, extra simulation has been performed to evaluate the effect of sample size on sound absorption consistency in function of frequency. Following these prescriptions, an experimental study was done to confirm these improvements on frequency dependent sound absorption and single rating numbers such as NRC (Noise Reduction Coefficient) and SAA (Sound Absorption Average).

Applicability and reliability of an experimental method measuring underwater acoustic radiation efficiency of floating box-type plate structures in a reverberant water tank

This paper aims to investigate the applicability and reliability of an experimental method measuring the underwater acoustic radiation efficiencies of floating plate structures in a reverberant water tank. For the purpose, an experimental method for estimating the acoustic power of a source in a reverberant sound field has been introduced and applied to measure the acoustic radiation efficiencies of two floating box-type plate structures excited by a logarithmic sine sweeping force in a water tank. The validity of the adopted experimental method has been examined by the comparison with numerical results obtained by a fully coupled vibroacoustic analysis with an infinite radiation boundary condition for the structures. From the results, it is confirmed that the experimental method for measuring the underwater acoustic radiation efficiency of floating plate structures in a reverberant water tank is reliably applicable for high-frequency ranges above the Schroeder frequency of the water tank.

FEM for the Acoustic Modeling of Eigenmodes: Case of the Cultural Heritage Monument of Neoria, Crete

Eigenfrequencies inside a space significantly affect its acoustic characteristics, especially below the Schroeder frequency in the low-frequency range. In Architectural Acoustics, accurate detection and visualization of eigenmodes can be particularly useful in practical applications. One of the most important landmarks in Chania, Greece, is Neoria, a cluster of 16th-century Venetian shipyards. One existing Neoria will be converted and used as a multipurpose hall. For this objective, acoustic modeling and various measurements were performed in the space. One of the purposes of the measures and modeling was the investigation of the eigenfrequencies and the eigenmodes of the area. Finite Element Method (FEM) was used for the acoustic modeling, while the acoustic measurements were performed in various positions according to ISO 3382-1. Impulse responses were measured, and frequency responses of the space were extracted using Fourier analysis. The measurements and the acoustic modeling results show that the frequencies with the most significant effect on the area are 86.1 Hz, 150.7 Hz, and 204.6 Hz. Eigenmodes of the frequencies are visualized with the application of FEM and especially the positions of nodes and antinodes, which can be utilized appropriately for the optimum placement of absorbers and diffusers in the space.

Retooling results of a massive database of home theater measurements

In 2010 the first results of a large-scale measurement program on home theaters was reported in an AES presentation. More than 1,000 rooms were measured for impulse response from each of a minimum of six loudspeakers to at least four (usually more) listening locations. However, computer processing capacity limited the initial report to 275 rooms. Room acoustics and sound system information was reported: RT and its std. dev. with frequency, room volume statistics for those rooms, and transient and steady-state frequency response for some example rooms were shown. From these few rooms an important item of interest emerged: the Schroeder frequency did not show as a boundary of response deviations between lower and higher frequencies—the average deviation from the average response was more constant with frequency than expected. This work will be briefly reviewd. Now in 2022 a new look at this data plus that which was gathered in the intervening years may be studied with much more advanced math tools such as machine learning so much more can be understood. This presentation is interim describing work to be done and soliciting comments from practitioners.

Evaluating the reverberation chamber as a small room: How room dimensions affect the generalization of testing results

Testing acoustic properties is performed in specially designed reverberation chambers according to relevant ASTM, ISO, and other international standards. Performing this testing is complicated by the fact that practical considerations limit the size of the reverberation rooms. Where the dimensions of the room are similar in size to the wavelength of the frequency of interest, modal behavior becomes dominant, and the statistical analysis of room acoustics based on the diffuse sound field theory is not sufficient to characterize the sound field. However, standard methods implement formulas which assume a diffuse sound field to compare results from different labs. Using the Maxon Acoustics Lab, a purpose-built floor ceiling test lab comprised of two stacked, 300 m3 reverberation chambers, we will examine the various physical criteria for evaluating a theoretically diffuse sound field. Discussion will include historical debates concerning the validity of comparing measurements in different labs and an analysis of the Maxxon Lab through the lens of various methods for evaluating the acoustic environment in comparison to the statistical analysis for room acoustics. Methods will include analysis of modal density and distribution, sampling of the sound field per ASTM standards E90, E492, and C423, the Schroeder frequency, and other statistical analyses.

Acoustic modal analysis of room responses from the perspective of state-space balanced realization with application to field interpolation.

Despite its importance in structural dynamics and vibration, modal analysis is rarely performed in acoustics due to the high modal density of sound fields. A novel acoustic modal analysis (AMA) approach is proposed in this paper for enclosed acoustic fields, such as a room, from the perspective of state-space formulation of control systems. A single-input-multiple-output (SIMO) state-space model is established in light of the balanced realization (BR), given impulse response measurements. The BR model is then converted to a modal form such that the modal parameters, including natural frequencies, damping ratios, and mode shapes, can be estimated. To reconstruct mode shapes, plane wave decomposition (PWD) and compressive sensing (CS) techniques are exploited to solve the underdetermined problem for a spatially sparse representation of mode shapes under the Schroeder frequency. As a result, a model of a continuous system can be "interpolated" for any arbitrary source-receiver positions on the basis of the estimated mode shapes. With the identified modal parameters, the low-frequency and early reflection part of room impulse responses (RIRs) can be synthesized for arbitrary source-sensor pairs. The proposed AMA acoustic field interpolation is validated by extensive simulations and experiments.

Time–frequency methods for characterization of room impulse responses and decay time measurement

In the low frequency range, the reverberation of a room should be characterized by the decay time, which can be determined experimentally by various methods. In order to make accurate and precise measurements, the differences between these methods and their advantages should be known and quantified. In this work, the performance of four linear and four adaptive time–frequency impulse response analysis methods was evaluated with the aim to find the most accurate method for characterizing the decays of the room modes. The methods were compared using generated Prony test signals with known modal properties and measured room impulse responses were also examined. The decay time using the selected time–frequency impulse response method was then estimated in three rooms and the measurement results were compared with interrupted sine tone methods. It has been shown that the choice of measurement setup and analysis parameters has an important influence on the estimation of the decay time and its modal frequency. The time–frequency impulse response method using the window width optimized Stockwell transform, the continuous wavelet transform, or the Morlet wave method were found to be the most robust methods, and their applicability is also enhanced by the fact that only a single measurement is required for the selected microphone position. The presented test methods could be used to evaluate the performance of other approaches for the evaluation of reverberation below the Schroeder frequency or the ability of measurement and signal processing techniques to characterize low frequency transient signals.

Experimental evaluation of a sound diffuse field in a ordinary room

This paper presents a methodology based on analysis of spatial correlation functions for assessing diffusivity of a sound field in reverberant and ordinary reflective rooms. A set of sound field measurements using a microphone array is performed in a room excited by a broadband signal and for many positions of loudspeakers. Each recorded signal is then broken down with a filter bank and the spatial correlation of each obtained band-limited signal is computed. Theoretical predictions of the spatial correlation function and the measured ones show very good agreement when spatial averaging and frequency conditions are met for producing a diffuse field. Evolution of the spatial correlation functions versus frequency and number of spatial averaging, i.e., number of sources positions, are examined in a reflective room. Results show that the proposed technique allows to determine the minimum number of required source positions and the actual lower frequency bound for considering that the field is diffuse. The latter is analogous to the Schroeder frequency for reverberant room. Interest of the presented methodology is its ability to experimentally assess in any reflective ordinary room the actual frequency range over which the field can be considered as diffuse. [Work funded by the AMIDEX Foundation.]

Loudspeaker configuration in reverberation rooms for sound absorption measurement using room mode determination

For Sabine absorption coefficient measurements in reverberation rooms, it is preferable to excite as many room modes as possible. But the number of excited room modes is significantly affected by many factors, among which one dominating factor is the loudspeaker location particularly at frequencies below the Schroeder frequency. In turn, this could significantly influence the statistical absorption coefficient obtained according to ISO 354. Therefore, this study aims to investigate the impact of loudspeaker configurations on the excitation of room modes. Frequency response functions are measured using a broad band steady state noise signal using four sound sources at various locations, and the numbers of excited room modes in the one-third octave bands are quantified and compared against those by Green's function simulations. A procedure for determining favorable loudspeaker positions based on the excited room modes is proposed, which can be a useful input to the working group of the ISO 354 standard.

Assessment of statistical features of the diffused acoustic field in a test section of a cavitation tunnel

Noise measurements in a cavitation tunnel is the preferred approach for predicting the underwater noise radiated by a propeller at sea. The accuracy of the prediction is linked to the similitude laws used to transform the model scale measurements to the full-scale ones. Besides, this accuracy is obviously linked to the accuracy of the model scale measurements. The knowledge of the acoustical features of the facility used to perform such measurements is then one of the key points of such measurements. The proposed communication presents the recent results obtained in the test section of the Large Cavitation Tunnel (GTH at Val-de-Reuil) to describe the statistics of the diffuse acoustic field. The results show that the diffuse field complies spatially very well with the Gumbel distribution for narrow band levels (expressed in dB for the power spectral density). The limit of the statistical domain (defined as the “Schroeder frequency”) is also estimated thanks to the experimental measurements performed inside the test section.

About acoustic field characteristics in the test section of a cavitation tunnel

Noise measurements in a cavitation tunnel are the common procedure widely used by most of the institutes for predicting the underwater radiated noise (URN) of a boat propeller. The accuracy of such a measurement is strongly linked to the “acoustic response” of the facility. The knowledge of this acoustic behaviour is then one of the key point to perform an accurate measurement. Due to the geometrical shape of a test-section, the acoustic propagation is similar to propagation in duct. This propagation could be described at low frequencies by modal propagation. For higher frequencies, the total number of propagating modes increases and the acoustics inside the duct becomes statistical. Schroeder described the limit between these two domains (modal/statistical) in 1954 for closed space applications like room acoustics. The formula derived in 1954 was completed and improved few years after to fit with results obtained by experiments and Monte Carlo simulations. Following the same effort, the principle of this limiting frequency (called here and henceforth Schroeder frequency) is derived to comply with particularities of the acoustic propagation in duct and to the environment of a test-section of a cavitation tunnel. Because it exists cavitation tunnel with rectangular or circular test-sections, the adaptation of the Schroeder equation is performed for both configurations. The description of the statistical features of the acoustic field is also derived. Finally, experiments have been carried out in our cavitation tunnel to assess the accuracy of the equation and more particularly the threshold used to delimit the two domains.

Common mathematical framework for stochastic reverberation models.

In the field of room acoustics, it is well known that reverberation can be characterized statistically in a particular region of the time-frequency domain (after the transition time and above Schroeder's frequency). Since the 1950s, various formulas have been established, focusing on particular aspects of reverberation: exponential decay over time, correlations between frequencies, correlations between sensors at each frequency, and time-frequency distribution. In this paper, the author introduces a stochastic reverberation model, which permits us to retrieve all these well-known results within a common mathematical framework. To the best of the author's knowledge, this is the first time that such a unification work is presented. The benefits are multiple: several formulas generalizing the classical results are established that jointly characterize the spatial, temporal, and spectral properties of late reverberation.

Assessment of the airborne sound insulation from mobility vibration measurements; a hybrid experimental numerical approach

A new measurement procedure is proposed to assess the airborne sound insulation of a partition under diffuse sound field excitation using mobility measurements combined with a numerical procedure. The advantage of this hybrid approach is that the diffuse acoustic field does not need to be physically created, thus avoiding problems related to the generation of such fields at low frequencies. Furthermore, the acoustic properties of both the source and receiving rooms will not affect the determination of the sound reduction index R. The proposed method is especially suited for frequencies below the so called Schroeder frequency of the room, and is complementary to the standardized measurement approaches as described in ISO 10140-2:2010. The measurement part of the proposed procedure involves the measurement of the mobility by forcing the partition along a grid of excitation points (e.g. by means of a shaker) and measuring its response along a grid of response points (e.g. by means of a scanning laser Doppler vibrometer). Using the resulting matrix of mobility transfer functions, the vibrational response of the partition excited by a diffuse acoustic field is numerically calculated, from which the radiated sound power is computed using the Rayleigh integral. Thus the reliance on source and receiving rooms used in standard sound insulation testing is removed entirely. The proposed method provides an estimate that only depends on the properties of the partition. The method was tested in an acoustic laboratory on a single layer glass plate of 1.35 × 1.54 m2, as well as on a funicular shell structure with dimensions of 3 × 3 m2. Comparisons with analytical models and standardized ISO10140-2:2010 measurements confirm the validity of the results.

New research on low-frequency membrane absorbers

Low-frequency (LF) room modes are one of the greatest issues for accurate sound recording and reproduction. Effective LF absorbers can mitigate modes in professional and consumer audio rooms. However, fiber- and foam-based absorbers act on sound velocity; membrane absorbers act on sound pressure (greatest at hard surfaces and corners). Velocity at hard surfaces is zero; thus, fiber and foam absorbers work far less effectively than membrane absorbers under 200Hz. Additionally, most independent testing laboratories are only large enough to accurately measure absorption results above 160Hz (per Schroeder frequency) but not below. Only one lab is large enough to be accurate down to 40Hz. A new LF membrane-based absorber product was designed to compliment the frequency range of an existing product. Both were separately tested for LF absorption down to 40Hz at the above-referenced lab. Ten LF absorber tests revealed that the type of absorber, and its location and orientation in a room, are critical to LF absorber effectiveness. Some unexpected results, however, showed clearly that without standardized laboratory absorption testing in a lab capable of accurately testing down to 40Hz, it is not possible to state conclusively that low-frequency absorber products perform as claimed.

Acoustic characterization of Penn State’s recording studio above and below the Schroeder frequency

In most performance spaces, rooms are large enough that most of the audible frequency range falls above the Schroeder frequency, whereas smaller spaces have a significant audible range below the Schroeder frequency. The goal of this study was to characterize the acoustic performance of the control room and live room of Penn State’s recording studio. Both rooms have important roles to play in the operation of a studio which depend on their acoustic character. COMSOL models for each room were created to understand the modal behavior below the Schroeder frequency and measured modes were visualized from frequency response measurement points in one plane at a one foot spacing. For frequencies above the Schroeder frequency, an ODEON model was created and validated against measured impulse responses, where metrics such as T60 and EDT have been considered. By comparing the computer simulations to the measurements, one is able to make recommendations for improvements and provide audio engineers with insight on how...

Experimental methods for determining the cross-over frequency above which phaseless geometrical methods for room acoustics computer simulation is suitable

The most used computer codes for simulating the sound propagation in closed spaces are based on geometrical acoustics. Such an approach is known to be suitable when sound waves interference does not play a major role. For estimating a cross-over frequency above which this condition is met, acousticians use a simple formula for the so called “Schroeder frequency,” which in turn is related to modal overlap. In this work, two methods for observing experimentally the cross-over frequency above which wave interference may be disconsidered are presented. One of them is related to the statistical independence of room impulse responses (in space), while the other one is based on the analysis of impulse responses' phase information within a frequency band. Preliminary results for a shoebox room, based on analytically generated impulse responses, and measurements performed in an auditorium are shown, and indicate similar frequency values, also comparable to the estimated Schroeder frequencies.