Reverberation Formula Research Articles

Revisiting reverberation formulae

In 1992, the author proposed a generalization of Sabine's formula that develops reverberation time over a series of powers of the reflection coefficient on the boundaries. Some years later, he reduced the development to just two terms that made it possible to monitor reverberation times up to large absorptions. The present paper revisits this development and justifies it with the help of free path statistics in 3D rectangular enclosures. It proves that Kuttruff's reverberation formula is a special case of the general formula, but diverges for large absorption. Reverting to Kuttruff's original integration process leads to a formula that does not diverge. The paper further explains the difference between Eyring-type reverberation times that vanish for total absorption, and Sabine-type reverberation times that never vanish even for total absorption, and proposes a simple scheme for evaluating the asymptotic free path statistics and thus improving reverberation time prediction. Lastly, the approach is extended to non-ergodic enclosures.

A tool to calculate the reverberation time of rooms with uneven absorption distribution

The reverberation time in rectangular rooms with uneven distribution of absorbing elements and surfaces, for example suspended ceilings, lead to a non-diffuse sound field. This implies longer reverberation times than predicted by sabines or eyrings reverberation formula. To enable a correct prediction of the reverberation times in such rooms, the only possibility was to apply room acoustic simulations, which require high effort to model the room and to perform the simulation. In 2021, a prediction method was published by Zhou et. al. to calculate the reverberation time in such rooms much more precisely than using sabines or eyrings formula, without the need for a room acoustic simulation. This method has been transformed into an online calculation tool, named "reverberate". In this contribution the methodology will be quickly reviewed and the online tool will be presented.

Development of Room Acoustic Prediction and Evaluation Tools for Designers

We develop room acoustic prediction and evaluation tools for designers to be used in halls and conference areas where the design of acoustical performance is essential. The designers, with no specialized knowledge of acoustics, are able to examine the tools at an early stage of the project. The interface tool consists of the 3D CAD software-Rhinoceros, and its plug-in tool-Grasshopper. The designers first build a room model on Rhinoceros and locate the sound source and the receiver points in the model. They then set the calculation conditions via Grasshopper, such as room use and type of interior finishes. Next, room sound propagation is calculated using the ray tracing method in the external python programs. Then, acoustic indices, such as the average sound absorption coefficient and clarity, are predicted. These indices are automatically evaluated according to the room use, and the evaluated results are visualized in the room model on Rhinoceros. This paper outlines these tools and exhibits the agreement of predicted and measured indices in a small hall. These tools are applied to a lecture room with complicated ceiling ribs, and an office with a stairwell, which are difficult to examine by the conventional reverberation formula.

Interpretation of Eyring's reverberation formula based on ultradiscrete method

Interpretation of Eyring's reverberation formula based on ultradiscrete method

A History of the Use of Time Intervals after the Direct Sound in Concert Hall Design before the Reverberation Formula of Sabine Became Generally Accepted

With the proposal of the D50 and the C80 two room acoustic parameters were introduced in which the time interval after the direct sound was taken into account. It is not widely known that comparable applications have a history which date back into the 17th century. Already B.C. Vitruvius had noticed that echoes occurred in theatres and that these were unwanted to create good hearing conditions. During the 17th and 18th century a theory on echoes arose in books about general physics and sound. This theory influenced the books about architectural acoustics written during the end of the 18th and the 19th century and consequently concert hall design at the end of the 19th century. Examples where the theory on echoes was used in the design are the Palais du Trocadero in Paris, the Neue Gewandhaus in Leipzig and the Salle Pleyel in Paris.

Comments on ’’Sabine’s reverberation time and ergodic auditoriums’’ [J. Acoust. Soc. Amer. 58, 643–655 (1975)

Joyce’s paper is, in fact, a confirmation of the author’s standpoint that, generally, Sabine’s formula does not hold. A reverberation formula from recent literature, referred to by Joyce, is shown to be false. Instead, a more correct formula is given here. Subject Classification: [43]55.20, [43]55.30, [43]55.55.

Some Further Remarks on the Fitzroy Multiaxial Reverberation Formula

New information is added relative to the author's multiaxial reverberation formula, originally presented as an invited paper at the 56th Meeting of the ASA in 1958 and later published [D. Fitzroy, J. Acoust. Soc. Am. 31, 893 (1959)]. It includes experiences with off-parallel boundaries and the allocations of sound absorption contributed by an audience where such absorption is effective along different axes. This paper also includes further results of field experience.

Reverberation Formula Which Seems to Be More Accurate with Nonuniform Distribution of Absorption

Reverberation measurements, made in rooms where the absorption is nonuniformly distributed, usually vary widely from predictions based on the currently employed formulas. Through extensive tests made in a large number of rooms, where distribution of sound absorption varies widely in uniformity, an empirically derived equation is submitted. Photographs of graphic recorder decay curves, showing contrasts in predicted slopes, seem to confirm the validity of the formula. The recorder curves are from field tests. The data submitted are a small portion of an accumulation of measurements, covering several years, in rooms of widely varying volumes, widely varying boundary ratios, and marked dissimilarity in amounts of absorption and ratios of absorption distribution.

Reverberation Formula Which Seems to be More Accurate with Nonuniform Distribution of Absorption, with Field Examples of its Application

Reverberation measurements, made in rooms where the absorption is nonuniformly distributed, usually vary widely from the predictions based on the usually employed formulas. Through extensive tests made in a large number of rooms, where distribution of sound absorption varies widely in uniformity, an empirically derived formula is submitted. Photographs of graphic recorder decay curves, showing contrasts in predicted slopes, seem to confirm the validity of the formula. The recorder curves are records taken in the field. The data submitted are a small portion of an accumulation of measurements, covering a period of some five to six years, in rooms of widely varying volumes, widely varying boundary ratios, and marked dissimilarity in amounts of absorption and ratios of absorption distribution. This paper also includes a discussion of the new semicircular California Masonic Memorial Temple 3300-seat auditorium, recently completed in San Francisco, together with experiences with other rooms of difficult shapes. (This is an expansion of a paper, originally planned for the Washington meeting but not delivered.)

Reverberation Formula Which Seems To Be More Accurate with Nonuniform Distribution of Absorption

Reverberation measurements, made in rooms where the absorption is nonuniformly distributed, usually vary widely from the predictions based on the usually employed formulas. Based on extensive tests made in a large number of rooms, where distribution of sound absorption varies widely in uniformity, an empirically derived formula is submitted, together with photographs of graphic recorder decay curves which seem to confirm the validity of the formula. The data submitted are a small portion of the accumulated measurements, covering a period of some five to six years, in rooms of widely varying volume, widely varying boundary ratios, and marked dissimilarity in amounts of absorption and ratios of absorption distribution.

Example of the Dependence of Sound Absorption on the Area of Sample

The sound absorption in a solid concrete room was measured by the reverberation method, first with the room entirely bare, then with a “border” of 114-in. Acousti-Celotex (46 ft2 in area) around the ceiling, then with a “border” added around the top of the walls (total acoustical tile 112 ft2), and finally with the entire ceiling and one-third of the side walls covered (265 ft2 of acoustical tile). The test sound, generated by mechanical impact, was filtered on reception into third-octave and half-octave bands. The volume of the rectangular room was 1350 ft3 and its area 793 ft2. With the border of 46 ft2 the coefficients determined in the customary fashion by the Sabine reverberation equation were near those published for 114 in. Acousti-Celotex. At 125 and 8000 cps the coefficients changed little with the area of sample. At 1000 cps, however, the measured coefficient dropped from 0.93 to 0.44, to 0.25 as the sample area was increased. This is possible evidence for a change in effective mean free path and for the need of an appropriate adjustment in the reverberation formula. The name Sabine absorption coefficient is suggested for the usual coefficient computed by the reverberation technique, to distinguish it from the energy absorption coefficient that does not exceed unity.

Steady State of Sound in a Room

A relationship based on Sabine's reverberation formula and the assumption that sound from a steady source in a room reaches a uniform steady state intensity is T = KE0V/Is where T is the reverberation time of the room, E0 is the steady state intensity, V is the volume, Is is the intensity within an absorbing chamber coupled to the steady source, and K is a constant. The constancy of K was tested in twelve rooms using warble tones centered at 256 and 512 c.p.s. The mean deviation of K was 33 percent at 256 and 25 percent at 512, but the value at the higher frequency was more than twice as great as that at the lower frequency. The procedure used does not yield useful values of T.

Sound Absorption Coefficients and Acoustic Impedance

The relation between normal acoustic impedance and the various types of absorption coefficient has been discussed and specific formulas for the coefficients in terms of acoustic impedance have been published. Contours of constant absorption coefficient plotted as a function of the real and imaginary parts of acoustic impedance plotted on a linear scale are families of circles for both the normal coefficient, the coefficient which in the Sabine reverberation formula usually determines the initial rate of sound decay for rectangular chambers and low frequencies, and for the familiar free-wave absorption coefficient for any single angle of incidence. Similar contours for the Sabine or statistical absorption coefficient, which determines the rate of decay for completely diffuse sound, are shown to be very nearly circular. These plots, together with measured impedance curves or the equivalent electrical circuits of a sound-absorbing structure, are useful in designing materials and mountings for optimum sound-absorbing properties. The curves give theoretical explanation of the empirical value found by Sabine for the average cosine of the angle of incidence for frequencies above 500 c.p.s. Graphs of the absorption coefficients plotted as functions of magnitude of impedance for various values of impedance angle show that for angles greater than 45° the treatment of a complex impedance as a real impedance of the same magnitude introduces an error of greater than −20 percent in computing absorption coefficients.

What is Measured in Sound Absorption Measurements?

Experimental work in various laboratories has shown that in the application of the reverberation theory to the measurement of sound absorption coefficients, certain of the fundamental assumptions of the theory are not realized and perhaps cannot be realized in practice. This paper is an attempt at a critical analysis of the points at which theory and practice diverge. What is sought as the absorption coefficient at a given frequency of a given material is a numerical value which, when applied in the reverberation formula, will give the effect on the rate of decay produced in any room by any area of that material. In small rooms, due to diffraction, with small test samples the equivalent absorption per unit area increases as the test sample area decreases, whereas with large areas localized on one surface the decibel rate of decay tends not to be linear. Both effects are functions of the absorptivity of the material and possibly of the wavelength of the sound. Various proposed methods looking to the standardization of the results of measurement in different reverberation chambers are considered.

A New Reverberation Formula

The assumption is made that the probabilities are that any ray of energy, undergoing repeated reflection, will strike the individual surfaces of a room in proportion to the ratios of their areas to the total surface. The resulting expression for the reverberation time employs the logarithm of the geometric mean of the reflection coefficients.

The Effect of Form on the Reverberation of Sound in Rooms

Measurements of the mean free path p of sound in three dimensional models of eleven typical auditoriums—such as rectangular rooms (with and without balconies), fan shaped theatres, country and city churches, and rooms with gabled, barreled or domed ceilings—indicate that p varies from about 3.8 V/S (V=volume, S=interior surface) in a large rectangular room with a low ceiling to 4.3 V/S in a cruciform type of church auditorium. The reverberation formula for the impressed vibrations takes the form t=kV−rS loge (1−α)+4mV where k depends upon the shape of the room and has a value of .046 in the large room with the low ceiling, .053 in the cruciform type of church auditorium, and about .049 for the conventional type of room; r is a probability factor which depends upon the location of the absorptive material in a room; and m is the attenuation constant of the air. The more exact theory of reverberation must describe also the rate of decay of the free vibrations in a room. Some measurements of reverberation have been made in a small room which exhibit gross violations of the generally accepted reverberation formula. These measurements reveal the necessity of considering the free vibrations in reverberation theory.

The Effect of Humidity Upon the Absorption of Sound in a Room

P. Sabine and E. Meyer have noted that for frequencies above 2,000 d.v. the rate of absorption of sound in a room increases as the relative humidity decreases. The writer has investigated and confirmed this effect during the past year and has obtained some quantitative data which have a bearing upon problems in architectural acoustics and sound signaling. Assuming that the intensity of a plane wave in air diminishes in accordance with e—mct where c is the velocity t the time and m the absorption coefficient in the air the reverberation formula becomes (in British units) t = .049V−S loge (1 − a) + 4mV. By making reverberation measurements, under different conditions of humidity, in two rooms having the same boundary material (painted concrete) but different mean free paths, it is possible to determine both m and a as functions of the frequency and humidity of the air. For frequencies below 512 d.v., m is negligible. It increases approximately with the square of the frequency. At 4,096 d.v., m increases from about .0015 at 70 per cent relative humdidity (21°C) to .0033 at 20 per cent relative humidity. At frequencies above 6,000 d.v., the absorption in the air in a room may be greater than the absorption by the boundaries. The absorptivity of painted concrete remains nearly constant above 512 d.v., and has a value of about .016.

An Error in the Derivation of the Reverberation Formula

Sabine's formula T = 55.3V/avs yields finite values for the period of reverberation even when the coefficient of absorption is unity. This formula is a first approximation to that developed by Norris T = −24V/vs Log10 (1−a); the latter yields T=0 for a=1. An error in the interpretation of the fundamental differential equation in the derivation of Sabine's formula is demonstrated. Another mode of attack permits a more correct analysis of a room under certain specified distributions of absorbing material.