Arthur van der Harten, CENG, M.IOA, M.ASA

Acoustic Distinctions || Pachyderm Acoustic

**The Limitations of Absorption Coefficients**

For the majority of the relatively short history of architectural acoustics as practice, the dissipation of sound energy by way of surface interaction has been characterized by a single value per unit of spectrum (usually the octave-band or third-octave band) two-digit precision measurement known as the absorption coefficient. It has been a staple of the practice of architectural acoustics since it was first introduced more than a century ago.

Prediction of the acoustical environment of an enclosed space with computer modeling is a relatively recent development, but is becoming common practice in the industry. These programs require the use of an absorption coefficient as input for their predictions. However, with the common use of computer modeling to simulate acoustics by way of geometrical or numerical methods, there is a growing sense that the single value absorption coefficient has a number of severe limitations:

- Precision: Coefficients range from 0.01 to 1. The problem with this becomes obvious if you take 10 Log(α) of the highest possible absorption coefficient that does not indicate complete absorption:

*10 Log (0.99) = -20 dB*

From measurements performed in an anechoic or hemi-anechoic chamber, it is clear that absorption can far exceed 20 decibel losses, while still reflecting some energy back – typically in the form of approximately -40 decibel scattered reflections from wedge-tips.

- Angular variation: The so-called random-incidence absorption coefficient provides one value that is an average of all angles of incidence. The random-incidence absorption coefficient is measured in a reverberation chamber, and derived from the Sabine equation. In the traditional method utilizing the Sabine equation, these coefficients are comprehensive. However, when used in the context of ray-tracing algorithms, applying the same coefficient for all angles of incidence is inaccurate for any finish assembly with entrained air. For example – an A-mounted porous absorber is locally reacting (has approximately uniform absorption over all angles of incidence) but the same absorber E-mounted has far more absorption at normal incidence than at grazing incidence.

- Finite vs. Infinite finish performance: The absorption coefficient (at least as originally conceptualized using the Sabine Equation) suffers from a presumption of conformity to unity. The standard absorption measurement frequently yields absorption values well in excess of 1. This is due to the wave-based interaction of sound with a finite sample – and has confounded practitioners to no end. Standard guidance ranges from applying the raw coefficient to Sabine predictions (which would lead to an unphysical result in geometrical algorithms), to rounding down to 1 (the so-called ‘practical absorption coefficient’ from the ISO standards). There have been a number of theories regarding the physics of this phenomenon. Methods devised to account for the added absorption of finite sample size in excess of the geometrically apparent absorptive sample leave much to be desired, but the most interesting, in the author’s opinion, is the Tomasson method. In this technique, the Field Impedance (Zf), or the effect of the air around the absorber, and especially near edges is determined.

**Futures of Enhanced Precision**

Until recently, the means to overcome these difficulties were obscure, and inaccessible to acousticians. However, academia has been uncovering a number of methods which begin to provide us a means to drop the outmoded absorption coefficient, in favor or more comprehensive techniques of acoustical material performance prediction. Porous absorbers can be characterized using Delaney-Bazley, Miki, or Biot methods. Perforated layers can be calculated using the method devised by Ingard. Through the use of transfer matrices, the impedance of multi-layer finishes can be devised for nearly any sound absorptive assembly. Admittedly, this is an area of ongoing research and verification, but for many possible absorptive finish systems, the results of these methods have the potential to far exceed the accuracy of random-incidence absorption coefficients.

**Figure 1: Absorption coefficients – 2 inches of 3pcf glass fiber A-mounted (left) and 2 inches of 3 pcf glass fiber E-400 mounted (right) –** **calculated in Pachyderm.**

**Figure 2: Plot of absorption coefficient of a typical acoustical ceiling tile – Cylindrical plot with frequency along linear axis and angle of incidence on radial domain (left), balloon plots of octave band summed absorption coefficients (right) – calculated in Pachyderm**

These alternative methods can be explored using Pachyderm, a publicly available open-source acoustics simulation project which aims to push the boundaries of acoustics simulation. One of the goals of the project is to address many of the shortcomings of simulation in acoustics.

One of the ways in which state-of-the-art acoustical simulation techniques have been extended is with the integration of Transfer Matrix Methods for use in geometrical acoustics simulations. Using transfer matrices, absorption coefficients can be calculated for all angles of incidence at all octave bands. The combination of geometrical acoustics simulation techniques with Transfer Matrix methods for finish absorption prediction can increase the accuracy of acoustics prediction for infinite finish samples with airspaces (such as the typical acoustical ceiling tile system). Using the methods implemented in Pachyderm, acousticians have an opportunity to collectively devise a better, more precise set of standards for the characterization of absorption using methods based on parameters such as flow resistivity, density, Young’s Modulus, Poisson’s Ratio, etc.

**Figure 3: Two views of a visualization of a spherical wave reflecting from a surface with acoustic absorption calculated by angle of incidence – simulated in Pachyderm**

**Future Work**

There are a number of systems in Pachyderm which are still under development. One such system is the integration of transfer matrix results with numerical methods. Impedance functions provide enough data to inform the finite volume method, also implemented in Pachyderm.

Furthermore, the implementation of Tomasson’s Finite Field Impedance method has the potential to lend to the accuracy of these methods – but the exact means to accomplish this has yet to be devised. Tomasson’s method is not without limitations, however. The results suffer for materials with airspaces, and there are many conceivable ways to account for the absorption beyond unity that is predicted by the Tomasson method.

**Figure 4: Balloon plots of absorption coefficients by octave band at various locations along the radius of a circular disc of 4 inch thick glass fiber rigid insulation – calculated in Pachyderm**

We are interested in collaborating on a future of acoustical prediction tools with less uncertainty. If interested, feel free to contact the author for more information about Pachyderm, or to discuss future work and additional approaches to improvement.