Maa’s equation for the number of normal modes of sound waves in rectangular rooms

The Reflections series takes a look back on historical articles from The Journal of the Acoustical Society of America that have had a significant impact on the science and practice of acoustics.

where V is the volume; S is the total interior surface; L is the sum of the length, width, and height of the room; f u is the upper limit frequency under which N is determined; and c is the sound speed.The first term represents the number of normal modes when the wavelength is orders of magnitude smaller than the room dimensions.In room-acoustics (and in other areas), this is termed the geometrical-acoustics regime, as discussed by Courant and Hilbert in 1924 (Ref. 2) and also Morse in the 1936 seminal book. 3The second and third terms are Maa's corrections to the geometrical-acoustics theory.They are based on clearly comprehensible physics in acoustic frequency-space and are more succinct than Bolt's derivation, 4 which was published side-by-side in the same JASA issue.Note, though, that Bolt's equation (labeled N B in Fig. 1) agrees quite well with that of Maa.Unsurprisingly, the high frequency theory, given by the first term in Eq. ( 1), fails to predict the number of normal modes 1,4 in this modal frequency range because of the large wavelengths on the order as the room dimensions.The contributions of Bolt (N B ) and Maa (labeled "3") provide an accurate theory to predict the number of normal modes using the room dimensions (V, S, L) and the upper limit frequency f u .

IMPACT OF THE ARTICLE
During the first half of the 20th century, when the reverberation and normal mode theories were developed and applied to room-acoustic investigations, particularly to reverberation chamber measurements of sound absorption, 6 there was great interest in understanding sound wave propagation within enclosures with rigid boundaries.Maa's equation was also generalizable for rooms of arbitrary geometries, as confirmed later by Morse. 7Bolt, 6 among others, pursued this line of research later, and displayed his preference for Maa's equation 1 over his own. 4When Schroeder explored the modal frequency responses in large rooms, he traced back to Bolt's work along this line to derive and justify "Schroeder's frequency," 8 which is a critical frequency below which distinct normal modes are recognizable. 9Room-acoustic normal mode theory has definitely found various applications that are still relevant today.For example, normal mode theory in small room-acoustics has a practical relevance in "critical listening rooms" such as recording studios. 10Indeed, the implications of acoustic modal theory often require that acoustic designers be readily able to estimate the acoustic modal parameters from the experimental measurements of room impulse responses in studio-type rooms.A recent work 11  The Reflections series takes a look back on historical articles from The Journal of the Acoustical Society of America that have had a significant impact on the science and practice of acoustics.

HISTORICAL BACKGROUND
Before joining F. V. Hunt in Harvard, Maa spent one academic semester with V. O. Knudsen at the University of California Los Angeles (UCLA).Maa's derivation of Eq. ( 1) was a result of discussions during his close friendship with Bolt, who derived equation of the modal number within the scope of his thesis research under Prof. Knudsen's supervision. 4To demonstrate independence of academic accomplishment, both Bolt and Maa agreed to present their derivations individually at the ASA 1938 Fall Meeting and, subsequently, published individual papers 1,4 side-by-side in the same JASA issue.They both used the term eigentones for what have since been called normal modes in the room-acoustics literature.During that ASA Meeting, it was suggested that normal modes represent a more specifically accurate term in room-acoustics. 1,4 NING XIANG Graduate Program in Architectural Acoustics, Rensselaer Polytechnic Institute, Troy, New York 12180, USA (Published online 1 December 2021)

ACKNOWLEDGMENT
The author is grateful to Dr. Leo Beranek, who generously presented his collection of professional acoustics books in 2010 to the author's team, and the collection now resides in Rensselaer's Architecture Library.Maa's friendship with R. H. Bolt during his research stay with V. Knudsen at UCLA, followed by his doctoral research with F. V. Hunt in Harvard as featured in this article, is extracted from Maa's memoir, which is documented in a book 7 has successfully applied a model-based Bayesian inference for this task.The number of normal modes of the sound waves of a reverberation chamber measured experimentally in the work of V. O. Knudsen (Ref.5) and predicted using different theories, including Bolt's equation (N B ), and Maa's equation (Â).Reprinted with permission from R. H. Bolt, J. Acoust.Soc.Am. 10, 228-234 (1939).Copyright 1939 Acoustical Society of America (Ref.4).

Article:
Distribution of eigentones in a rectangular chamber at low frequency range Author: Dah-You Maa Publication Date: January 1939 (JASA 10, 334); https://doi.org/10.1121/1.1915981Maa's equation for the number of normal modes of sound waves in rectangular rooms among Beranek's book collection (Acoustics Today, p. 58, Fall 2014).Dr. Leo Beranek received the book in 2005 personally from Maa with his handwritten dedication "To Dr. Leo Beranek for 65 years' friendship-Maa, Dah-you-on March 1st 2005."