James B. Lee

Reverberation theory obscures real physics in concert halls

Wallace Sabine posited sound in a room to be a uniform field, in equilibrium, varying but slowly with respect to time required to traverse the space: the “reverberant” field. It easy to demonstrate that such is not so, especially in rooms like occupied concert halls. But then, Sabine designed the Boston Symphony Hall, a paradigm of acoustic excellence. If that hall be regarded as a physics experiment, never has it been replicated by Sabine’s followers, even with extensive emendations to his concept of reverberation. The real physics of concert halls involves non-equilibrium manifestation of physical acoustics with respect to bounding surfaces, particularly proximity effects on stage and resonant scattering about the audience.

Wigner cross-products encode relative phases

Two distinct kinds of elements comprise the Wigner time-frequency distribution. First are elements occurring at the actual times and frequencies present in the signal: call these “actual” elements. Second are elements occurring at the arithmetic mean of the actual elements: call these “ephemeral” elements. The actual elements essentially are a power spectrum resolved in time and frequency; always they are positive. The ephemeral elements, sometimes called “cross-products,” can be positive, negative, both, or zero, depending on the relative phases of the actual elements; they encode phase information necessary to complete time-frequency representation of all information in the original signal.

Nonstationary signals, holograms, and concert halls

Sabine’s analysis of sound in rooms involves a “mean-free-path,” 4V/S, scaling linearly; for concert halls this measure is about 30 m, implying that perception in the time domain will be an issue. It is not possible to average signals over the space within a concert hall, for it is not possible to average speech and music over time. Speech and music are nonstationary signals, therefore concert halls cannot be ergodic systems. The necessary and sufficient condition for proper acoustics in large spaces is that every listener receive the same full information at some time. It is fulfilled if the space is a time-varying diffuse volumetric hologram with direct sound from the orchestral region acting as reference beam. If the space contains specularly reflected components those will be heard as echoes, representing information about size and shape of the hall; this “architectural information” develops at the expense of “musical information.” Therefore, diffuse reflection is essential in the body of a concert hall.

Sound has size: Stages of concert halls

Sabine characterized whole concert halls by ratio of volume to absorption, which devolves into a “reverberation time” of logarithmic decay of intensity of sound. These times, on the order of seconds, are generic but indeterminate functions of architecture. The critical time for human perception, the aural integration period, where perception of time (intervals) passes over to inverse time (frequencies), is two orders of magnitude shorter, and can be used to scale and define the “orchestral region” of a hall. In 25 ms sound travels 8.5 m, and 8.5 m is the wavelength of 40 Hz, near low E, the lowest orchestral note. If, by analogy with optics, 8.5 m is thought of as a “coherence length,” it affects the time domain by promoting ensemble and the frequency domain by enhancing bass tones through proximity effects. A primary property of human hearing so scales the stages of concert halls.

Mie scattering obviates echoes in concert halls

Mie-resonant-scattering from objects (of sizes similar to impinging waves) embedded in a plane surface has two aspects: it distributes sound into a pattern approximating a cosine law of diffuse reflection; it deconstructs phase relations among components comprising acoustic signals. The former is much better known than the latter. But in large rooms like concert halls, phase-preserving specular reflections constitute echoes, compromising musical information. Deconstructing phases of reflected musical signals by diffuse reflection confines temporal information to direct sound, presenting only the frequency spectrum in reflected sounds.

Back to vitruvius

There is no generally accepted theory of acoustics of concert halls based on the physics of sound as waves. Lord Rayleigh, founder of the science, remarked but little about sound in rooms. Wallace Clement Sabine made an analogy of sound in rooms to the kinetic theory of gases; although this is theoretically untenable, Sabine designed the superb concert hall which is the subject of this session. Leo Leroy Beranek published the first systematic examination of concert halls, demonstrating that subjective evaluations of acoustic quality are consistent: everyone agrees which are good and which are poor; nonetheless Beranek encountered severe difficulty in applying his experimental results. What aspects of the physics of Rayleigh explicate the data of Beranek and the design of Sabine?

Diagnosing scattering with the Wigner distribution

The Wigner distribution resolves the power of a signal in both the time domain and the frequency domain. It also is a complete representation because it displays the relative phases of a signal’s components as cross‐products between them. This means that if one records the time‐series of an impulse response in a room and computes a Wigner distribution from it, the kind of scattering, predominantly resonant or predominantly specular, is readily apparent, because the phase response of each is radically different. Specular scattering preserves the relative phases among components comprising the initial impulse and so preserves the impulse itself; resonant scattering randomizes the phases among components and so converts an impulse into a wide band of frequencies spread smoothly in time. Thus specular scattering tends to promote echoes and resonant scattering tends to promote diffusion. Graphic examples of each are given in the literature.

Here and there, near and far: How proximity and separation affect scattering in concert halls

Classical theory of scattering deals with plane waves, waves which are so far from their source that they form straight fronts of sound, with pressure amplitude and particle velocity exactly in phase. If such waves are much larger than objects they encounter inverse fourth power (Rayleigh) scattering obtains; if the waves are about the same size as the objects resonant (Mie) scattering occurs; if waves are much shorter than the objects specular (Ufimtsev) scattering is the rule. These all affect sound in the far field. But if sources are closer than a wavelength to objects their waves encounter the plane approximation is not valid; pressure amplitude and particle velocity are not in phase, so resonance‐like phenomena occur. These occur on stages of concert halls: bass instruments producing waves 2 m or longer always are close to the floor; some, like tympani and viols, can be close to vertical surfaces too. This sort of scat‐tering enhances fundamentals of notes with respect to the overtones, strongly aff...

Lord Rayleigh revisited: Can vortices in tubes be sources of noise?

Energy can flow through tubes filled with air governed by the laws of fluid dynamics, even in the violent form of shock waves. Energy also can flow through such systems as oscillations of small amplitude, which are governed by the equations of linear acoustics. What happens in the regime between? Calculations performed upon complaint of excessive low‐frequency noise during alternate blasting for two long parallel railway tunnels indicated that particle velocities were a substantial fraction of the speed of sound at the tunnels’ mouths—between the regimes of shock waves and linear resonances. A Strouhal calculation suggested that vortices were the likely source: A sequence of counter‐rotating vortices was being propelled out of the tunnels’ mouths, generating a powerful monopole source at 12 Hz. Modern noise consultants were baffled, but Lord Rayleigh had solved the system a century before, as a compound nonlinear eigenvalue problem, necessarily taking viscosity into account. The blasting engineers drove s...

Is diffusion in concert halls a property of the volume or of the surface

For nearly a century, the principal concept of the theory of concert halls has been that of a nearly uniform field of energy, generated by many reflections, that varies but slowly with time. This supposed uniformity and isotropy gave rise to the term “good diffusion,” a property of the volume; it is difficult to quantify. An alternate approach is to apply the photometric concept of diffuse reflection at the surfaces: This is readily quantified, permits calculation of intensity at the audience, and posits nothing about the properties of the volume. Microcomputer programs to implement the concept are at hand and have been applied to a study of Boston Symphony Hall, renowned for its highly diffuse ceiling and walls. Modeling techniques, also borrowed from photometry, are available too: 1/50th scale architectural models are eminently suitable; note that models employing mirrors—specular reflectors—are not accurate representations of good concert halls.