SOUND IN A CLOSED ENVIRONMENT - INTRODUCTION

I'd like to introduce with these notes an extract of Dr. Eng. Tommaso Giunti's degree thesis regarding, among other issues, acoustic properties of listening rooms.

Every introductory music or science textbook represents 'sound' as a wave configuration not unlike those forming on a water surface hit by a stone.

Since this representation is intuitive, it's 'good enough' and when some is speaking we can imagine waves spreading from the mouth and reaching in some way our ears.

This model is quite a reduction of the true nature of sound, a really strange 'animal' when cornered in a closed space.

Fundamental sound properties are, from a physical standpoint, frequency and loudness. Some in the audience is already pointlessy screaming to the computer: 'Timbre, by gosh! What about timbre?'

Timbre is made up of a sum of several sounds characterized by different frequencies (height) and intensities (volume). Moreover sound velocity does not depend on these parameters: it depends only by the means the wave is traveling in. In standard atmosphere conditions sound velocity is about 1220 km/h ( 340 meters/s) that is certainly fast, but surely not instantaneous...

In a realistic room (stereo living room, auditorium, theatre, sweeper cabinet) sound behaves in a different manner depending on its frequency. This behavior has been analyzed by Schroeder who defined a limit frequency, peculiar of each room, that marks the low frequency region behavior and the high frequency one.

Bigger and more absorbing rooms (with a small reverberation time) have a lower limit frequency while smaller and little absorbing ones show a higher one.

 So the sound behavior can be studied distinguishing three spectral regions:

1. At low frequencies, under the Schroeder frequency, the room 'fills up' with steady-state sound waves and acoustic behavior is dominated by standing wave presence like an oscillating string presenting maximum and minimum amplitude points seemingly steady along its length. Imagine a quantity of strings in the room, picked in different ways and with different tensions, representing the complex configuration of the low frequencies in the environment.

2. The second spectral region, above Schroeder frequency, is a transition band covering about 2 octaves. The sound behavior is a hybrid between the first and the third region.

3. At high frequencies the sound behaves like light hitting a mirror: sound waves bounce like billiard balls, attenuating after a series of reflections.  

Modal Theory

In the first region, the low frequency one, sound field is dominated by standing waves and by environmental resonance. A closed room can be defined as a 'resonant cavity'. This cavity presents frequencies where sound intensity reinforces itself markedly.

Acoustic pressure waves mix with reflected waves. If interference is positive, a tone at that frequency will be reinforced in respect to other frequencies.

Resonant frequencies can be calculated using the geometric dimensions of the room and some accessory parameters which assume integer values.

Geometric Acoustics

Beyond Schroeder frequency the resonant model is inadequate to describe the sound field and sound waves must be considered as light rays propagating into space along rectilinear trajectories, specularly reflected when they hit an obstacle. Literature suggests this kind of approach, instead of the wave based one.

Sound ray reflection on a rigid surface is such that the reflected ray and the incident one lie on the same plane and form equal angles with the surface perpendicular.

As the billiard balls...the difference with the first region is apparent.

The sum of sound reflections in a given room quickly creates configurations of an unimaginable complexity. Luckily every reflection attenuates sound energy in a consistent manner, otherwise echoes in a concert room would end accumulating enough energy to break human eardrums.

On the contrary the sound energy is dissipated into the room walls, with obvious advantages for concert comprehensibility (and for listeners' health).

These geometric acoustics principles allow us to study the sound ray behavior even in a closed room (ie. a living room): we can consider each reflection as happening on a mirror, given an even surface, that is to say a surface free from macroscopic irregularities.

Every reflection generates the birth of a 'phantom' source. A reflected sound is perceived as it was produced by the reflecting surface; so if listeners have behind them highly reflective surfaces, some instruments may 'move' in their perception from the sound stage towards the ticket office. In this case, fire the architect.

The overall sound perceived in the listening spot is the sum of the direct sound with each reflection happening in the room.

The description of the listening environment according to the geometric acoustics criteria gets highly complicated in the case of non-rectangular rooms; more so if the room contains objects and furniture, rendering impractical this way without using approximations.

Amplitude response of a room at medium-high frequencies fluctuates in ways difficult to model and renders impractical the making of large 'sweet listening spots'. There will be optimal listening areas and areas in which the listener will lose much of the performance.

This is true for big auditoriums as well as for domestic listening environments.