# Room acoustics

• Sound radiated by a point source is reflected by the rigid wall

• Image sound sources due to reflection:
The reflections act as if they were radiated from another identical point source behind the wall (at the same distance from the wall as the real sound source).

• Sound field is built up from the direct sound and a vector sum of contributions of all the images.

• "images of images"

## Audio spectrum

• Different approaches for different frequency ranges in small room.

• Wavelength large compared to room dimensions:

• no resonance boost

• Wavelength comparable to room dimensions:

• wave acoustics, standing waves, modes

• resonance boost of loudness

• Wavelength too short for wave acoustics and too long to be considered as a ray:

• diffusion and diffraction of sound dominate

• Wavelength substantially smaller than room dimensions:

• sound rays: "The angel of incidence = the angel of the reflection"

## Modes

• Room resonances, natural frequencies, standing waves

• Two parallel reflective walls (of infinite extent) and air between them can be considered as a resonant resonant system.

• Frequency of resonance fr = c/2L, where c is the speed of sound and L is spacing between the walls.

• Resonance also at multiples of fr, at frequencies 2fr, 3fr, ... (harmonic frequencies)

• Modes and their intercation affects room response.

• Wave acoustics in rectangular rooms

• From wave equation (stated by Rayleigh in 1869) after several steps solutions for sound in rectangular enclosures.

• Calculation of the permissible frequencies corresponding to the modes of the rectangular enclosures:

where p,q,r = 0,1,2,3, ...., c is the speed of sound, L the length of the room, W width, and H height. sqrt means square root.

• Axial modes involve reflections from two surfaces of the room.

• Tangential modes from four surfaces.
1/2 of the energy of axial modes.

• Oblique modes from six surfaces.
1/4 of the energy of axial modes.

• Axial modes have the greatest affect to the room response, and at low frequency.
But tangential and oblique modes have some effect on the room's sound pressure pattern.

• Number of normal modes increase with frequency
=> Smoother room response in high frequencies.

• Modes decay at different rates.

• Mode decay depends on how absorbing material is distributed in the room.

• Reverberation in each octave bands depends on average of the decay of the several modes.

• The higher the octave center frequency, the more modes included.

• Each resonant mode is effective over a narrow band of frequencies.

• This bandwidth is determined by the amount of absorption in the room and is about 5-Hz wide for typical audio rooms.

• The bandwidth of the modes increase when the reverberation time shortens.

• A strong signal component may force closely adjacent modes into vibrating at the excitation frequency.

• When the excitating force is removed:
=> Adjacent modes decay at their natural frequencies.

• During decay period, new frequencies are radiated into the room.
=> A brief change of pitch.

## Echo

• Reflected waves of the sound

• same waveform

• lower level and delayed

• any time when the level of echo signal momentarily attains the original sound level, the echo becomes audible

## HAAS effect

• For short delays (0-50 ms), echoes are perceived as part of the direct sound.

• The reverberation acts to increase the loudness of the sound

• The reverberation also changes the color of the sound ("liveliness of the room").

• Echoes with in 0-50 ms are not perceived as discrete, unless their level is increased.

• Human auditory system suppresses early reflections, occurring less than 50 ms after the direct sound.

## Sound field in time domain

• 1: Direct sound reaches the audience first.

• 2: Early reflected sound reaches the audience after the direct sound

• Early reflections are needed to give the feeling that someone is present at a space.

• 3: Reverberation reaches the audience from every direction (as a result of complex reflections).

• Reverberation creates the sense that performance is taking place in a large space.

• Pitch change during reverberant decay
<= shift of energy between normal modes
<= perceptual dependency of pitch on sound intensity

## Reverberation

• Reverberation = the tailing off of sound in an enclosure because of multiple reflections from boundaries.

• Sound pressure builds up as reflected components arrive later

• Sound decays exponentially after the source ceases.

• It takes finite time because of the finite speed of sound, losses at reflections, damping effect of the air, and divergence.

• Reverberation can be "good" or "bad" depending on its degree or on the circumstances.
Examples:

• Symphony orchestra was recorded in an anechoic room and it sounded terrible (thin, weak, without resonance).
=> Music requires reverberation.

• Speech is more intelligble in rooms having lower reverberation times (long reverberation time maskes last consonants in the speech).

## Reverberation time

Here are notes Nick Zacharov made on Reverberation time.

• Time needed for the mean-square sound pressure in a room to decay from steady state value by 60 dB after the sound source is suddenly turned off.

• Reverberation time is different for different frequencies.

• For high frequencies shorter reverberation time, mainly due to better high frequency absorption in the surface.

• There are equations to approximate reverberation time in a room. (Sabine, Eyring, Fizroy, ...)

## The sabine Equation

• Reverberation time (T, in seconds) is directly proportional to the volume of the room (V, [m^3]) and inversely proportional to the room surface area (A, [m^2]) and the average coefficient of sound absorption (s):

• Developed at the turn of the century empirically

## Comb filter effect

• Delayd signal (echo) is summed to the direct signal
=> the same effect as the signal were filtered with filter whose transfer function is where d is delay.

• Amplitude response of this kind of filter is looking like "comb" having deep notches equally spaced.

## Spaciousness

• Sound reflected from surfaces important to perceived spaciousness

• reflected sound signals incoherent with direct sound

• proper level of reflections

• reflections arrive less than 50-100 ms after direct sound (or they will sound like echoes)

• reflections arrive from lateral directions

## Summary of room acoustics

• If music or speech is percived in some place, room acoustics should be considered.

• feeling of the space

• sound direction

• reflections

• In outdoor acoustics (i.e. free field), sound can travel unaltered in all directions

• reflection

• refraction

• diffraction

• absorption

• Indoor acoustics determines how sound acts in eclosed spaces

• reflective surfaces
=> sound energy better contained i.e. loudness is increased

• different approaches for different frequency ranges:
room modes / sound rays

• room modes: axial, tangential, oblique
colorations of sound
modal frequencies make up the room acoustics

• Echo

• reflected wave of sufficient amplitude

• in small rooms, echoes exist physically, but not subjectively

• increase loudness

• Haas effect: our hearing mechanism integrates the sound intensities over short time intervals

• Reverberation

• One part of a sound field

• good in moderate quantities, bad in excess
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