Introduction (not presented
but provided for clarity)

The Sabine type formulas of decay rates are
derived for diffuse sound fields. This restricts their use typically
to 300 Hz and above. Standing wave modes dominate the lower frequency
range form of acoustic energy storage. Dissipation of this energy
from the room occurs in two forms: transmission out of the room
and absorption within the room.

Rooms used for acoustic work frequently have
heavier than usual walls to increase isolation from exterior noise.
This results in less opportunity for transmission type of energy
loss from the room which increases its dependence on internal acoustic
absorption to provide sufficient decay rates.

Absorption of acoustic energy is by means
of friction effects applied to kinetic energy components of the
sound waves. This friction is usually “wall friction,”
where the reflecting wave is locally transformed by the stiff and
heavy wall impedance. The surface normal component of the waves’
kinetic energy density converts to extra pressure and the tangential
component is exposed to opportunities for surface frictional dissipation.

There are three types of low frequency wave
containment in a room: Longitudinal, tangential and oblique. The
decay rates of these are not the same. The longitudinal modes are
one dimensional, axial standing waves and present the lowest amount
of kinetic energy density to the wall surfaces, hence they have
the longest decay rates. The tangential modes impact two pairs of
wall surfaces and the oblique impacts all three pairs of walls.
The tangential and oblique modes produce about twice the decay rate
as the longitudinal mode because their grazing impact on wall surfaces
provides for more wall friction. Sabine type equations also account
for this type of activity.

Bass traps are discrete devices as contrasted
with a wall surface. Their performance depends on their placement
relative to the energy distribution of the various modes of vibration.
At a particular location, the trap may provide significant absorption
at one frequency, and minimal absorption at another. Traps located
in the tri corners of a room contact pressure fluctuations associated
with each room resonance.

Corner loaded bass traps pull energy out
of the standing wave with each pressure change that occurs. Low
frequency presents pressure changes at a slower rate than would
be by a higher frequency. Calculations of decay rates that are based
on this understanding are derived by distributing the energy in
the room into the number of pressure zones that exist for the particular
mode, then dissipating a fraction of that energy each half cycle,
depending on the number of traps located in these pressure zones.

This new method of calculation predicts the
number and frequency response of the bass traps required to attain
specified decay rate frequency response of a room. Calculation and
measurements in test chambers are found to agree. For example, a
2000 ft3 chamber with each of its 8 tri corners loaded with an efficient
bass trap produces an RT-60 of 0.3 seconds at 113 Hz.

The formula developed to handle this viewpoint
decay rates includes a term which counts the number of fluctuating
pressure zones in a room. Its appearance is very similar to the
equation that predicts modal density. Another curious effect noticed
with very efficient bass traps is the saturation effect of absorption.
Decay rates are proportional to the amount of absorption in a corner,
but they become less sensitive with higher absorption and reach
a limit, indicating that a finite rate of energy can be withdrawn
from a resonant field, i.e., no more than all the energy contained
in the half wave length held by the corner can be extracted per
half cycle, in spite of the “amount” of absorption available.

Room
Acoustics and Low Frequency Damping

The
quality, “Q,” of a resonant system identifies its response
characteristic. High-Q systems are sharply resonant. They are easy
to drive and have a strong response at the resonant frequency (Fo).
Low-Q systems respond less strongly and over an extended frequency
range. A flat response system has zero Q.

The
frequency response curve of a speaker may be flat from 20-20,000
Hz in the test chamber, a room without reflections. Place the speaker
in a real room with a microphone at

the
listening position. Measure again the response. A series of peaks
and valleys are recorded. Move the speaker or mic and a different
curve is developed. A room has many resonant frequencies. Which
of them are stimulated is dependent on speaker placement. Each peak
and null in the spectrum identifies a resonant condition.

Any physical resonance will have a pressure
distribution in space. The microphone at a pressure peak will register
a strong signal. Move the mic ¼ wavelength to a node and
no signal is received. In either case resonance is evident.

Definitions
of “Q”

The “Q” of a system can be measured
from its frequency response curve. The ratio of the resonance center
frequency to the bandwidth that accompanies the ½ power or
3 dB down point comprises one definition of the “Q”
of a system.

Usually
room response curves are presented dB vs. log frequency format. Resonances
occur at different center frequencies. If the “Q” is the
same, the response curve shape is the same no matter which center
frequency is chosen. The “Q” of an average room lies between
10 and 40. The “QP of a free piano string is 1000.

Resonant
systems with slight resistance have High-Q responses. Add energy dissipations
(resistance) to lower the "Q". Another definition of "Q"
is 2pi times the ratio of the energy of the system to the energy lost
per cycle.

Decay
Relations

Ordinary resonances decay out following an
exponential curve in time. The time constant (T) of the decay is
the time required for the system to drop to 1/e of the original
energy level.

The
exponential decay equation can be used to develop the definition
of “Q” for the system. If the exponent is a small fraction,
less than 1/10, then a simple approximation arises. “Q”
equals 2p times the resonant frequency times the decay constant.

The
traditional presentation of decay measurements is the RT60; the
time required for the energy to drop 60 dB. The exponential curve
appears as a straight line in its dB vs. time plot.

By
combining the dB level version of energy with the exponential version,
the RT60 is resolved to be 13.8 times the decay constant.

“Q”
and Decay Constants

The resonance response Q can be expressed
in the traditional measure of decay, RT60. It is developed by combining
the lightly damped Q relations with the RT60 decay constant relationship.

The
result of the previous analysis is the linear relationship between
the resonant frequency of a listening room and its “Q”
for a fixed RT60. For example, a room may well have an RT60 of 1 second
at a resonant frequency of 90 Hz. This means that the room has a “Q”
of 50 for that resonance. A current spec for listening rooms is an
RT60 of .5 seconds. If this applies to room resonance modes, their
“Q” varies from 5 to 100 in the 20 to 400 Hz range.

Resonant
Bandwidth Relations

The “Q” of the resonant mode
is linear with frequency for a constant RT60. By referring to the
half power bandwidth relationship, the bandwidth is definable in
terms of RT60. For a constant RT60 the bandwidth is constant.

The frequency response of a listening room
can be taken with a linear frequency sweep. This will show the fixed
bandwidth resonances to have the same shape regardless of center
frequency.

If
it is determined that the ”Q” of some mode needs to
be reduced, the proper resistance needs to be added. The energy
relations for “Q” yield the required (dQ) addition based
on initial Qi and final Qf values.

Example

The bandwidth of the 100 Hz room resonance
mode may be found to be 3 Hz giving an initial Qi of 33. The desirable
bandwidth might be 5 Hz for a “Q” of 20. The correction
required has a strength of 50. It is developed by adding the proper
amount of absorption to the resonant mode.

The
initial RT60 of the room is .73 seconds. The additional absorption
added is sufficient to establish alone in the room an RT60 of 1.1
seconds. The result of the total absorption produces an RT60 of .44
seconds.

In
order to provide the correction (dQ), a fraction of total energy (F)
must be removed from the resonant mode each cycle. The Sabine type
equations do not apply here. They are based on absorptive surfaces
exposed to diffuse sound fields and are valid above 300 Hz. Here is
low frequency absorption and it is related to the volume and position
of the absorption relative to that of the standing wave.

Resonant
Decay by Discrete Absorption

A
basic view of energy absorption allows a fraction (F) of the energy
remaining in a system to be removed at a regular rate (1/N times
a second). This leads to the exponential decay relations whose “RT60”
expression is well known. If the fraction is less than 20%, the
system is “lightly damped,” and the log term can be
simplified in approximation.

The decay equation is very general. It remains
only to define the rate and fraction of energy absorption for any
particular system and the RT60 can be predicted.

One
Dimension Resonance Decay

The
“Impedance Tube” provides a device in which standing
waves can be generated and then their decay monitored. The absorption
device is located at one end of a tube while the sound source is
at the other.

Work is done at the absorption each time
there is excess pressure. This occurs twice each cycle, once when
the pressure goes positive, and then again when it goes negative.
The rate of absorption is twice the resonant frequency.

The
fraction of energy lost by each absorption depends on the position
and number of traps in the resonant field. A trap located at one end
of the impedance tube (A) experiences pressure pulses and can absorb
energy. The same trap located at a pressure node (B) experiences no
pressure change and does no work.

The
single trap at the end of the tube has access to one-half the total
energy in the tube. There are two pressure zones, ¼ wavelength
in size for the first harmonic.

The
second harmonic has its energy split amongst four ¼ wavelength
zones. The trap has access to only ¼ the total energy stored
in the resonant condition.

The
third harmonic has six discrete pressure zones. The trap only works
1/6 of the total energy in the field. The relative size of the trap
to the zone increases with higher mode (j) numbers, so its efficiency
increases.

Multiple
traps in a resonant field increase the fraction of energy removed
each pressure pulse. Two properly placed traps in the third mode or
harmonic has access to 2/6 or 1/3 of the system’s energy.

The
total number of ¼ wavelength pressure zones is twice the mode
number. The fraction of energy lost per pressure pulse is the ratio
of trapped zones (J) to the total number of zones (2L) times an efficiency
term.

The
RT60 equation can be written for one dimension trapping. For small
absorption, the approximation is made.

The
simple Sabine decay formula for one dimension is a classic derivation.
A pulse is injected into the impedance tube. Absorption is located
at the tube end. The fraction of energy lost upon impact is the absorption
coefficient (a).

The
PZT decay formula can be converted into a form like the Sabine. Any
frequency of resonance belongs to one of a harmonic series. It is
the multiple of the mode number (L) and fundamental frequency (fo).
Since absorption is only at one end of the tube for both cases, only
one pressure zone is trapped.

The
efficiency term (n) in PZT analysis and the absorption coefficient
(a) in Sabine calculations have the same physical definition. It is
the ratio of energy lost to initial energy. For the one dimension
systems, PZT rationale results in the same conclusion as does the
classic Sabine analysis.

Two
Dimensional Decay Rates

The
two dimensional physical space is outlined by an X and Y dimension.
Each resonant mode is identified by a “mode number,”
a set of two whole numbers (L,M). If one of the mode numbers is
zero, the one dimensional model develops.

The
standard equation for the frequency of a resonant mode has two components.
They can be converted into wave numbers by dividing each mode number
by its associated physical length. The mode frequency equation can
be rewritten in terms of wave numbers.

The
primitive cell in two dimensions is the (1,1) mode. Positive pressure
in opposite corners with negative pressure in the other two marks
the energy distribution at one moment. A half cycle later the polarity
reverses. Between these moments are complimentary patterns of kinetic
energy distribution.

There
are a total of 4 quarter wavelength zones in the pressure distribution
of the primitive cell. They are in the corners. All the energy in
the resonant cell is found within these four zones twice each cycle.
80% of a zone is found contained within the radius, 1/6 of the wavelength
from the corner.

Higher
mode numbers are simply more such cells packed into the same space.
A (2,1) mode has two cells in the X axis and one cell in the Y. A
(2,2) mode is two cells wide by two cells high. The total number of
cells is the product of the two mode numbers.

The
total number of pressure zones (K) will be four times the number of
cells in a mode. If some number (J) of them are absorptively trapped,
the fraction of pressure zones trapped is known if the efficiency
term is included.

The
RT60 formula derived for PZT methods is general and can be applied
to this two dimensional case. For light absorption, a further simplification
results.

Three
Dimensional Modes

The
three dimensional model of Pressure Zone Trapping also has a primitive
cell, (1,1,1). It has eight corners, each containing a quarter wavelength
pressure zone. If all eight zones were placed together a complete
sphere would be formed.

Harmonics
of the fundamental are built in terms of complete cells. The (1,1,2)
will be one cell high, one cell wide, and two cells deep. It will
have 8 x 2 or 16 pressure zones. The (1,1,3) mode is one by one by
three cells in configuration and has 8 x 3 or 24 pressure zones. The
(2,2,2) mode is accordingly two by two by two cells for a total of
eight and 8 x 8 or 64 pressure zones. The total number of pressure
zones for any (L,M,N) mode is 8(LMN). They momentarily hold all the
energy of the resonant field two times per cycle for any standing
wave mode in a three dimensional field.

The
basic Pressure Zone Trapping formula still applies. The more complicated
term for frequency, well known and dependent on three terms, can be
substituted. The value for absorption coefficient remains the fraction
of energy absorbed per absorption event. It is the fraction of trapped
zones times the efficiency term.

The
formal RT60 equation can be simplified if the absorption coefficient
is less than 1/5 by approximation. The complete RT60 equation is
written by substituting terms for frequency and fraction of energy.
This formal equation can be simplified if the absorption coefficient
(F) is less than 1/5 in the log term.

The
RT60 equation can be further developed. The room volume (Vr) term
is introduced which converts the three mode numbers into wave numbers.

Wave
Number Space

Wave number
space is a three dimensional coordinate system with A, B, and C
axes. Each point (P) in this space defines a resonant mode for the
room. This is not a continuous field space. It is more like a crystal;
discrete points set apart at specific distances.

The
mode point is at the tip of the resultant vector (D) whose magnitude
is the sum of the squares of the components. It is also at the far
corner of a rectangle whose volume (V) is known by the products of
its components.

The
frequency and RT60 formulas can be rewritten in terms of this wave
number space geometry.

This
listening room already has a decay time. Frequently improvement in
the decay rate is desired. The minimum upgrade is to trap one zone
for each 500 cubic feet of room volume. The resulting RT60 is a simple
expression but is only valid for an absolutely rigid room whose only
absorption is due to the trapped zones.

Example

Consider a room 18 by 24 by 8 feet high.
We can look at mode (2,2,1). The wave numbers (1/9, 1/12, 1/8) are
easily calculated along with the volume and diagonal wave number
in space. The decay time for that mode is 0.3 sec. This assumes
one 100% efficient absorption device per 500 cubic feet of room
volume.

How Many
Traps

The
efficiency term (n) is defined as the ratio of energy absorbed to
the energy presented. The ¼ wavelength pressure zone contains
a discrete quantity of energy in a definable volume. The trap occupies
part of that quadrant with its own volume (V). 80% of the zone’s
energy lies within 1/6 wavelength radius from the corner. The ratio
of PZT volume to the 1/8 spherical section volume comprises the
geometric efficiency (E). This is further reduced by the mechanical
efficiency of the trap (a) itself; typically 50%.

The
RT60 equation can be fitted with this efficiency term. Additional
substitutions and reductions provide the RT60 to have an inverse frequency
dependency. Recall the Sabine equations to not be directly frequency
dependent. There appears the dimensionless ratio in wave number space
of the modal volume to the cubed modal length. This ratio is largest
for symmetric modes (1, 1, 7) or (2, 2, 2) and smallest for the eccentric
modes as (1, 2, 6). It is always less than unity and a mean value
of 1/3 is chosen.

The
use of traps sufficient to remedy a room’s poor low end ranges
from one trap per 500 cubic feet to one trap per 250 cubic feet
of room volume. This simplifies further the RT60 equation. The trap
volume can be resolved for the 500 cubic foot ratio to be inversely
dependent on both RT60 and frequency.

The typical acoustic efficiency is 50% for
these three commercial traps. Their volume levels cross extended
through the frequency range call out the RT60 vs. frequency plot
for the 250 cubic foot or 500 cubic foot rate. For example, a 4
cubic foot trap provides 2 seconds at 20 Hz, 1 second at 50 Hz and
½ second at 90 Hz RT60 times.

Conversely,
for a particular resonant frequency, room volume and required RT60,
the number (J) of trapped volumes can be calculated.

Example

A
room of 2,000 cubic feet needs an RT60 of 1/2 second at 50 Hz and
tubes having a volume of 4 cubic feet each will be used. A total
of 7 traps must be placed in the pressure zones of that mode resonance.

By
utilizing PZT methods, an absorptive treatment for low frequency resonance
can be specified. The (dQ) change in room Q is easily approximated.
The volume (Vt) of traps required to produce that change can also
be defined.

Example

The 2,000 cubic foot room needed a Q adjustment
of 50. The volume of PZT adjustment is 12 cubic feet.

The
listening room is the last link in the audio chain. It is an acoustic
coupler loaded with resonances. Hundreds of rooms have been developed
into satisfactory listening environments by using the 500 cubic
feet per trap rule. The average trap volume is 2.5 cubic feet. A
correction in Quality of 60 is what the average acoustic treatment
produces. Serious listening rooms usually require a correction in
Quality of 30. This means the average (Q=40) listening room must
have its Q cut in half and a serious room must have a Q equal to
1/3 its untreated Q.

A frequently asked question involves the
number of traps required to reduce an existing RT60. PZT allows
the answer without resorting to Sabine formulas.

Examples

A
2000 cubic foot room has an RT60 of 1.3 sec. at 50 Hz. We wish to
reduce it to 0.7 sec. using 4 cubic foot traps. Calculations show
4.4 traps will lower the RT60 as required.

A
2000 cubic foot soft room with an RT60 of 0.5 seconds needs to be
reduced to 0.3 seconds. Using 4 cubic foot traps, calculations show
9 are needed.

If
RT60 equipment is not available, a slow sine sweep frequency response
will suffice. Measure the 3 dB down bandwidth dF. Substitute its relation
for initial RT60. The desired RT60 is often specified and doesn’t
need conversion to final bandwidth.

Reverb
Chamber

Absorption
is usually measured in reverb chambers using RT60 values and the
Sabine absorption formula. PZT equations can be rearranged into
the same format. The distinctive frequency dependence of PZT absorption
is clear. This relation connects standard Sabine lab methods to
PZT theory.

Conclusion

A listening room does not have an acoustically
flat response. Most rooms can play better when their Q is reduced
by a factor of 2 or 3. Room color is damped out from the listening
ambience. It is the Q not the EQ that distinguishes the listening
room from a standard room. Pink noise is an appropriate test signal
for EQ settings. Pure tone, not 1/3 octave sweeps or RT60 are required
to monitor the room Q.

The Pressure Zone Trap (PZT) approach provides
a rational view of discrete absorptive devices in the resonant field.
It allows specifications to reduce the RT60, or Q of the room to
acceptable levels.