# Engineering Acoustics/Bass Reflex Enclosure Design

## Introduction

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![](Bassreflex-Gehäuse_(enclosure).png "Bassreflex-Gehäuse_(enclosure).png")

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Bass-reflex enclosures improve the low-frequency response of loudspeaker
systems. Bass-reflex enclosures are also called \"vented-box design\" or
\"ported-cabinet design\". A bass-reflex enclosure includes a vent or
port between the cabinet and the ambient environment. This type of
design, as one may observe by looking at contemporary loudspeaker
products, is still widely used today. Although the construction of
bass-reflex enclosures is fairly simple, their design is not simple, and
requires proper tuning. This reference focuses on the technical details
of bass-reflex design. General loudspeaker information can be found
here.

## Effects of the Port on the Enclosure Response

Before discussing the bass-reflex enclosure, it is important to be
familiar with the simpler sealed enclosure system performance. As the
name suggests, the sealed enclosure system attaches the loudspeaker to a
sealed enclosure (except for a small air leak included to equalize the
ambient pressure inside). Ideally, the enclosure would act as an
acoustical compliance element, as the air inside the enclosure is
compressed and rarified. Often, however, an acoustic material is added
inside the box to reduce standing waves, dissipate heat, and other
reasons. This adds a resistive element to the acoustical lumped-element
model. A non-ideal model of the effect of the enclosure actually adds an
acoustical mass element to complete a series lumped-element circuit
given in Figure 1. For more on sealed enclosure design, see the Sealed
Box Subwoofer
Design
page.

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*Figure 1. Sealed enclosure acoustic circuit.*

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</center>
```
In the case of a bass-reflex enclosure, a port is added to the
construction. Typically, the port is cylindrical and is flanged on the
end pointing outside the enclosure. In a bass-reflex enclosure, the
amount of acoustic material used is usually much less than in the sealed
enclosure case, often none at all. This allows air to flow freely
through the port. Instead, the larger losses come from the air leakage
in the enclosure. With this setup, a lumped-element acoustical circuit
has the following form.

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```
![](Vented_box_ckt.gif "Vented_box_ckt.gif")\
*Figure 2. Bass-reflex enclosure acoustic circuit.*

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</center>
```
In this figure, $Z_{RAD}$ represents the radiation impedance of the
outside environment on the loudspeaker diaphragm. The loading on the
rear of the diaphragm has changed when compared to the sealed enclosure
case. If one visualizes the movement of air within the enclosure, some
of the air is compressed and rarified by the compliance of the
enclosure, some leaks out of the enclosure, and some flows out of the
port. This explains the parallel combination of $M_{AP}$, $C_{AB}$, and
$R_{AL}$. A truly realistic model would incorporate a radiation
impedance of the port in series with $M_{AP}$, but for now it is
ignored. Finally, $M_{AB}$, the acoustical mass of the enclosure, is
included as discussed in the sealed enclosure case. The formulas which
calculate the enclosure parameters are listed in Appendix
B.

It is important to note the parallel combination of $M_{AP}$ and
$C_{AB}$. This forms a Helmholtz resonator (click here for more
information).
Physically, the port functions as the "neck" of the resonator and the
enclosure functions as the "cavity." In this case, the resonator is
driven from the piston directly on the cavity instead of the typical
Helmholtz case where it is driven at the "neck." However, the same
resonant behavior still occurs at the enclosure resonance frequency,
$f_{B}$. At this frequency, the impedance seen by the loudspeaker
diaphragm is large (see Figure 3 below). Thus, the load on the
loudspeaker reduces the velocity flowing through its mechanical
parameters, causing an anti-resonance condition where the displacement
of the diaphragm is a minimum. Instead, the majority of the volume
velocity is actually emitted by the port itself instead of the
loudspeaker. When this impedance is reflected to the electrical circuit,
it is proportional to $1/Z$, thus a minimum in the impedance seen by the
voice coil is small. Figure 3 shows a plot of the impedance seen at the
terminals of the loudspeaker. In this example, $f_B$ was found to be
about 40 Hz, which corresponds to the null in the voice-coil impedance.

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```
![](_Za0_Zvc_plots.gif "_Za0_Zvc_plots.gif")\
*Figure 3. Impedances seen by the loudspeaker diaphragm and voice coil.*

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</center>
```
## Quantitative Analysis of Port on Enclosure

The performance of the loudspeaker is first measured by its velocity
response, which can be found directly from the equivalent circuit of the
system. As the goal of most loudspeaker designs is to improve the bass
response (leaving high-frequency production to a tweeter), low frequency
approximations will be made as much as possible to simplify the
analysis. First, the inductance of the voice coil, $\it{L_E}$, can be
ignored as long as $\omega \ll R_E/L_E$. In a typical loudspeaker,
$\it{L_E}$ is of the order of 1 mH, while $\it{R_E}$ is typically
8$\Omega$, thus an upper frequency limit is approximately 1 kHz for this
approximation, which is certainly high enough for the frequency range of
interest.

Another approximation involves the radiation impedance, $\it{Z_{RAD}}$.
It can be shown \[1\] that this value is given by the following equation
(in acoustical ohms):

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```
$Z_{RAD} = \frac{\rho_0c}{\pi a^2}\left[\left(1 - \frac{J_1(2ka)}{ka}\right) + j\frac{H_1(2ka)}{ka}\right]$

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```
Where $J_1(x)$ and $H_1(x)$ are types of Bessel functions. For small
values of *ka*,

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$J_1(2ka) \approx ka$

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and

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$H_1(2ka) \approx \frac{8(ka)^2}{3\pi}$

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$\Rightarrow Z_{RAD} \approx j\frac{8\rho_0\omega}{3\pi^2a} = jM_{A1}$

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Hence, the low-frequency impedance on the loudspeaker is represented
with an acoustic mass $M_{A1}$ \[1\]. For a simple analysis, $R_E$,
$M_{MD}$, $C_{MS}$, and $R_{MS}$ (the transducer parameters, or
*Thiele-Small* parameters) are converted to their acoustical
equivalents. All conversions for all parameters are given in Appendix
A.
Then, the series masses, $M_{AD}$, $M_{A1}$, and $M_{AB}$, are lumped
together to create $M_{AC}$. This new circuit is shown below.

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```
![](VB_LF_ckt.gif "VB_LF_ckt.gif")\
*Figure 4. Low-Frequency Equivalent Acoustic Circuit*

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</center>
```
Unlike sealed enclosure analysis, there are multiple sources of volume
velocity that radiate to the outside environment. Hence, the diaphragm
volume velocity, $U_D$, is not analyzed but rather
$U_0 = U_D + U_P + U_L$. This essentially draws a "bubble" around the
enclosure and treats the system as a source with volume velocity $U_0$.
This "lumped" approach will only be valid for low frequencies, but
previous approximations have already limited the analysis to such
frequencies anyway. It can be seen from the circuit that the volume
velocity flowing *into* the enclosure, $U_B = -U_0$, compresses the air
inside the enclosure. Thus, the circuit model of Figure 3 is valid and
the relationship relating input voltage, $V_{IN}$ to $U_0$ may be
computed.

In order to make the equations easier to understand, several parameters
are combined to form other parameter names. First, $\omega_B$ and
$\omega_S$, the enclosure and loudspeaker resonance frequencies,
respectively, are:

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$\omega_B = \frac{1}{\sqrt{M_{AP}C_{AB}}}$

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$\omega_S = \frac{1}{\sqrt{M_{AC}C_{AS}}}$

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</td>
```
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```
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```
Based on the nature of the derivation, it is convenient to define the
parameters $\omega_0$ and *h*, the Helmholtz tuning ratio:

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$\omega_0 = \sqrt{\omega_B\omega_S}$

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$h = \frac{\omega_B}{\omega_S}$

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```
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```
A parameter known as the *compliance ratio* or *volume ratio*, $\alpha$,
is given by:

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```
$\alpha = \frac{C_{AS}}{C_{AB}} = \frac{V_{AS}}{V_{AB}}$

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</center>
```
Other parameters are combined to form what are known as *quality
factors*:

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$Q_L = R_{AL}\sqrt{\frac{C_{AB}}{M_{AP}}}$

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$Q_{TS} = \frac{1}{R_{AE}+R_{AS}}\sqrt{\frac{M_{AC}}{C_{AS}}}$

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```
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This notation allows for a simpler expression for the resulting transfer
function \[1\]:

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```
$\frac{U_0}{V_{IN}} = G(s) = \frac{(s^3/\omega_0^4)}{(s/\omega_0)^4+a_3(s/\omega_0)^3+a_2(s/\omega_0)^2+a_1(s/\omega_0)+1}$

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```
where

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$a_1 = \frac{1}{Q_L\sqrt{h}}+\frac{\sqrt{h}}{Q_{TS}}$

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$a_2 = \frac{\alpha+1}{h}+h+\frac{1}{Q_L Q_{TS}}$

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$a_3 = \frac{1}{Q_{TS}\sqrt{h}}+\frac{\sqrt{h}}{Q_L}$

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```
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```
## Development of Low-Frequency Pressure Response

It can be shown \[2\] that for $ka < 1/2$, a loudspeaker behaves as a
spherical source. Here, *a* represents the radius of the loudspeaker.
For a 15" diameter loudspeaker in air, this low frequency limit is about
150 Hz. For smaller loudspeakers, this limit increases. This limit
dominates the limit which ignores $L_E$, and is consistent with the
limit that models $Z_{RAD}$ by $M_{A1}$.

Within this limit, the loudspeaker emits a volume velocity $U_0$, as
determined in the previous section. For a simple spherical source with
volume velocity $U_0$, the far-field pressure is given by \[1\]:

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```
$p(r) \simeq j\omega\rho_0 U_0 \frac{e^{-jkr}}{4\pi r}$

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```
It is possible to simply let $r = 1$ for this analysis without loss of
generality because distance is only a function of the surroundings, not
the loudspeaker. Also, because the transfer function magnitude is of
primary interest, the exponential term, which has a unity magnitude, is
omitted. Hence, the pressure response of the system is given by \[1\]:

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```
$\frac{p}{V_{IN}} = \frac{\rho_0s}{4\pi}\frac{U_0}{V_{IN}} = \frac{\rho_0Bl}{4\pi S_DR_EM_AS}H(s)$

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```
Where $H(s) = sG(s)$. In the following sections, design methods will
focus on $|H(s)|^2$ rather than $H(s)$, which is given by:

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$|H(s)|^2 = \frac{\Omega^8}{\Omega^8 + \left(a^2_3 - 2a_2\right)\Omega^6 + \left(a^2_2 + 2 - 2a_1a_3\right)\Omega^4 + \left(a^2_1 - 2a_2\right)\Omega^2 + 1}$

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$\Omega = \frac{\omega}{\omega_0}$

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This also implicitly ignores the constants in front of $|H(s)|$ since
they simply scale the response and do not affect the shape of the
frequency response curve.

## Alignments

A popular way to determine the ideal parameters has been through the use
of alignments. The concept of alignments is based upon well investigated
electrical filter theory. Filter development is a method of selecting
the poles (and possibly zeros) of a transfer function to meet a
particular design criterion. The criteria are the desired properties of
a magnitude-squared transfer function, which in this case is $|H(s)|^2$.
From any of the design criteria, the poles (and possibly zeros) of
$|H(s)|^2$ are found, which can then be used to calculate the numerator
and denominator. This is the "optimal" transfer function, which has
coefficients that are matched to the parameters of $|H(s)|^2$ to compute
the appropriate values that will yield a design that meets the criteria.

There are many different types of filter designs, each which have
trade-offs associated with them. However, this design approach is
limited because of the structure of $|H(s)|^2$. In particular, it has
the structure of a fourth-order high-pass filter with all zeros at *s* =
0. Therefore, only those filter design methods which produce a low-pass
filter with only poles will be acceptable methods to use. From the
traditional set of algorithms, only Butterworth and Chebyshev low-pass
filters have only poles. In addition, another type of filter called a
quasi-Butterworth filter can also be used, which has similar properties
to a Butterworth filter. These three algorithms are fairly simple, thus
they are the most popular. When these low-pass filters are converted to
high-pass filters, the $s \rightarrow 1/s$ transformation produces $s^8$
in the numerator.

More details regarding filter theory and these relationships can be
found in numerous resources, including \[5\].

## Butterworth Alignment

The Butterworth algorithm is designed to have a *maximally flat* pass
band. Since the slope of a function corresponds to its derivatives, a
flat function will have derivatives equal to zero. Since as flat of a
pass band as possible is optimal, the ideal function will have as many
derivatives equal to zero as possible at *s* = 0. Of course, if all
derivatives were equal to zero, then the function would be a constant,
which performs no filtering.

Often, it is better to examine what is called the *loss function*. Loss
is the reciprocal of gain, thus

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$|\hat{H}(s)|^2 = \frac{1}{|H(s)|^2}$

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The loss function can be used to achieve the desired properties, then
the desired gain function is recovered from the loss function.

Now, applying the desired Butterworth property of maximal pass-band
flatness, the loss function is simply a polynomial with derivatives
equal to zero at *s* = 0. At the same time, the original polynomial must
be of degree eight (yielding a fourth-order function). However,
derivatives one through seven can be equal to zero if \[3\]

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```
$|\hat{H}(\Omega)|^2 = 1 + \Omega^8 \Rightarrow |H(\Omega)|^2 = \frac{1}{1 + \Omega^8}$

```{=html}
</center>
```
With the high-pass transformation $\Omega \rightarrow 1/\Omega$,

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<center>
```
$|H(\Omega)|^2 = \frac{\Omega^8}{\Omega^8 + 1}$

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</center>
```
It is convenient to define $\Omega = \omega/\omega_{3dB}$, since
$\Omega = 1 \Rightarrow |H(s)|^2 = 0.5$ or -3 dB. This definition allows
the matching of coefficients for the $|H(s)|^2$ describing the
loudspeaker response when $\omega_{3dB} = \omega_0$. From this matching,
the following design equations are obtained \[1\]:

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$a_1 = a_3 = \sqrt{4+2\sqrt{2}}$

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$a_2 = 2+\sqrt{2}$

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## Quasi-Butterworth Alignment

The quasi-Butterworth alignments do not have as well-defined of an
algorithm when compared to the Butterworth alignment. The name
"quasi-Butterworth" comes from the fact that the transfer functions for
these responses appear similar to the Butterworth ones, with (in
general) the addition of terms in the denominator. This will be
illustrated below. While there are many types of quasi-Butterworth
alignments, the simplest and most popular is the 3rd order alignment
(QB3). The comparison of the QB3 magnitude-squared response against the
4th order Butterworth is shown below.

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$\left|H_{QB3}(\omega)\right|^2 = \frac{(\omega/\omega_{3dB})^8}{(\omega/\omega_{3dB})^8 + B^2(\omega/\omega_{3dB})^2 + 1}$

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<td>
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$\left|H_{B4}(\omega)\right|^2 = \frac{(\omega/\omega_{3dB})^8}{(\omega/\omega_{3dB})^8 + 1}$

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```
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```
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</table>
```
Notice that the case $B = 0$ is the Butterworth alignment. The reason
that this QB alignment is called 3rd order is due to the fact that as
*B* increases, the slope approaches 3 dec/dec instead of 4 dec/dec, as
in 4th order Butterworth. This phenomenon can be seen in Figure 5.

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```
![](QB3_gradient.GIF "QB3_gradient.GIF")\
*Figure 5: 3rd-Order Quasi-Butterworth Response for*$0.1 \leq B \leq 3$

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```
Equating the system response $|H(s)|^2$ with $|H_{QB3}(s)|^2$, the
equations guiding the design can be found \[1\]:

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$B^2 = a^2_1 - 2a_2$

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$a_2^2 + 2 = 2a_1a_3$

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$a_3 = \sqrt{2a_2}$

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$a_2 > 2 + \sqrt{2}$

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</table>
```
## Chebyshev Alignment

The Chebyshev algorithm is an alternative to the Butterworth algorithm.
For the Chebyshev response, the maximally-flat passband restriction is
abandoned. Now, a *ripple*, or fluctuation is allowed in the pass band.
This allows a steeper transition or roll-off to occur. In this type of
application, the low-frequency response of the loudspeaker can be
extended beyond what can be achieved by Butterworth-type filters. An
example plot of a Chebyshev high-pass response with 0.5 dB of ripple
against a Butterworth high-pass response for the same $\omega_{3dB}$ is
shown below.

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<center>
```
![](Butt_vs_Cheb_HP.gif "Butt_vs_Cheb_HP.gif")\
*Figure 6: Chebyshev vs. Butterworth High-Pass Response.*

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</center>
```
The Chebyshev response is defined by \[4\]:

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<center>
```
$|\hat{H}(j\Omega)|^2 = 1 + \epsilon^2C^2_n(\Omega)$\

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$C_n(\Omega)$ is called the *Chebyshev polynomial* and is defined by
\[4\]:

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$C_n(\Omega) = \big\lbrace$

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<td>
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$\rm{cos}[\it{n}\rm{cos}^{-1}(\Omega)]$

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</td>
```
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<td>
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$|\Omega| < 1$

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</td>
```
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</tr>
```
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```
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<td>
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$\rm{cosh}[\it{n}\rm{cosh}^{-1}(\Omega)]$

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```
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<td>
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$|\Omega| > 1$

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</td>
```
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```
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</table>
```
Fortunately, Chebyshev polynomials satisfy a simple recursion formula
\[4\]:

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$C_0(x) = 1$

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$C_1(x) = x$

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$C_n(x) = 2xC_{n-1} - C_{n-2}$

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```
For more information on Chebyshev polynomials, see the Wolfram
Mathworld: Chebyshev
Polynomials
page.

When applying the high-pass transformation to the 4th order form of
$|\hat{H}(j\Omega)|^2$, the desired response has the form \[1\]:

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```
$|H(j\Omega)|^2 = \frac{1+\epsilon^2}{1+\epsilon^2C^2_4(1/\Omega)}$

```{=html}
</center>
```
The parameter $\epsilon$ determines the ripple. In particular, the
magnitude of the ripple is $10\rm{log}[1+\epsilon^2]$ dB and can be
chosen by the designer, similar to *B* in the quasi-Butterworth case.
Using the recursion formula for $C_n(x)$,

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```
$C_4\left(\frac{1}{\Omega}\right) = 8\left(\frac{1}{\Omega}\right)^4 - 8\left(\frac{1}{\Omega}\right)^2 + 1$\

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</center>
```
Applying this equation to $|H(j\Omega)|^2$ \[1\],

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```
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<tr>
```
```{=html}
<td colspan=2>
```
$\Rightarrow |H(\Omega)|^2 = \frac{\frac{1 + \epsilon^2}{64\epsilon^2}\Omega^8}{\frac{1 + \epsilon^2}{64\epsilon^2}\Omega^8 + \frac{1}{4}\Omega^6 + \frac{5}{4}\Omega^4 - 2\Omega^2 + 1}$

```{=html}
</td>
```
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</tr>
```
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<tr>
```
```{=html}
<td>
```
$\Omega = \frac{\omega}{\omega_n}$

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</td>
```
```{=html}
<td>
```
$\omega_n = \frac{\omega_{3dB}}{2}\sqrt{2 + \sqrt{2 + 2\sqrt{2+\frac{1}{\epsilon^2}}}}$

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</td>
```
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</tr>
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```
Thus, the design equations become \[1\]:

```{=html}
<table align=center cellpadding=15>
```
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<tr>
```
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<td>
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$\omega_0 = \omega_n\sqrt[8]{\frac{64\epsilon^2}{1+\epsilon^2}}$

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</td>
```
```{=html}
<td>
```
$k = \rm{tanh}\left[\frac{1}{4}\rm{sinh}^{-1}\left(\frac{1}{\epsilon}\right)\right]$

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<td>
```
$D = \frac{k^4 + 6k^2 + 1}{8}$

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</td>
```
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</tr>
```
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<tr>
```
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<td>
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$a_1 = \frac{k\sqrt{4 + 2\sqrt{2}}}{\sqrt[4]{D}},$

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</td>
```
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<td>
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$a_2 = \frac{1 + k^2(1+\sqrt{2})}{\sqrt{D}}$

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</td>
```
```{=html}
<td>
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$a_3 = \frac{a_1}{\sqrt{D}}\left[1 - \frac{1 - k^2}{2\sqrt{2}}\right]$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Choosing the Correct Alignment

With all the equations that have already been presented, the question
naturally arises, "Which one should I choose?" Notice that the
coefficients $a_1$, $a_2$, and $a_3$ are not simply related to the
parameters of the system response. Certain combinations of parameters
may indeed invalidate one or more of the alignments because they cannot
realize the necessary coefficients. With this in mind, general
guidelines have been developed to guide the selection of the appropriate
alignment. This is very useful if one is designing an enclosure to suit
a particular transducer that cannot be changed.

The general guideline for the Butterworth alignment focuses on $Q_L$ and
$Q_{TS}$. Since the three coefficients $a_1$, $a_2$, and $a_3$ are a
function of $Q_L$, $Q_{TS}$, *h*, and $\alpha$, fixing one of these
parameters yields three equations that uniquely determine the other
three. In the case where a particular transducer is already given,
$Q_{TS}$ is essentially fixed. If the desired parameters of the
enclosure are already known, then $Q_L$ is a better starting point.

In the case that the rigid requirements of the Butterworth alignment
cannot be satisfied, the quasi-Butterworth alignment is often applied
when $Q_{TS}$ is not large enough.. The addition of another parameter,
*B*, allows more flexibility in the design.

For $Q_{TS}$ values that are too large for the Butterworth alignment,
the Chebyshev alignment is typically chosen. However, the steep
transition of the Chebyshev alignment may also be utilized to attempt to
extend the bass response of the loudspeaker in the case where the
transducer properties can be changed.

In addition to these three popular alignments, research continues in the
area of developing new algorithms that can manipulate the low-frequency
response of the bass-reflex enclosure. For example, a 5th order
quasi-Butterworth alignment has been developed \[6\]; its advantages
include improved low frequency extension, and much reduced driver
excursion at low frequencies and typically bi-amping or tri-amping,
while its disadvatages include somewhat difficult mathematics and
electronic complication (electronic crossovers are typically required).
Another example \[7\] applies root-locus techniques to achieve results.
In the modern age of high-powered computing, other researchers have
focused their efforts in creating computerized optimization algorithms
that can be modified to achieve a flatter response with sharp roll-off
or introduce quasi-ripples which provide a boost in sub-bass frequencies
\[8\].

Back to Engineering Acoustics

## References

\[1\] Leach, W. Marshall, Jr. *Introduction to Electroacoustics and
Audio Amplifier Design*. 2nd ed. Kendall/Hunt, Dubuque, IA. 2001.

\[2\] Beranek, L. L. *Acoustics*. 2nd ed. Acoustical Society of America,
Woodbridge, NY. 1993.

\[3\] DeCarlo, Raymond A. "The Butterworth Approximation." Notes from
ECE 445. Purdue University. 2004.

\[4\] DeCarlo, Raymond A. "The Chebyshev Approximation." Notes from ECE
445. Purdue University. 2004.

\[5\] VanValkenburg, M. E. *Analog Filter Design*. Holt, Rinehart and
Winston, Inc. Chicago, IL. 1982.

\[6\] Kreutz, Joseph and Panzer, Joerg. \"Derivation of the
Quasi-Butterworth 5 Alignments.\" *Journal of the Audio Engineering
Society*. Vol. 42, No. 5, May 1994.

\[7\] Rutt, Thomas E. \"Root-Locus Technique for Vented-Box Loudspeaker
Design.\" *Journal of the Audio Engineering Society*. Vol. 33, No. 9,
September 1985.

\[8\] Simeonov, Lubomir B. and Shopova-Simeonova, Elena.
\"Passive-Radiator Loudspeaker System Design Software Including
Optimization Algorithm.\" *Journal of the Audio Engineering Society*.
Vol. 47, No. 4, April 1999.

## Appendix A: Equivalent Circuit Parameters

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Name

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Electrical Equivalent

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Mechanical Equivalent

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Acoustical Equivalent

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Voice-Coil Resistance

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$R_E$

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$R_{ME} = \frac{(Bl)^2}{R_E}$

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$R_{AE} = \frac{(Bl)^2}{R_ES^2_D}$

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Driver (Speaker) Mass

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$M_{MD}$

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$M_{AD} = \frac{M_{MD}}{S^2_D}$

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Driver (Speaker) Suspension Compliance

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$L_{CES} = (Bl)^2C_{MS}$

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$C_{MS}$

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$C_{AS} = S^2_DC_{MS}$

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Driver (Speaker) Suspension Resistance

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$R_{ES} = \frac{(Bl)^2}{R_{MS}}$

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$R_{MS}$

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$R_{AS} = \frac{R_{MS}}{S^2_D}$

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Enclosure Compliance

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$L_{CEB} = \frac{(Bl)^2C_{AB}}{S^2_D}$

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$C_{MB} = \frac{C_{AB}}{S^2_D}$

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$C_{AB}$

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Enclosure Air-Leak Losses

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$R_{EL} = \frac{(Bl)^2}{S^2_DR_{AL}}$

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$R_{ML} = S^2_DR_{AL}$

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$R_{AL}$

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Acoustic Mass of Port

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$C_{MEP} = \frac{S^2_DM_{AP}}{(Bl)^2}$

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$M_{MP} = S^2_DM_{AP}$

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$M_{AP}$

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Enclosure Mass Load

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$M_{AB}$

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Low-Frequency Radiation Mass Load

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$M_{A1}$

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Combination Mass Load

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$C_{MEC} = \frac{S^2_DM_{AC}}{(Bl)^2}$\
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$M_{MC} = S^2_D(M_{AB} + M_{A1}) + M_{MD}$

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$M_{AC} = M_{AD} + M_{AB} + M_{A1}$\
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## Appendix B: Enclosure Parameter Formulas

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![](_Vented_enclosure.gif "_Vented_enclosure.gif")\
*Figure 7: Important dimensions of bass-reflex enclosure.*

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$C_{AB} = \frac{V_{AB}}{\rho_0c^2_0}$

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$B = \frac{d}{3}\left(\frac{S_D}{S_B}\right)^2\sqrt{\frac{\pi}{S_D}} + \frac{8}{3\pi}\left[1 - \frac{S_D}{S_B}\right]$

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$\rho_0 \leq \rho_{eff} \leq \rho_0\left(1 - \frac{V_{fill}}{V_B}\right) + \rho_{fill}\frac{V_{fill}}{V_B}$

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$V_{AB} = V_B\left[1-\frac{V_{fill}}{V_B}\right]\left[1 + \frac{\gamma - 1}{1 + \gamma\left(\frac{V_B}{V_{fill}} - 1\right)\frac{\rho_0c_{air}}{\rho_{fill}c_{fill}}}\right]$

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$c_{air} =$specific heat of air at constant isovolumetric process (about
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$\rho_0 =$mean density of air (about $1.3 \frac{\rm kg}{\rm m^3}$ at 300
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$c_0 =$ speed of sound in air (about 344 m/s)

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$\rho_{eff}$ = effective density of enclosure. If little or no filling
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