[<< wikibooks] Engineering Acoustics/Filter Design and Implementation
== Introduction ==
Acoustic filters, or mufflers, are used in a number of applications requiring the suppression or attenuation of sound.  Although the idea might not be familiar to many people, acoustic mufflers make everyday life much more pleasant.  Many common appliances, such as refrigerators and air conditioners, use acoustic mufflers to produce a minimal working noise.  The application of acoustic mufflers is mostly directed to machine components or areas where there is a large amount of radiated sound such as high pressure exhaust pipes, gas turbines, and rotary pumps.
Although there are a number of applications for acoustic mufflers, there are really only two main types which are used.  These are absorptive and reactive mufflers.  Absorptive mufflers incorporate sound absorbing materials to attenuate the radiated energy in gas flow.  Reactive mufflers use a series of complex passages to maximize sound attenuation while meeting set specifications, such as pressure drop, volume flow, etc.  Many of the more complex mufflers today incorporate both methods to optimize sound attenuation and provide realistic specifications.
In order to fully understand how acoustic filters attenuate radiated sound, it is first necessary to briefly cover some basic background topics.  For more information on wave theory and other material necessary to study acoustic filters please refer to the references below.

== Basic Wave Theory ==
Although not fundamentally difficult to understand, there are a number of alternate techniques used to analyze wave motion which could seem overwhelming to a novice at first.  Therefore, only 1-D wave motion will be analyzed to keep most of the mathematics as simple as possible.  This analysis is valid, with not much error, for the majority of pipes and enclosures encountered in practice.

=== Plane-Wave Pressure Distribution in Pipes ===
The most important equation used is the wave equation in 1-D form (See [1],[2], 1-D Wave Equation, Vibrations of Strings for information).
Therefore, it is reasonable to suggest, if plane waves are propagating, that the pressure distribution in a pipe is given by:

p

=

P
i

e

j
[
ω
t
−
k
x
]

+

P
r

e

j
[
ω
t
+
k
x
]

{\displaystyle \mathbf {p} =\mathbf {Pi} e^{j[\omega t-kx]}+\mathbf {Pr} e^{j[\omega t+kx]}}

where Pi and Pr are incident and reflected wave amplitudes respectively.  Also note that bold notation is used to indicate the possiblily of complex terms.  The first term represents a wave travelling in the +x direction and the second term, -x direction.
Since acoustic filters or mufflers typically attenuate the radiated sound power as much as possible, it is logical to assume that if we can find a way to maximize the ratio between reflected and incident wave amplitude then we will effectively attenuated the radiated noise at certain frequencies.  This ratio is called the reflection coefficient and is given by:

R

=

(

P
r

P
i

)

{\displaystyle \mathbf {R} =\left({\frac {\mathbf {Pr} }{\mathbf {Pi} }}\right)}

It is important to point out that wave reflection only occurs when the impedance of a pipe changes.  It is possible to match the end impedance of a pipe with the characteristic impedance of a pipe to get no wave reflection.  For more information see [1] or [2].
Although the reflection coefficient isn't very useful in its current form since we want a relation describing sound power, a more useful form can be derived by recognizing that the power intensity coefficient is simply the magnitude of reflection coefficient square [1]:

R

π

=

|

R

|

2

{\displaystyle R_{\pi }=\left|\mathbf {R} \right|^{2}}

As one would expect, the power reflection coefficient must be less than or equal to one.  Therefore, it is useful to define the transmission coefficient as it said:

T

π

=

(

1
−

R

π

)

{\displaystyle T_{\pi }=\left(1-R_{\pi }\right)}

which is the amount of power transmitted.  This relation comes directly from conservation of energy.  When talking about the performance of mufflers, typically the power transmission coefficient is specified.

== Basic Filter Design ==
For simple filters, a long wavelength approximation can be made to make the analysis of the system easier.  When this assumption is valid (e.g. low frequencies) the components of the system behave as lumped acoustical elements.  Equations relating the various properties are easily derived under these circumstances, see Lumped Elements for further information.
The following derivations assume long wavelength.  Practical applications for most conditions are given later.

=== Low-Pass Filter ===

These are devices that attenuate the radiated sound power at higher frequencies.  This means the power transmission coefficient is approximently 1 across the band pass at low frequencies(see figure to right).
This is equivalent to an expansion in a pipe, with the volume of gas located in the expansion having an acoustic compliance (see figure to right).  Continuity of acoustic impedance (see Java Applet at: Acoustic Impedance Visualization) at the junction, see [1], gives a power transmission coefficient of:

T

π

=

(

1

1
+

(

S

1

−
S

2
S

)

k
L

)

{\displaystyle T_{\pi }=\left({\frac {1}{1+\left({\frac {S_{1}-S}{2S}}\right)kL}}\right)}

where k is the wavenumber (see Wave Properties), L &

S

1

{\displaystyle S_{1}}
are length and area of expansion respectively, and S is the area of the pipe.
The cut-off frequency is given by:

f

c

=

(

S
c

π
L
(

S

1

−
S
)

)

{\displaystyle f_{c}=\left({\frac {Sc}{\pi L(S_{1}-S)}}\right)}

=== High-Pass Filter ===

These are devices that attenuate the radiated sound power at lower frequencies.  Like before, this means the power transmission coefficient is approximately 1 across the band pass at high frequencies (see figure to right).
This is equivalent to a short side branch (see figure to right) with a radius and length much smaller than the wavelength (lumped element assumption).  This side branch acts like an acoustic mass and applies a different acoustic impedance to the system than the low-pass filter.  Again using continuity of acoustic impedance at the junction yields a power transmission coefficient of the form [1]:

T

π

=

(

1

1
+

(

π

a

2

2
S
L
k

)

2

)

{\displaystyle T_{\pi }=\left({\frac {1}{1+\left({\frac {\pi a^{2}}{2SLk}}\right)^{2}}}\right)}

where a and L are the radius and effective length of the small tube, respectively, and S is the area of the pipe.
The cut-off frequency is given by:

f

c

=

(

c

a

2

2
S
L

)

{\displaystyle f_{c}=\left({\frac {ca^{2}}{2SL}}\right)}

=== Band-Stop Filter ===

These are devices that attenuate the radiated sound power over a certain frequency range (see figure to right).  Like before, the power transmission coefficient is approximently 1 in the band pass region.
Since the band-stop filter is essentially a cross between a low and high pass filter, one might expect to create one by using a combination of both techniques.  This is true in that the combination of a lumped acoustic mass and compliance gives a band-stop filter.  This can be realized as a helmholtz resonator (see Helmholtz Resonator or figure to right).  Again, since the impedance of the helmholtz resonator can be easily determined, continuity of acoustic impedance at the junction can give the power transmission coefficient as [1]:

T

π

=

(

1

1
+

(

c

/

2
S

ω
L

/

S

b

−

c

2

/

ω
V

)

2

)

{\displaystyle T_{\pi }=\left({\frac {1}{1+\left({\frac {c/2S}{\omega L/S_{b}-c^{2}/\omega V}}\right)^{2}}}\right)}

where

S

b

{\displaystyle S_{b}}
is the area of the neck, L is the effective length of the neck, V is the volume of the helmholtz resonator, and S is the area of the pipe.  It is interesting to note that the power transmission coefficient is zero when the frequency is that of the resonance frequency of the helmholtz.  This can be explained by the fact that at resonance the volume velocity in the neck is large with a phase such that all the incident wave is reflected back to the source [1].
The zero power transmission coefficient location is given by:

f

c

=

(

c

2
π

)

(

S

b

L
V

)

{\displaystyle f_{c}=\left({\frac {c}{2\pi }}\right){\sqrt {\left({\frac {S_{b}}{LV}}\right)}}}

This frequency value has powerful implications.  If a system has the majority of noise at one frequency component, the system can be "tuned" using the above equation, with a helmholtz resonator, to perfectly attenuate any transmitted power (see examples below).

=== Design ===
If the long wavelength assumption is valid, typically a combination of methods described above are used to design a filter.  A specific design procedure is outlined for a helmholtz resonator, and other basic filters follow a similar procedure (see 1).
Two main metrics need to be identified when designing a helmholtz resonator [3]:
(1) - Resonance frequency desired:

f

c

=

c

2
π

C

o

V

{\displaystyle f_{c}={\frac {c}{2\pi }}{\frac {\sqrt {C_{o}}}{V}}}
where

C

o

=

S
L

{\displaystyle C_{o}={\frac {S}{L}}}
.
(2) - Transmission loss:

C

o

V

2
S

=
c
o
n
s
t

{\displaystyle {\frac {\sqrt {C_{o}V}}{2S}}=const}
based on TL level.  This constant is found from a TL graph (see HR pp. 6).
This will result in two equations with two unknowns which can be solved for the unknown dimensions of the helmholtz resonator.  It is important to note that flow velocities degrade the amount of transmission loss at resonance and tend to move the resonance location upwards [3].
In many situations, the long wavelength approximation is not valid and alternative methods must be examined.  These are much more mathematically rigorous and require a complete understanding acoustics involved.  Although the mathematics involved are not shown, common filters used are given in the section that follows.

== Actual Filter Design ==
As explained previously, there are two main types of filters used in practice: absorptive and reactive.  The benefits and drawback of each will be briefly expained, along with their relative applications (see Absorptive Mufflers.

=== Absorptive ===
These are mufflers which incorporate sound absorbing materials to transform acoustic energy into heat.  Unlike reactive mufflers which use destructive interference to minimize radiated sound power, absorptive mufflers are typically straight through pipes lined with multiple layers of absorptive materials to reduce radiated sound power.  The most important property of absorptive mufflers is the attenuation constant.  Higher attenuation constants lead to more energy dissipation and lower radiated sound power.

==== Examples ====

There are a number of applications for absorptive mufflers.  The most well known application is in racecars, where engine performance is desired.  Absorptive mufflers don't create a large amount of back pressure (as in reactive mufflers) to attenuate the sound, which leads to higher muffler performance.  It should be noted however, that the radiate sound is much higher.  Other applications include plenum chambers (large chambers lined with absorptive materials, see picture below), lined ducts, and ventilation systems.

=== Reactive ===
Reactive mufflers use a number of complex passages (or lumped elements) to reduce the amount of acoustic energy transmitted.  This is accomplished by a change in impedance at the intersections, which gives rise to reflected waves (and effectively reduces the amount of transmitted acoustic energy).  Since the amount of energy transmitted is minimized, the reflected energy back to the source is quite high.  This can actually degrade the performance of engines and other sources.  Opposite to absorptive mufflers, which dissipate the acoustic energy, reactive mufflers keep the energy contained within the system.  See Reactive Mufflers for more information.

==== Examples ====

Reactive mufflers are the most widely used mufflers in combustion engines 1.  Reactive mufflers are very efficient in low frequency applications (especially since simple lumped element analysis can be applied).  Other application areas include: harsh environments (high temperature/velocity engines, turbines, etc.), specific frequency attenuation (using a helmholtz like device, a specific frequency can be toned to give total attenuation of radiated sound power), and a need for low radiated sound power (car mufflers, air conditioners, etc.).

=== Performance ===
There are 3 main metrics used to describe the performance of mufflers; Noise Reduction, Insertion Loss, and Transmission Loss.  Typically when designing a muffler, 1 or 2 of these metrics is given as a desired value.

==== Noise Reduction (NR) ====
Defined as the difference between sound pressure levels on the source and receiver side.  It is essentially the amount of sound power reduced between the location of the source and termination of the muffler system (it doesn't have to be the termination, but it is the most common location) [3].

N
R
=

(

L

p
1

−

L

p
2

)

{\displaystyle NR=\left(L_{p1}-L_{p2}\right)}

where

L

p
1

{\displaystyle L_{p1}}
and

L

p
2

{\displaystyle L_{p2}}
is sound pressure levels at source and receiver respectively.  Although NR is easy to measure, pressure typically varies at source side due to standing waves [3].

==== Insertion Loss (IL) ====
Defined as difference of sound pressure level at the receiver with and without sound attenuating barriers.  This can be realized, in a car muffler, as the difference in radiated sound power with just a straight pipe to that with an expansion chamber located in the pipe.  Since the expansion chamber will attenuate some of the radiate sound power, the pressure at the receiver with sound attenuating barriers will be less.  Therefore, a higher insertion loss is desired [3].

I
L
=

(

L

p
,
w
i
t
h
o
u
t

−

L

p
,
w
i
t
h

)

{\displaystyle IL=\left(L_{p,without}-L_{p,with}\right)}

where

L

p
,
w
i
t
h
o
u
t

{\displaystyle L_{p,without}}
and

L

p
,
w
i
t
h

{\displaystyle L_{p,with}}
are pressure levels at receiver without and with a muffler system respectively.  Main problem with measuring IL is that the barrier or sound attenuating system needs to be removed without changing the source [3].

==== Transmission Loss (TL) ====
Defined as the difference between the sound power level of the incident wave to the muffler system and the transmitted sound power.  For further information see Transmission Loss [3].

T
L
=
10
l
o
g

(

1
τ

)

{\displaystyle TL=10log\left({\frac {1}{\tau }}\right)}
with

τ
=

(

I

t

I

i

)

{\displaystyle \tau =\left({\frac {I_{t}}{I_{i}}}\right)}

where

I

t

{\displaystyle I_{t}}
and

I

i

{\displaystyle I_{i}}
are the transmitted and incident wave power respectively.  From this expression, it is obvious the problem with measure TL is decomposing the sound field into incident and transmitted waves which can be difficult to do for complex systems (analytically).

==== Examples ====
(1) - For a plenum chamber (see figure below):

T
L
=
−
10
l
o
g

(

S

(

c
o
s
θ

2
π

d

2

+

1
−
α

α

S

w

)

)

{\displaystyle TL=-10log\left(S\left({\frac {cos\theta }{2\pi d^{2}}}+{\frac {1-\alpha }{\alpha S_{w}}}\right)\right)}
in dB
where

α

{\displaystyle \alpha }
is average absorption coefficient.

(2) - For an expansion (see figure below):

N
R
=
10
l
o
g

[

1
2

|

e

−
i
k

x

s

+

(

1
−
S

1
+
S

)

e

i
k

x

s

|

2

(

1
+
S

)

2

]

{\displaystyle NR=10log\left[{\frac {1}{2}}\left|e^{-ikx_{s}}+\left({\frac {1-S}{1+S}}\right)e^{ikx_{s}}\right|^{2}\left(1+S\right)^{2}\right]}

I
L
=
10
l
o
g

[

(

1
+
S

)

2

4

]

{\displaystyle IL=10log\left[{\frac {\left(1+S\right)^{2}}{4}}\right]}

T
L
=
10
l
o
g

[

(

1
+
S

)

2

4
S

]

{\displaystyle TL=10log\left[{\frac {\left(1+S\right)^{2}}{4S}}\right]}

where

S
=

(

A

2

A

1

)

{\displaystyle S=\left({\frac {A_{2}}{A_{1}}}\right)}

(3) - For a helmholtz resonator (see figure below):

T
L
=
10
l
o
g

[

1
+

(

(

c

2

S

b

)

ω
L

/

S
−

(

c

2

ω
V

)

)

2

]

{\displaystyle TL=10log\left[1+\left({\frac {\left({\frac {c}{2S_{b}}}\right)}{\omega L/S-\left({\frac {c^{2}}{\omega V}}\right)}}\right)^{2}\right]}
in dB

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