# Acoustics/Acoustics in Violins

For a detailed anatomy of the violin, please refer to Atelierla
Bussiere.

![](Violin_front_view.jpg "Violin_front_view.jpg")![](backview.jpg "backview.jpg")

## How does a violin make sound?

### General concept

When a violinist bows a string, which can produce vibrations with
abundant harmonics, the vibrations of the strings are structurally
transmitted to the bridge and the body of the instrument through the
bridge. The bridge transmits the vibrational energy produced by the
strings to the body through its feet, further triggering the vibration
of body. The vibration of the body determines sound radiation and sound
quality, along with the resonance of the cavity.

![](Acoustics_in_violins_procedure.jpg "Acoustics_in_violins_procedure.jpg")

### String

The vibration pattern of the strings can be easily be observed. To the
naked eye, the string appears to move back and forth in a parabolic
shape (see figure), which resembles the first mode of free vibration of
a stretched string. The vibration of strings was first investigated by
Hermann Von
Helmholtz,
the famous mathematician and physicist in 19th century. A surprising
scenario was discovered that the string actually moves in an inverse "V"
shape rather than parabolas (see figure). What we see is just an
envelope of the motion of the string. To honor his findings, the motion
of bowed strings had been called "Helmholtz motion."

![](String.jpg "String.jpg")
![](Helmholtzmotion.jpg "Helmholtzmotion.jpg")

## Bridge

The primary role of the bridge is to transform the motion of vibrating
strings into periodic driving forces by its feet to the top plate of the
violin body. The configuration of the bridge can be referred to the
figure. The bridge stands on the belly between f holes, which have two
primary functions. One is to connect the air inside the body with
outside air, and the other one is to make the belly between f holes move
more easily than other parts of the body. The fundamental frequency of a
violin bridge was found to be around 3000 Hz when it is on a rigid
support, and it is an effective energy-transmitting medium to transmit
the energy from the string to body at frequencies from 1 kHz to 4 kHz,
which is in the range of keen sensitivity of human hearing. In order to
darken the sound of violin, the player attaches a mute on the bridge.
The mute is actually an additional mass which reduces the fundamental
frequency of the bridge. As a result, the sound at higher frequencies is
diminished since the force transferred to the body has been decreased.
On the other hand, the fundamental frequency of the bridge can be raised
by attaching an additional stiffness in the form of tiny wedges, and the
sound at higher frequencies will be amplified accordingly.

The sound post connects the flexible belly to the much stiffer back
plate. The sound post can prevent the collapse of the belly due to high
tension force in the string, and, at the same time, couples the
vibration of the plate. The bass bar under the belly extends beyond the
f holes and transmits the force of the bridge to a larger area of the
belly. As can be seen in the figure, the motion of the treble foot is
restricted by the sound post, while, conversely, the foot over bass bar
can move up and down more easily. As a result, the bridge tends to move
up and down, pivoting about the treble foot. The forces appearing at the
two feet remain equal and opposite up to 1 kHz. At higher frequencies,
the forces become uneven. The force on the soundpost foot predominates
at some frequencies, while it is the bass bar foot at some.

![](crossview.jpg "crossview.jpg")

### Body

The body includes top plate, back plate, the sides, and the air inside,
all of which serve to transmit the vibration of the bridge into the
vibration of air surrounding the violin. For this reason, the violin
needs a relatively large surface area to push enough amount of air back
and forth. Thus, the top and back plates play important roles in the
mechanism. Violin makers have traditionally paid much attention to the
vibration of the top and back plates of the violin by listening to the
tap tones, or, recently, by observing the vibration mode shapes of the
body plates. The vibration modes of an assembled violin are, however,
much more complicated.

The vibration modes of top and back plates can be easily observed in a
similar technique first performed by Ernest Florens Friedrich Chaldni
(1756--1827), who is often respectfully referred "the father of
acoustics." First, the fine sand is uniformly sprinkled on the plate.
Then, the plate can be resonated, either by a powerful sound wave tuned
to the desired frequencies, by being bowed by a violin bow, or by being
excited mechanically or electromechanically at desired frequencies.
Consequently, the sand disperses randomly due to the vibration of plate.
Some of the sand falls outside the region of plate, while some of the
sand is collected by the nodal regions, which have relatively small
movement, of the plate. Hence, the mode shapes of the plate can be
visualized in this manner, which can be referred to the figures in the
reference site, Violin
Acoustics. The first
seven modes of the top and back plates of violin are presented, with
nodal lines depicted by using black sands.

The air inside the body is also important, especially in the range of
lower frequencies. It is like the air inside a bottle when you blow into
the neck, or, as known as Helmholtz resonance, which has its own modes
of vibration. The air inside the body can communicate with air outside
through the f holes, and the outside air serves as medium carrying waves
from the violin.

See www.violinbridges.co.uk for more
articles on bridges and acoustics.

### Sound radiation

A complete description of sound radiation of a violin should include the
information about radiation intensity as functions both of frequency and
location. The sound radiation can be measured by a microphone connected
to a pressure level meter which is rotatably supported on a stand arm
around the violin, while the violin is fastened at the neck by a clip.
The force is introduced into the violin by using a miniature impact
hammer at the upper edge of the bridge in the direction of bowing. The
detail can be referred to Martin Schleske, master studio for
violinmaking.
The radiation intensity of different frequencies at different locations
can be represented by directional characteristics, or acoustic maps. The
directional characteristics of a violin can be shown in the figure in
the website of Martin
Schleske, where
the radial distance from the center point represents the absolute value
of the sound level (re 1Pa/N) in dB, and the angular coordinate of the
full circle indicates the measurement point around the instrument.
According to the directional characteristics of violins, the principal
radiation directions for the violin in the horizontal plane can be
established. For more detail about the principal radiation direction for
violins at different frequencies, please refer to reference (Meyer
1972).

## References and other links

-   Violin Acoustics
-   Paul Galluzzo\'s
    Homepage
-   Martin Schleske, master studio for
    violinmaking
-   Atelierla Bussiere
-   Fletcher, N. H., and Rossing, T. D., *The physics of musical
    instrument*, Springer-Verlag, 1991
-   Meyer, J., \"Directivity of bowed stringed instruments and its
    effect on orchestral sound in concert halls\", J. Acoustic. Soc.
    Am., 51, 1972, pp. 1994--2009


# Acoustics/Microphone Technique

![](Acoustics_microphone_technique.JPG "Acoustics_microphone_technique.JPG")

## General technique

1.  A microphone should be used whose frequency response will suit the
    frequency range of the voice or instrument being recorded.
2.  Vary microphone positions and distances until you achieve the
    monitored sound that you desire.
3.  In the case of poor room acoustics, place the microphone very close
    to the loudest part of the instrument being recorded or isolate the
    instrument.
4.  Personal taste is the most important component of microphone
    technique. Whatever sounds right to you, *is* right.

## Types of microphones

### Dynamic microphones

These are the most common general-purpose microphones. They do not
require power to operate. If you have a microphone that is used for live
performance, it is probably a dynamic mic.

They have the advantage that they can withstand very high sound pressure
levels (high volume) without damage or distortion, and tend to provide a
richer, more intense sound than other types. Traditionally, these mics
did not provide as good a response on the highest frequencies
(particularly above 10 kHz), but some recent models have come out that
attempt to overcome this limitation.

In the studio, dynamic mics are often used for high sound pressure level
instruments such as drums, guitar amps and brass instruments. Models
that are often used in recording include the Shure SM57 and the
Sennheiser MD421.

### Condenser microphones

These microphones are often the most expensive microphones a studio
owns. They require power to operate, either from a battery or phantom
power, provided using the mic cable from an external mixer or pre-amp.
These mics have a built-in pre-amplifier that uses the power. Some
vintage microphones have a tube amplifier, and are referred to as tube
condensers.

While they cannot withstand the very high sound pressure levels that
dynamic mics can, they provide a flatter frequency response, and often
the best response at the highest frequencies. Not as good at conveying
intensity, they are much better at providing a balanced accurate sound.

Condenser mics come with a variety of sizes of transducers. They are
usually grouped into smaller format condensers, which often are long
cylinders about the size of a nickel coin in diameter, and larger format
condensers, the transducers of which are often about an inch in diameter
or slightly larger.

In the studio, condenser mics are often used for instruments with a wide
frequency range, such as an acoustic piano, acoustic guitar, voice,
violin, cymbals, or an entire band or chorus. On louder instruments they
do not use close miking with condensers. Models that are often used in
recording include the Shure SM81 (small format), AKG C414 (large format)
and Neumann U87 (large format).

### Ribbon microphones

Ribbon microphones are often used as an alternative to condenser
microphones. Some modern ribbon microphones do not require power, and
some do. The first ribbon microphones, developed at RCA in the 1930s,
required no power, were quite fragile and could be destroyed by just
blowing air through them. Modern ribbon mics are much more resiliant,
and can be used with the same level of caution as condenser mics.

Ribbon microphones provide a warmer sound than a condenser mic, with a
less brittle top end. Some vocalists (including Paul McCartney) prefer
them to condenser mics. In the studio they are used on vocals, violins,
and even drums. Popular models for recording include the Royer R121 and
the AEA R84.

## Working distance

### Close miking

When miking at a distance of 1 inch to about 1 foot from the sound
source, it is considered close miking. This technique generally provides
a tight, present sound quality and does an effective job of isolating
the signal and excluding other sounds in the acoustic environment.

#### Bleed

Bleeding occurs when the signal is not properly isolated and the
microphone picks up another nearby instrument. This can make the mixdown
process difficult if there are multiple voices on one track. Use the
following methods to prevent leakage:

-   Place the microphones closer to the instruments.
-   Move the instruments farther apart.
-   Put some sort of acoustic barrier between the instruments.
-   Use directional microphones.

#### A B miking

The A B miking distance rule (ratio 3 - 1) is a general rule of thumb
for close miking. To prevent phase anomalies and bleed, the microphones
should be placed at least three times as far apart as the distance
between the instrument and the microphone.

!A B Miking

### Distant miking

Distant miking refers to the placement of microphones at a distance of 3
feet or more from the sound source. This technique allows the full range
and balance of the instrument to develop and it captures the room sound.
This tends to add a live, open feeling to the recorded sound, but
careful consideration needs to be given to the acoustic environment.

### Accent miking

Accent miking is a technique used for solo passages when miking an
ensemble. A soloist needs to stand out from an ensemble, but placing a
microphone too close will sound unnaturally present compared the distant
miking technique used with the rest of the ensemble. Therefore, the
microphone should be placed just close enough to the soloist so that the
signal can be mixed effectively without sounding completely excluded
from the ensemble.

### Ambient miking

Ambient miking is placing the microphones at such a distance that the
room sound is more prominent than the direct signal. This technique is
used to capture audience sound or the natural reverberation of a room or
concert hall.

## Stereo and surround technique

### Stereo

Stereo miking is simply using two microphones to obtain a stereo
left-right image of the sound. A simple method is the use of a spaced
pair, which is placing two identical microphones several feet apart and
using the difference in time and amplitude to create the image. Great
care should be taken in the method as phase anomalies can occur due to
the signal delay. This risk of phase anomaly can be reduced by using the
X/Y method, where the two microphones are placed with the grills as
close together as possible without touching. There should be an angle of
90 to 135 degrees between the mics. This technique uses only amplitude,
not time, to create the image, so the chance of phase discrepancies is
unlikely.

!Spaced Pair !X/Y
Method

### Surround

To take advantage of 5.1 sound or some other surround setup, microphones
may be placed to capture the surround sound of a room. This technique
essentially stems from stereo technique with the addition of more
microphones. Because every acoustic environment is different, it is
difficult to define a general rule for surround miking, so placement
becomes dependent on experimentation. Careful attention must be paid to
the distance between microphones and potential phase anomalies.

## Placement for varying instruments

### Amplifiers

When miking an amplified speaker, such as for electric guitars, the mic
should be placed 2 to 12 inches from the speaker. Exact placement
becomes more critical at a distance of less than 4 inches. A brighter
sound is achieved when the mic faces directly into the center of the
speaker cone and a more mellow sound is produced when placed slightly
off-center. Placing off-center also reduces amplifier noise.

A bigger sound can often be achieved by using two mics. The first mic
should be a dynamic mic, placed as described in the previous paragraph.
Add to this a condenser mic placed at least 3 times further back
(remember the 3:1 rule), which will pickup the blended sound of all
speakers, as well as some room ambience. Run the mics into separate
channels and combine them to your taste.

### Brass instruments

High sound-pressure levels are produced by brass instruments due to the
directional characteristics of mid to mid-high frequencies. Therefore,
for brass instruments such as trumpets, trombones, and tubas,
microphones should face slightly off of the bell\'s center at a distance
of one foot or more to prevent overloading from wind blasts.

### Guitars

Technique for acoustic guitars is dependent on the desired sound.
Placing a microphone close to the sound hole will achieve the highest
output possible, but the sound may be bottom-heavy because of how the
sound hole resonates at low frequencies. Placing the mic slightly
off-center at 6 to 12 inches from the hole will provide a more balanced
pickup. Placing the mic closer to the bridge with the same working
distance will ensure that the full range of the instrument is captured.

A technique that some engineers use places a large-format condenser mic
12-18 inches away from the 12th fret of the guitar, and a small-format
condenser very close to the strings nearby. Combining the two signals
can produce a rich tone.

### Pianos

Ideally, microphones would be placed 4 to 6 feet from the piano to allow
the full range of the instrument to develop before it is captured. This
isn\'t always possible due to room noise, so the next best option is to
place the microphone just inside the open lid. This applies to both
grand and upright pianos.

### Percussion

One overhead microphone can be used for a drum set, although two are
preferable. If possible, each component of the drum set should be miked
individually at a distance of 1 to 2 inches as if they were their own
instrument. This also applies to other drums such as congas and bongos.
For large, tuned instruments such as xylophones, multiple mics can be
used as long as they are spaced according to the 3:1 rule. Typically,
dynamic mics are used for individual drum miking, while small-format
condensers are used for the overheads.

### Voice

Standard technique is to put the microphone directly in front of the
vocalist\'s mouth, although placing slightly off-center can alleviate
harsh consonant sounds (such as \"p\") and prevent overloading due to
excessive dynamic range.

### Woodwinds

A general rule for woodwinds is to place the microphone around the
middle of the instrument at a distance of 6 inches to 2 feet. The
microphone should be tilted slightly towards the bell or sound hole, but
not directly in front of it.

## Sound Propagation

It is important to understand how sound propagates due to the nature of
the acoustic environment so that microphone technique can be adjusted
accordingly. There are four basic ways that this occurs:

### Reflection

Sound waves are reflected by surfaces if the object is as large as the
wavelength of the sound. It is the cause of echo (simple delay),
reverberation (many reflections cause the sound to continue after the
source has stopped), and standing waves (the distance between two
parallel walls is such that the original and reflected waves in phase
reinforce one another).

### Absorption

Sound waves are absorbed by materials rather than reflected. This can
have both positive and negative effects depending on whether you desire
to reduce reverberation or retain a live sound.

### Diffraction

Objects that may be between sound sources and microphones must be
considered due to diffraction. Sound will be stopped by obstacles that
are larger than its wavelength. Therefore, higher frequencies will be
blocked more easily than lower frequencies.

### Refraction

Sound waves bend as they pass through mediums with varying density. Wind
or temperature changes can cause sound to seem like it is literally
moving in a different direction than expected.

## Sources

-   Huber, Dave Miles, and Robert E. Runstein. *Modern Recording
    Techniques*. Sixth Edition. Burlington: Elsevier, Inc., 2005.
-   Shure, Inc. (2003). *Shure Product Literature.* Retrieved November
    28, 2005, from
    <http://www.shure.com/scripts/literature/literature.aspx>.


# Acoustics/Microphone Design and Operation

![](Acoustics_microphone_design_and_operation.JPG "Acoustics_microphone_design_and_operation.JPG")

## Introduction

Microphones are devices which convert pressure fluctuations into
electrical signals. There are two main methods of accomplishing this
task that are used in the mainstream entertainment industry. They are
known as dynamic microphones and condenser microphones. Piezoelectric
crystals can also be used as microphones but are not commonly used in
the entertainment industry. For further information on piezoelectric
transducers Click
Here.

## Dynamic microphones

This type of microphone converts pressure fluctuations into electrical
current. These microphones work by means of the principle known as
Faraday's Law. The principle states that when an electrical conductor is
moved through a magnetic field, an electrical current is induced within
the conductor. The magnetic field within the microphone is created using
permanent magnets and the conductor is produced in two common
arrangements.

!Figure 1: Sectional View of Moving-Coil Dynamic
Microphone{width="300"}

The first conductor arrangement is made of a coil of wire. The wire is
typically copper and is attached to a circular membrane or piston
usually made from lightweight plastic or occasionally aluminum. The
impinging pressure fluctuation on the piston causes it to move in the
magnetic field and thus creates the desired electrical current. Figure 1
provides a sectional view of a moving-coil microphone.

!Figure 2: Dynamic Ribbon
Microphone{width="300"}

The second conductor arrangement is a ribbon of metallic foil suspended
between magnets. The metallic ribbon is what moves in response to a
pressure fluctuation and in the same manner, an electrical current is
produced. Figure 2 provides a sectional view of a ribbon microphone. In
both configurations, dynamic microphones follow the same principles as
acoustical transducers. For further information about transducers Click
Here.

## Condenser microphones

This type of microphone converts pressure fluctuations into electrical
potentials through the use of changing an electrical capacitor. This is
why condenser microphones are also known as capacitor microphones. An
electrical capacitor is created when two charged electrical conductors
are placed at a finite distance from each other. The basic relation that
describes capacitors is:

**Q=C\*V**

where Q is the electrical charge of the capacitor's conductors, C is the
capacitance, and V is the electric potential between the capacitor's
conductors. If the electrical charge of the conductors is held at a
constant value, then the voltage between the conductors will be
inversely proportional to the capacitance. Also, the capacitance is
inversely proportional to the distance between the conductors. Condenser
microphones utilize these two concepts.

!Figure 3: Sectional View of Condenser
Microphone{width="600"}

The capacitor in a condenser microphone is made of two parts: the
diaphragm and the back plate. Figure 3 shows a section view of a
condenser microphone. The diaphragm is what moves due to impinging
pressure fluctuations and the back plate is held in a stationary
position. When the diaphragm moves closer to the back plate, the
capacitance increases and therefore a change in electric potential is
produced. The diaphragm is typically made of metallic coated Mylar. The
assembly that houses both the back plate and the diaphragm is commonly
referred to as a capsule.

To keep the diaphragm and back plate at a constant charge, an electric
potential must be presented to the capsule. There are various ways of
performing this operation. The first of which is by simply using a
battery to supply the needed DC potential to the capsule. A simplified
schematic of this technique is displayed in figure 4. The resistor
across the leads of the capsule is very high, in the range of 10 mega
ohms, to keep the charge on the capsule close to constant.

!Figure 4: Internal Battery Powered Condenser
Microphone{width="500"}

Another technique of providing a constant charge on the capacitor is to
supply a DC electric potential through the microphone cable that carries
the microphones output signal. Standard microphone cable is known as XLR
cable and is terminated by three pin connectors. Pin one connects to the
shield around the cable. The microphone signal is transmitted between
pins two and three. Figure 5 displays the layout of dynamic microphone
attached to a mixing console via XLR cable.

!Figure 5: Dynamic Microphone Connection to Mixing Console via XLR
Cable{width="700"}

**Phantom Supply/Powering** (Audio Engineering Society, DIN 45596): The
first and most popular method of providing a DC potential through a
microphone cable is to supply +48 V to both of the microphone output
leads, pins 2 and 3, and use the shield of the cable, pin 1, as the
ground to the circuit. Because pins 2 and 3 see the same potential, any
fluctuation of the microphone powering potential will not affect the
microphone signal seen by the attached audio equipment. This
configuration can be seen in figure 6. The +48 V will be stepped down at
the microphone using a transformer and provide the potential to the back
plate and diaphragm in a similar fashion as the battery solution. In
fact, 9, 12, 24, 48 or 52 V can be supplied, but 48 V is the most
frequent.

!Figure 6: Condenser Microphone Powering
Techniques{width="600"}

The second method of running the potential through the cable is to
supply 12 V between pins 2 and 3. This method is referred to as
**T-powering** (also known as Tonaderspeisung, AB powering; DIN 45595).
The main problem with T-powering is that potential fluctuation in the
powering of the capsule will be transmitted into an audio signal because
the audio equipment analyzing the microphone signal will not see a
difference between a potential change across pins 2 and 3 due to a
pressure fluctuation and one due to the power source electric potential
fluctuation.

Finally, the diaphragm and back plate can be manufactured from a
material that maintains a fixed charge. These microphones are termed
electrets. In early electret designs, the charge on the material tended
to become unstable over time. Recent advances in science and
manufacturing have allowed this problem to be eliminated in present
designs.

## Conclusion

Two branches of microphones exist in the entertainment industry. Dynamic
microphones are found in the moving-coil and ribbon configurations. The
movement of the conductor in dynamic microphones induces an electric
current which is then transformed into the reproduction of sound.
Condenser microphones utilize the properties of capacitors. Creating the
charge on the capsule of condenser microphones can be accomplished by
battery, phantom powering, T-powering, and by using fixed charge
materials in manufacturing.

## References

-   Sound Recording Handbook. Woram, John M. 1989.
-   Handbook of Recording Engineering Fourth Edition. Eargle, John.
    2003.

## Microphone manufacturer links

-   AKG
-   Audio
    Technica
-   Audix
-   While Bruel & Kjær produces
    microphones for measurement purposes,
    DPA is the equipment sold for
    recording purposes
-   Electrovoice
-   Josephson Engineering
-   Neumann
    (currently a subsidiary of Sennheiser)
-   Rode
-   Schoeps
-   Sennheiser
-   Shure
-   Wharfedale


# Acoustics/Acoustic Loudspeaker

![](Acoustics_loudspeakers.JPG "Acoustics_loudspeakers.JPG")

The purpose of the acoustic transducer is to convert electrical energy
into acoustic energy. Many variations of acoustic transducers exist,
although the most common is the moving coil-permanent magnet transducer.
The classic loudspeaker is of the moving coil-permanent magnet type.

The classic electrodynamic loudspeaker driver can be divided into three
key components:

1.  The Magnet Motor Drive System
2.  The Loudspeaker Cone System
3.  The Loudspeaker Suspension

!Figure 1 Cut-away of a moving coil-permanent magnet
loudspeaker

## The Magnet Motor Drive System

The main purpose of the Magnet Motor Drive System is to establish a
symmetrical magnetic field in which the voice coil will operate. The
Magnet Motor Drive System is comprised of a front focusing plate,
permanent magnet, back plate, and a pole piece. In figure 2, the
assembled drive system is illustrated. In most cases, the back plate and
the pole piece are built into one piece called the yoke. The yoke and
the front focusing plate are normally made of a very soft cast iron.
Iron is a material that is used in conjunction with magnetic structures
because the iron is easily saturated when exposed to a magnetic field.
Notice in figure 2, that an air gap was intentionally left between the
front focusing plate and the yoke. The magnetic field is coupled through
the air gap. The magnetic field strength (B) of the air gap is typically
optimized for uniformity across the gap. \[1\]

```{=html}
<center>
```
Figure 2 Permanent Magnet Structure

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</center>
```
When a coil of wire with a current flowing is placed inside the
permanent magnetic field, a force is produced. B is the magnetic field
strength, $l$ is the length of the coil, and $I$ is the current flowing
through the coil. The electro-magnetic force is given by the expression
of Laplace :

```{=html}
<center>
```
$d\underline F  = I\underline {dl}  \times \underline B$

```{=html}
</center>
```
$\underline B$ and $\underline {dl}$ are orthogonal, so the force is
obtained by integration on the length of the wire (Re is the radius of a
spire, n is the number of spires and $\underline e_x$ is on the axis of
the coil):

```{=html}
<center>
```
$\underline F  = 2\pi R_eBnI\underline e_x$

```{=html}
</center>
```
This force is directly proportional to the current flowing through the
coil.

```{=html}
<center>
```
![](Magnet2.gif "Magnet2.gif")

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</center>
```
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<center>
```
Figure 3 Voice Coil Mounted in Permanent Magnetic Structure

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</center>
```
The coil is excited with the AC signal that is intended for sound
reproduction, when the changing magnetic field of the coil interacts
with the permanent magnetic field then the coil moves back and forth in
order to reproduce the input signal. The coil of a loudspeaker is known
as the voice coil.

## The loudspeaker cone system

On a typical loudspeaker, the cone serves the purpose of creating a
larger radiating area allowing more air to be moved when excited by the
voice coil. The cone serves a piston that is excited by the voice coil.
The cone then displaces air creating a sound wave. In an ideal
environment, the cone should be infinitely rigid and have zero mass, but
in reality neither is true. Cone materials vary from carbon fiber,
paper, bamboo, and just about any other material that can be shaped into
a stiff conical shape. The loudspeaker cone is a very critical part of
the loudspeaker. Since the cone is not infinitely rigid, it tends to
have different types of resonance modes form at different frequencies,
which in turn alters and colors the reproduction of the sound waves. The
shape of the cone directly influences the directivity and frequency
response of the loudspeaker. When the cone is attached to the voice
coil, a large gap above the voice coil is left exposed. This could be a
problem if foreign particles make their way into the air gap of the
voice coil and the permanent magnet structure. The solution to this
problem is to place what is known as a dust cap on the cone to cover the
air gap. Below a figure of the cone and dust cap are shown.

```{=html}
<center>
```
![](loud_cone.gif "loud_cone.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 6 Cone and Dust Cap attached to Voice Coil

```{=html}
</center>
```
The speed of the cone can be expressed with an equation of a mass-spring
system with a damping coefficient \\xi :

```{=html}
<center>
```
$m\frac{{dv}}{{dt}} + \xi v + k\int {v \, dt}  = Bli$

```{=html}
</center>
```
The current intensity $i$ and the speed $v$ can also be related by this
equation ($U$ is the voltage, $R$ the electrical resistance and $L_b$
the inductance) :

```{=html}
<center>
```
$L_b \frac{{di}}{{dt}} + Ri = U - Blv$

```{=html}
</center>
```
By using a harmonic solution, the expression of the speed is :

```{=html}
<center>
```
$v = \frac{{Bli}}{{\xi  + j\left(m\omega  - \frac{k}{\omega}\right)}}$

```{=html}
</center>
```
The electrical impedance can be determined as the ratio of the voltage
on the current intensity :

```{=html}
<center>
```
$Z = \frac{U}{i} = R + jL\omega  + \frac{{B^2 l^2 }}{{\xi  + j \left(m\omega  - \frac{k}{\omega}\right)}}$

```{=html}
</center>
```
The frequency response of the loudspeaker is provided in Figure 7.

```{=html}
<center>
```
![](Electreson.gif "Electreson.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 7 Electrical impedance

```{=html}
</center>
```
A phenomena of electrical resonance is observable around the frequency
of 100 Hz. Besides, the inductance of the coil makes the impedance
increase from the frequency of 400 Hz. So the range of frequency where
the loudspeaker is used is 100 -- 4000 Hz

## The loudspeaker suspension

Most moving coil loudspeakers have a two piece suspension system, also
known as a flexure system. The combination of the two flexures allows
the voice coil to maintain linear travel as the voice coil is energized
and provide a restoring force for the voice coil system. The two piece
system consists of large flexible membrane surrounding the outside edge
of the cone, called the surround, and an additional flexure connected
directly to the voice coil, called the spider. The surround has another
purpose and that is to seal the loudspeaker when mounted in an
enclosure. Commonly, the surround is made of a variety of different
materials, such as, folded paper, cloth, rubber, and foam. Construction
of the spider consists of different woven cloth or synthetic materials
that are compressed to form a flexible membrane. The following two
figures illustrate where the suspension components are physically at on
the loudspeaker and how they function as the loudspeaker operates.

```{=html}
<center>
```
![](loud_suspension.gif "loud_suspension.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 8 Loudspeaker Suspension System

```{=html}
</center>
```
```{=html}
<center>
```
![](loudspk.gif "loudspk.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 9 Moving Loudspeaker

```{=html}
</center>
```
## Modeling the loudspeaker as a lumped system

Before implementing a loudspeaker into a specific application, a series
of parameters characterizing the loudspeaker must be extracted. The
equivalent circuit of the loudspeaker is key when developing enclosures.
The circuit models all aspects of the loudspeaker through an equivalent
electrical, mechanical, and acoustical circuit. Figure 9 shows how the
three equivalent circuits are connected. The electrical circuit is
comprised of the DC resistance of the voice coil, $R_e$, the imaginary
part of the voice coil inductance, $L_e$, and the real part of the voice
coil inductance, $R_{evc}$. The mechanical system has electrical
components that model different physical parameters of the loudspeaker.
In the mechanical circuit, $M_m$, is the electrical capacitance due to
the moving mass, $C_m$, is the electrical inductance due to the
compliance of the moving mass, and $R_m$, is the electrical resistance
due to the suspension system. In the acoustical equivalent circuit,
$M_a$ models the air mass and $R_a$ models the radiation impedance\[2\].
This equivalent circuit allows insight into what parameters change the
characteristics of the loudspeaker. Figure 10 shows the electrical input
impedance as a function of frequency developed using the equivalent
circuit of the loudspeaker.

```{=html}
<center>
```
![](Eq_circuit.gif "Eq_circuit.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 9 Loudspeaker Analogous Circuit

```{=html}
</center>
```
```{=html}
<center>
```
![](Freq_resp.gif "Freq_resp.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 10 Electrical Input Impedance

```{=html}
</center>
```
## The acoustical enclosure

### Function of the enclosure

The loudspeaker emits two waves : a front wave and a back wave. With a
reflection on a wall, the back wave can be added with the front wave and
produces destructive interferences. As a result, the sound pressure
level in the room is not uniform. At certain positions, the interaction
is additive, and the sound pressure level is higher. On the contrary,
certain positions offer destructive interaction between the waves and
the sound pressure level is lower.

```{=html}
<center>
```
![](louds_without_baffle.gif "louds_without_baffle.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 11 Loudspeaker without baffle producing destructive interferences

```{=html}
</center>
```
The solution is to put a baffle round the loudspeaker in order to
prevent the back wave from interfering with the front wave. The sound
pressure level is uniform in the room and the quality of the loudspeaker
is higher.

```{=html}
<center>
```
![](loudspeakers_baffled.gif "loudspeakers_baffled.gif")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 12 Loudspeakers with infinite baffle and enclosure

```{=html}
</center>
```
### Loudspeaker-external fluid interaction

The external fluid exerts a pressure on the membrane of the loudspeaker
cone. This additive force can be evaluate as an additive mass and an
additive damping in the equation of vibration of the membrane.

```{=html}
<center>
```
$- (m + MS)\omega ^2 \xi  + i\omega (C + RS)\xi  + K\xi  = fe^{i\omega t}$

```{=html}
</center>
```
When the fluid is the air, this additive mass and additive damping are
negligible. For example, at the frequency of 1000 Hz, the additive mass
is 3g.

### Loudspeaker-internal fluid interaction

The volume of air in the enclosure constitutes an additive stiffness.
This is called the acoustic load. In low frequencies, this additive
stiffness can be four times the stiffness of the loudspeaker cone. The
internal air stiffness is very high because of the boundary conditions
inside the enclosure. The walls impose a condition of zero airspeed that
makes the stiffness increase.

```{=html}
<center>
```
Figure 13 Stiffness of the loudspeaker cone and stiffness of the
internal air

```{=html}
</center>
```
The stiffness of the internal air (in red) is fourth time higher than
the stiffness of the loudspeaker cone (in blue). That is why the design
of the enclosure is relevant in order to improve the quality of the
sound and avoid a decrease of the sound pressure level in the room at
some frequencies.

## References

1.  The Loudspeaker Design Cookbook 5th Edition; Dickason, Vance., Audio
    Amateur Press, 1997.
2.  Beranek, L. L. Acoustics. 2nd ed. Acoustical Society of America,
    Woodbridge, NY. 1993.


# Acoustics/Sealed Box Subwoofer Design

A sealed or closed box baffle is the most basic but often the cleanest
sounding sub-woofer box design. The sub-woofer box in its most simple
form, serves to isolate the back of the speaker from the front, much
like the theoretical infinite baffle. The sealed box provides simple
construction and controlled response for most sub-woofer applications.
The slow low end roll-off provides a clean transition into the extreme
frequency range. Unlike ported boxes, the cone excursion is reduced
below the resonant frequency of the box and driver due to the added
stiffness provided by the sealed box baffle.

Closed baffle boxes are typically constructed of a very rigid material
such as MDF (medium density fiber board) or plywood .75 to 1 inch thick.
Depending on the size of the box and material used, internal bracing may
be necessary to maintain a rigid box. A rigid box is important to design
in order to prevent unwanted box resonance.

As with any acoustics application, the box must be matched to the
loudspeaker driver for maximum performance. The following will outline
the procedure to tune the box or maximize the output of the sub-woofer
box and driver combination.

## Closed baffle circuit

The sealed box enclosure for sub-woofers can be modelled as a lumped
element system if the dimensions of the box are significantly shorter
than the shortest wavelength reproduced by the sub-woofer. Most
sub-woofer applications are crossed over around 80 to 100 Hz. A 100 Hz
wave in air has a wavelength of about 11 feet. Sub-woofers typically
have all dimensions much shorter than this wavelength, thus the lumped
element system analysis is accurate. Using this analysis, the following
circuit represents a sub-woofer enclosure system.

![](Circuit_schema.jpg "Circuit_schema.jpg")

where all of the following parameters are in the mechanical mobility
analog

:   $V_e$ - voltage supply
:   $R_e$ - electrical resistance
:   $M_m$ - driver mass
:   $C_m$ - driver compliance
:   $R_m$ - resistance
:   $R_{Ar}$ - rear cone radiation resistance into the air
:   $X_{Af}$ - front cone radiation reactance into the air
:   $R_{Br}$ - rear cone radiation resistance into the box
:   $X_{Br}$ - rear cone radiation reactance into the box

## Driver parameters

In order to tune a sealed box to a driver, the driver parameters must be
known. Some of the parameters are provided by the manufacturer, some are
found experimentally, and some are found from general tables. For ease
of calculations, all parameters will be represented in the SI units
meter/kilogram/second. The parameters that must be known to determine
the size of the box are as follows:

:   $f_0$ - driver free-air resonance
:   $C_{MS}$ - mechanical compliance of the driver
:   $S_D$ - effective area of the driver

#### Resonance of the driver

The resonance of the driver is usually either provided by the
manufacturer or must be found experimentally. It is a good idea to
measure the resonance frequency even if it is provided by the
manufacturer to account for inconsistent manufacturing processes.

The following diagram shows verga and the setup for finding resonance:

Where voltage $V_1$ is held constant and the variable frequency source
is varied until $V_2$ is a maximum. The frequency where $V_2$ is a
maximum is the resonance frequency for the driver.

#### Mechanical compliance

By definition compliance is the inverse of stiffness or what is commonly
referred to as the spring constant. The compliance of a driver can be
found by measuring the displacement of the cone when known masses are
place on the cone when the driver is facing up. The compliance would
then be the displacement of the cone in meters divided by the added
weight in Newtons.

#### Effective area of the driver

The physical diameter of the driver does not lead to the effective area
of the driver. The effective diameter can be found using the following
diagram:

![](Effective_area.jpg "Effective_area.jpg")

From this diameter, the area is found from the basic area of a circle
equation.

## Acoustic compliance

From the known mechanical compliance of the cone, the acoustic
compliance can be found from the following equation:

$V_{as} = P C^2 C_{ms} S_d^2$

Where $P$ is air density and $C$ the speed of sound at a given
temperature and pressure.

From the driver acoustic compliance, the box acoustic compliance is
found. This is where the final application of the sub-woofer is
considered. The acoustic compliance of the box will determine the
percent shift upwards of the resonant frequency. If a large shift is
desire for high SPL applications, then a large ratio of driver to box
acoustic compliance would be required. If a flat response is desired for
high fidelity applications, then a lower ratio of driver to box acoustic
compliance would be required. Specifically, the ratios can be found in
the following figure using line (b) as reference.

$C_{AS} = C_{AB}r$

$r$ - driver to box acoustic compliant ratio

![](Compliance.jpg "Compliance.jpg")

## Sealed box design

#### Volume of box

The volume of the sealed box can now be found from the box acoustic
compliance. The following equation is used to calculate the box volume

V~B~= C~AB~&gam

#### Box dimensions

From the calculated box volume, the dimensions of the box can then be
designed. There is no set formula for finding the dimensions of the box,
but there are general guidelines to be followed. If the driver was
mounted in the center of a square face, the waves generated by the cone
would reach the edges of the box at the same time, thus when combined
would create a strong diffracted wave in the listening space. In order
to best prevent this, the driver should be either be mounted offset of a
square face, or the face should be rectangular.

The face of the box which the driver is set in should not be a square.


# Acoustics/Bass-Reflex Enclosure Design

```{=html}
<div style="float:right;margin:0 0 1em 1em;">
```
![](Bassreflex-Gehäuse_(enclosure).png "Bassreflex-Gehäuse_(enclosure).png")

```{=html}
</div>
```
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.

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 form shown in the diagram below.

![](Vented_box_ckt.gif "Vented_box_ckt.gif")

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.

![](Za0_Zvc_plots.gif "Za0_Zvc_plots.gif")

## 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):

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

```{=html}
<table align=center width=50% cellpadding=10>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$J_1(2ka) \approx ka$

```{=html}
</td>
```
```{=html}
<td>
```
and

```{=html}
</td>
```
```{=html}
<td>
```
$H_1(2ka) \approx \frac{8(ka)^2}{3\pi}$

```{=html}
</td>
```
```{=html}
<td>
```
$\Rightarrow Z_{RAD} \approx j\frac{8\rho_0\omega}{3\pi^2a} = jM_{A1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
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.

![](VB_LF_ckt.gif "VB_LF_ckt.gif")

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:

```{=html}
<table align=center width=40%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\omega_B = \frac{1}{\sqrt{M_{AP}C_{AB}}}$

```{=html}
</td>
```
```{=html}
<td>
```
$\omega_S = \frac{1}{\sqrt{M_{AC}C_{AS}}}$

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

```{=html}
<table align=center width=25%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\omega_0 = \sqrt{\omega_B\omega_S}$

```{=html}
</td>
```
```{=html}
<td>
```
$h = \frac{\omega_B}{\omega_S}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
A parameter known as the *compliance ratio* or *volume ratio*, $\alpha$,
is given by:

{C\_{AB}} = \\frac{V\_{AS}}{V\_{AB}}`</math>`{=html}}}

Other parameters are combined to form what are known as *quality
factors*:

```{=html}
<table align=center width=45%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$Q_L = R_{AL}\sqrt{\frac{C_{AB}}{M_{AP}}}$

```{=html}
</td>
```
```{=html}
<td>
```
$Q_{TS} = \frac{1}{R_{AE}+R_{AS}}\sqrt{\frac{M_{AC}}{C_{AS}}}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
This notation allows for a simpler expression for the resulting transfer
function \[1\]:

= 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}`</math>`{=html}}}

where

```{=html}
<table align=center width=70%>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$a_1 = \frac{1}{Q_L\sqrt{h}}+\frac{\sqrt{h}}{Q_{TS}}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2 = \frac{\alpha+1}{h}+h+\frac{1}{Q_L Q_{TS}}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_3 = \frac{1}{Q_{TS}\sqrt{h}}+\frac{\sqrt{h}}{Q_L}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## 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 \[1\]:

{4\\pi r}`</math>`{=html} }}

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 \[1\]:

= \\frac{\\rho_0s}{4\\pi}\\frac{U_0}{V\_{IN}} = \\frac{\\rho_0Bl}{4\\pi
S_DR_EM_AS}H(s)`</math>`{=html} }}

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:

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$|H(s)|^2 = \frac{\Omega^8}{\Omega^8 + \left(a^2_3 - 2a_2\right)\Omega^6 + \left(a^2_2 + 2 - 2a_1a_3\right)\Omega^4 + \left(a^2_1 - 2a_2\right)\Omega^2 + 1}$

```{=html}
</td>
```
```{=html}
<td>
```
$\Omega = \frac{\omega}{\omega_0}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
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 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 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

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 \[3\]

With the high-pass transformation $\Omega \rightarrow 1/\Omega$,

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 \[1\]:

```{=html}
<table align=center cellspacing=20>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$a_1 = a_3 = \sqrt{4+2\sqrt{2}}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2 = 2+\sqrt{2}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## 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.

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\left|H_{QB3}(\omega)\right|^2 = \frac{(\omega/\omega_{3dB})^8}{(\omega/\omega_{3dB})^8 + B^2(\omega/\omega_{3dB})^2 + 1}$

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

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</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.

![](QB3_gradient.GIF "QB3_gradient.GIF")

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

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$B^2 = a^2_1 - 2a_2$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2^2 + 2 = 2a_1a_3$

```{=html}
</td>
```
```{=html}
<td>
```
$a_3 = \sqrt{2a_2}$

```{=html}
</td>
```
```{=html}
<td>
```
$a_2 > 2 + \sqrt{2}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</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.

![](Butt_vs_Cheb_HP.gif "Butt_vs_Cheb_HP.gif")

The Chebyshev response is defined by \[4\]:

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

```{=html}
<table align=center>
```
```{=html}
<tr>
```
```{=html}
<td valign=center rowspan=2>
```
$C_n(\Omega) = \big\lbrace$

```{=html}
</td>
```
```{=html}
<td>
```
$\cos[n \cos^{-1}(\Omega)]$

```{=html}
</td>
```
```{=html}
<td>
```
$|\Omega| < 1$

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

```{=html}
</td>
```
```{=html}
<td>
```
$|\Omega| > 1$

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

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<td>
```
$C_0(x) = 1$

```{=html}
</td>
```
```{=html}
<td>
```
$C_1(x) = x$

```{=html}
</td>
```
```{=html}
<td>
```
$C_n(x) = 2xC_{n-1} - C_{n-2}$

```{=html}
</td>
```
```{=html}
</table>
```
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 \[1\]:

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 $C_n(x)$,

Applying this equation to $|H(j\Omega)|^2$ \[1\],

```{=html}
<table align=center cellpadding=15>
```
```{=html}
<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>
```
```{=html}
</tr>
```
```{=html}
<tr>
```
```{=html}
<td>
```
$\Omega = \frac{\omega}{\omega_n}$

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

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
Thus, the design equations become \[1\]:

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

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

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

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

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

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

```{=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 \[6\]. 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 \[8\].

## 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

```{=html}
<table align=center border=2>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Name

```{=html}
</th>
```
```{=html}
<th>
```
Electrical Equivalent

```{=html}
</th>
```
```{=html}
<th>
```
Mechanical Equivalent

```{=html}
</th>
```
```{=html}
<th>
```
Acoustical Equivalent

```{=html}
</th>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Voice-Coil Resistance

```{=html}
</th>
```
```{=html}
<td>
```
$R_E$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{ME} = \frac{(Bl)^2}{R_E}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{AE} = \frac{(Bl)^2}{R_ES^2_D}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Driver (Speaker) Mass

```{=html}
</th>
```
```{=html}
<td>
```
See $C_{MEC}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{MD}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AD} = \frac{M_{MD}}{S^2_D}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Driver (Speaker) Suspension Compliance

```{=html}
</th>
```
```{=html}
<td>
```
$L_{CES} = (Bl)^2C_{MS}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{MS}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{AS} = S^2_DC_{MS}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Driver (Speaker) Suspension Resistance

```{=html}
</th>
```
```{=html}
<td>
```
$R_{ES} = \frac{(Bl)^2}{R_{MS}}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{MS}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{AS} = \frac{R_{MS}}{S^2_D}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Enclosure Compliance

```{=html}
</th>
```
```{=html}
<td>
```
$L_{CEB} = \frac{(Bl)^2C_{AB}}{S^2_D}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{MB} = \frac{C_{AB}}{S^2_D}$

```{=html}
</td>
```
```{=html}
<td>
```
$C_{AB}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Enclosure Air-Leak Losses

```{=html}
</th>
```
```{=html}
<td>
```
$R_{EL} = \frac{(Bl)^2}{S^2_DR_{AL}}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{ML} = S^2_DR_{AL}$

```{=html}
</td>
```
```{=html}
<td>
```
$R_{AL}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Acoustic Mass of Port

```{=html}
</th>
```
```{=html}
<td>
```
$C_{MEP} = \frac{S^2_DM_{AP}}{(Bl)^2}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{MP} = S^2_DM_{AP}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AP}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Enclosure Mass Load

```{=html}
</th>
```
```{=html}
<td>
```
See $C_{MEC}$

```{=html}
</td>
```
```{=html}
<td>
```
See $M_{MC}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AB}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Low-Frequency Radiation Mass Load

```{=html}
</th>
```
```{=html}
<td>
```
See $C_{MEC}$

```{=html}
</td>
```
```{=html}
<td>
```
See $M_{MC}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{A1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<th>
```
Combination Mass Load

```{=html}
</th>
```
```{=html}
<td>
```
$C_{MEC} = \frac{S^2_DM_{AC}}{(Bl)^2}$\
$= \frac{S^2_D(M_{AB} + M_{A1}) + M_{MD}}{(Bl)^2}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{MC} = S^2_D(M_{AB} + M_{A1}) + M_{MD}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AC} = M_{AD} + M_{AB} + M_{A1}$\
$= \frac{M_{MD}}{S^2_D} + M_{AB} + M_{A1}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```
## Appendix B: Enclosure Parameter Formulas

![](Vented_enclosure.gif "Vented_enclosure.gif")

Based on these dimensions \[1\],

```{=html}
<table align=center cellpadding=5>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$C_{AB} = \frac{V_{AB}}{\rho_0c^2_0}$

```{=html}
</td>
```
```{=html}
<td>
```
$M_{AB} = \frac{B\rho_{eff}}{\pi a}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$B = \frac{d}{3}\left(\frac{S_D}{S_B}\right)^2\sqrt{\frac{\pi}{S_D}} + \frac{8}{3\pi}\left[1 - \frac{S_D}{S_B}\right]$

```{=html}
</td>
```
```{=html}
<td>
```
$\rho_0 \leq \rho_{eff} \leq \rho_0\left(1 - \frac{V_{fill}}{V_B}\right) + \rho_{fill}\frac{V_{fill}}{V_B}$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td colspan=2>
```
$V_{AB} = V_B\left[1-\frac{V_{fill}}{V_B}\right]\left[1 + \frac{\gamma - 1}{1 + \gamma\left(\frac{V_B}{V_{fill}} - 1\right)\frac{\rho_0c_{air}}{\rho_{fill}c_{fill}}}\right]$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$V_B = hwd$ (inside enclosure volume)

```{=html}
</td>
```
```{=html}
<td>
```
$S_B = wh$ (inside area of the side the speaker is mounted on)

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$c_{air} =$specific heat of air at constant volume

```{=html}
</td>
```
```{=html}
<td>
```
$c_{fill} =$specific heat of filling at constant volume ($V_{filling}$)

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$\rho_0 =$mean density of air (about 1.3 kg/$\rm m^3$)

```{=html}
<td>
```
$\rho_{fill} =$ density of filling

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td>
```
$\gamma =$ ratio of specific heats for air (1.4)

```{=html}
</td>
```
```{=html}
<td>
```
$c_0 =$ speed of sound in air (about 344 m/s)

```{=html}
</tr>
```
```{=html}
<tr align=center>
```
```{=html}
<td colspan=2>
```
$\rho_{eff}$ = effective density of enclosure. If little or no filling
(acceptable assumption in a bass-reflex system but not for sealed
enclosures), $\rho_{eff} \approx \rho_0$

```{=html}
</td>
```
```{=html}
</tr>
```
```{=html}
</table>
```


# Acoustics/Polymer-Film Acoustic Filters

```{=html}
<center>
```
![](Acoustics_polymer_film_acoustic_filters.JPG "Acoustics_polymer_film_acoustic_filters.JPG")

```{=html}
</center>
```
## Introduction

Acoustic filters are used in many devices such as mufflers, noise
control materials (absorptive and reactive), and loudspeaker systems to
name a few. Although the waves in simple (single-medium) acoustic
filters usually travel in gases such as air and carbon-monoxide (in the
case of automobile mufflers) or in materials such as fiberglass,
polyvinylidene fluoride (PVDF) film, or polyethylene (Saran Wrap), there
are also filters that couple two or three distinct media together to
achieve a desired acoustic response. General information about basic
acoustic filter design can be perused at the following wikibook page
Acoustic Filter Design &
Implementation.
The focus of this article will be on acoustic filters that use
multilayer air/polymer film-coupled media as its acoustic medium for
sound waves to propagate through; concluding with an example of how
these filters can be used to detect and extrapolate audio frequency
information in high-frequency \"carrier\" waves that carry an audio
signal. However, before getting into these specific type of acoustic
filters, we need to briefly discuss how sound waves interact with the
medium(media) in which it travels and how these factors can play a role
when designing acoustic filters.

## Changes in Media Properties Due to Sound Wave Characteristics

As with any system being designed, the filter response characteristics
of an acoustic filter are tailored based on the frequency spectrum of
the input signal and the desired output. The input signal may be
infrasonic (frequencies below human hearing), sonic (frequencies within
human hearing range), or ultrasonic (frequencies above human hearing
range). In addition to the frequency content of the input signal, the
density, and, thus, the characteristic impedance of the medium (media)
being used in the acoustic filter must also be taken into account. In
general, the characteristic impedance $Z_0 \,$ for a particular medium
is expressed as\...

```{=html}
<center>
```
`     `$Z_0 = \pm \rho_0 c  \,$`  `$(Pa \cdot s/m)$`  `

```{=html}
</center>
```
where

```{=html}
<center>
```
`     `$\pm \rho_0  \,$` = (equilibrium) density of medium  `$(kg/m^3)\,$\
`     `$c \,$` = speed of sound in medium  `$(m/s) \,$`  `\
`     `

```{=html}
</center>
```
The characteristic impedance is important because this value
simultaneously gives an idea of how fast or slow particles will travel
as well as how much mass is \"weighting down\" the particles in the
medium (per unit area or volume) when they are excited by a sound
source. The speed in which sound travels in the medium needs to be taken
into consideration because this factor can ultimately affect the time
response of the filter (i.e. the output of the filter may not radiate or
attenuate sound fast or slow enough if not designed properly). The
intensity $I_A \,$ of a sound wave is expressed as\...

```{=html}
<center>
```
`    `$I_A = \frac{1}{T}\int_{0}^{T} pu\quad dt = \pm \frac{P^2}{2\rho_0c}  \,$`    `$(W/m^2) \,$`. `

```{=html}
</center>
```
$I_A  \,$ is interpreted as the (time-averaged) rate of energy
transmission of a sound wave through a unit area normal to the direction
of propagation, and this parameter is also an important factor in
acoustic filter design because the characteristic properties of the
given medium can change relative to intensity of the sound wave
traveling through it. In other words, the reaction of the particles
(atoms or molecules) that make up the medium will respond differently
when the intensity of the sound wave is very high or very small relative
to the size of the control area (i.e. dimensions of the filter, in this
case). Other properties such as the elasticity and mean propagation
velocity (of a sound wave) can change in the acoustic medium as well,
but focusing on frequency, impedance, and/or intensity in the design
process usually takes care of these other parameters because most of
them will inevitably be dependent on the aforementioned properties of
the medium.

## Why Coupled Acoustic Media in Acoustic Filters?

In acoustic transducers, media coupling is employed in acoustic
transducers to either increase or decrease the impedance of the
transducer, and, thus, control the intensity and speed of the signal
acting on the transducer while converting the incident wave, or initial
excitation sound wave, from one form of energy to another (e.g.
converting acoustic energy to electrical energy). Specifically, the
impedance of the transducer is augmented by inserting a solid structure
(not necessarily rigid) between the transducer and the initial
propagation medium (e.g. air). The reflective properties of the inserted
medium is exploited to either increase or decrease the intensity and
propagation speed of the incident sound wave. It is the ability to
alter, and to some extent, control, the impedance of a propagation
medium by (periodically) inserting (a) solid structure(s) such as thin,
flexible films in the original medium (air) and its ability to
concomitantly alter the frequency response of the original medium that
makes use of multilayer media in acoustic filters attractive. The
reflection factor and transmission factor $\hat{R} \,$ and $\hat{T} \,$,
respectively, between two media, expressed as\...

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$\hat{R} = \frac{\text{pressure of reflected portion of incident wave}}{\text{pressure of incident wave}} = \frac{\rho c - Z_\mathrm{in}}{\rho c + Z_\mathrm{in}}$

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and

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$\hat{T} = \frac{\text{pressure of transmitted portion of incident wave}}{\text{pressure of incident wave}} = 1 + \hat{R}$,

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are the tangible values that tell how much of the incident wave is being
reflected from and transmitted through the junction where the media
meet. Note that $Z_{in} \,$ is the (total) input impedance seen by the
incident sound wave upon just entering an air-solid acoustic media
layer. In the case of multiple air-columns as shown in Fig. 2,
$Z_\mathrm{in} \,$ is the aggregate impedance of each air-column layer
seen by the incident wave at the input. Below in Fig. 1, a simple
illustration explains what happens when an incident sound wave
propagating in medium (1) and comes in contact with medium (2) at the
junction of the both media (x=0), where the sound waves are represented
by vectors.

As mentioned above, an example of three such successive air-solid
acoustic media layers is shown in Fig. 2 and the electroacoustic
equivalent circuit for Fig. 2 is shown in Fig. 3 where
$L = \rho_s h_s \,$ = (density of solid material)(thickness of solid
material) = unit-area (or volume) mass, $Z = \rho c = \,$ characteristic
acoustic impedance of medium, and $\beta = k = \omega/c = \,$
wavenumber. Note that in the case of a multilayer, coupled acoustic
medium in an acoustic filter, the impedance of each air-solid section is
calculated by using the following general purpose impedance ratio
equation (also referred to as transfer matrices)\...

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$\frac{Z_a}{Z_0} = \frac{\left( \frac{Z_b}{Z_0} \right) + j\ \tan(kd)}{1 + j\ \left( \frac{Z_b}{Z_0} \right) \tan(kd)} \,$

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where $Z_b \,$ is the (known) impedance at the edge of the solid of an
air-solid layer (on the right) and $Z_a \,$ is the (unknown) impedance
at the edge of the air column of an air-solid layer.

## Effects of High-Intensity, Ultrasonic Waves in Acoustic Media in Audio Frequency Spectrum

When an ultrasonic wave is used as a carrier to transmit audio
frequencies, three audio effects are associated with extrapolating the
audio frequency information from the carrier wave: (a) beating effects,
(b) parametric array effects, and (c) radiation pressure.

Beating occurs when two ultrasonic waves with distinct frequencies
$f_1 \,$ and $f_2 \,$ propagate in the same direction, resulting in
amplitude variations which consequently make the audio signal
information go in and out of phase, or "beat", at a frequency of
$f_1 - f_2 \,$.

Parametric array 1
effects occur when the intensity of an ultrasonic wave is so high in a
particular medium that the high displacements of particles (atoms) per
wave cycle changes properties of that medium so that it influences
parameters like elasticity, density, propagation velocity, etc. in a
non-linear fashion. The results of parametric array effects on
modulated, high-intensity, ultrasonic waves in a particular medium (or
coupled media) is the generation and propagation of audio frequency
waves (not necessarily present in the original audio information) that
are generated in a manner similar to the nonlinear process of amplitude
demodulation commonly inherent in diode circuits (when diodes are
forward biased).

Another audio effect that arises from high-intensity ultrasonic beams of
sound is a static (DC) pressure called radiation pressure. Radiation
pressure is similar to parametric array effects in that amplitude
variations in the signal give rise to audible frequencies via amplitude
demodulation. However, unlike parametric array effects, radiation
pressure fluctuations that generate audible signals from amplitude
demodulation can occur due to any low-frequency modulation and not just
from pressure fluctuations occurring at the modulation frequency
$\omega_M \,$ or beating frequency $f_1 - f_2 \,$.

## An Application of Coupled Media in Acoustic Filters

Figs. 1 - 3 were all from a research paper entitled New Type of
Acoustics Filter Using Periodic Polymer Layers for Measuring Audio
Signal Components Excited by Amplitude-Modulated High_Intensity
Ultrasonic
Waves
submitted to the Audio Engineering Society (AES) by Minoru Todo, Primary
Innovator at Measurement Specialties, Inc., in the October 2005 edition
of the AES Journal. Figs. 4 and 5 below, also from this paper, are
illustrations of test setups referred to in this paper. Specifically,
Fig. 4 is a test setup used to measure the transmission (of an incident
ultrasonic sound wave) through the acoustic filter described by Figs. 1
and 2. Fig. 5 is a block diagram of the test setup used for measuring
radiation pressure, one of the audio effects mentioned in the previous
section. It turns out that out of all of the audio effects mentioned in
the previous section that are caused by high-intensity ultrasonic waves
propagating in a medium, sound waves produced from radiated pressure are
the hardest to detect when microphones and preamplifiers are used in the
detection/receiver system. Although nonlinear noise artifacts occur due
to overloading of the preamplifier present in the detection/receiver
system, the bulk of the nonlinear noise comes from the inherent
nonlinear noise properties of microphones. This is true because all
microphones, even specialized measurement microphones designed for audio
spectrum measurements that have sensitivity well beyond the threshold of
hearing, have nonlinearities artifacts that (periodically) increase in
magnitude with respect to increase at ultrasonic frequencies. These
nonlinearities essentially mask the radiation pressure generated because
the magnitude of these nonlinearities are orders of magnitude greater
than the radiation pressure. The acoustic (low-pass) filter referred to
in this paper was designed in order to filter out the \"detrimental\"
ultrasonic wave that was inducing high nonlinear noise artifacts in the
measurement microphones. The high-intensity, ultrasonic wave was
producing radiation pressure (which is audible) within the initial
acoustic medium (i.e. air). By filtering out the ultrasonic wave, the
measurement microphone would only detect the audible radiation pressure
that the ultrasonic wave was producing in air. Acoustic filters like
these could possibly be used to detect/receive any high-intensity,
ultrasonic signal that may carry audio information which may need to be
extrapolated with an acceptable level of fidelity.

## References

\[1\] Minoru Todo, \"New Type of Acoustic Filter Using Periodic Polymer
Layers for Measuring Audio Signal Components Excited by
Amplitude-Modulated High-Intensity Ultrasonic Waves,\" Journal of Audio
Engineering Society, Vol. 53, pp. 930--41 (2005 October)

\[2\] Fundamentals of Acoustics; Kinsler et al, John Wiley & Sons, 2000

\[3\] ME 513 Course Notes, Dr. Luc Mongeau, Purdue University

\[4\]
<http://www.ieee-uffc.org/archive/uffc/trans/Toc/abs/02/t0270972.htm>

Created by Valdez L. Gant


# Acoustics/Noise in Hydraulic Systems

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![](Acoustics_noise_in_hydraulic_systems.JPG "Acoustics_noise_in_hydraulic_systems.JPG")

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## Noise in Hydraulic Systems

Hydraulic systems are the most preferred source of power transmission in
most of the industrial and mobile equipments due to their power density,
compactness, flexibility, fast response and efficiency. The field
hydraulics and pneumatics is also known as \'Fluid Power Technology\'.
Fluid power systems have a wide range of applications which include
industrial, off-road vehicles, automotive systems, and aircraft. But,
one of the main problems with the hydraulic systems is the noise
generated by them. The health and safety issues relating to noise have
been recognized for many years and legislation is now placing clear
demands on manufacturers to reduce noise levels \[1\]. Hence, noise
reduction in hydraulic systems demands lot of attention from the
industrial as well as academic researchers. It needs a good
understanding of how the noise is generated and propagated in a
hydraulic system in order to reduce it.

## Sound in fluids

The speed of sound in fluids can be determined using the following
relation.

$c = \sqrt {\frac{K}{\rho}}$ where K - fluid bulk modulus, $\rho$- fluid
density, c - velocity of sound

Typical value of bulk modulus range from **2e9 to 2.5e9 N/m2**. For a
particular oil, with a density of **889 kg/m3**,

speed of sound $c = \sqrt {\frac{2e9}{889}}= 1499.9 m/s$

## Source of Noise

The main source of noise in hydraulic systems is the pump which supplies
the flow. Most of the pumps used are positive displacement pumps. Of the
positive displacement pumps, axial piston swash plate type is mostly
preferred due to their reliability and efficiency.

The noise generation in an axial piston pump can be classified under two
categories (i) fluidborne noise and (ii) Structureborne noise

## Fluidborne Noise (FBN)

Among the positive displacement pumps, highest levels of FBN are
generated by axial piston pumps and lowest levels by screw pumps and in
between these lie the external gear pump and vane pump \[1\]. The
discussion in this page is mainly focused on **axial piston swash plate
type pumps**. An axial piston pump has a fixed number of displacement
chambers arranged in a circular pattern separated from each other by an
angular pitch equal to $\phi = \frac {360}{n}$ where n is the number of
displacement chambers. As each chamber discharges a specific volume of
fluid, the discharge at the pump outlet is sum of all the discharge from
the individual chambers. The discontinuity in flow between adjacent
chambers results in a kinematic flow ripple. The amplitude of the
kinematic ripple can be theoretical determined given the size of the
pump and the number of displacement chambers. The kinematic ripple is
the main cause of the fluidborne noise. The kinematic ripples is a
theoretical value. The actual **flow ripple** at the pump outlet is much
larger than the theoretical value because the **kinematic ripple** is
combined with a **compressibility component** which is due to the fluid
compressibility. These ripples (also referred as flow pulsations)
generated at the pump are transmitted through the pipe or flexible hose
connected to the pump and travel to all parts of the hydraulic circuit.

The pump is considered an ideal flow source. The pressure in the system
will be decided by resistance to the flow or otherwise known as system
load. The flow pulsations result in pressure pulsations. The pressure
pulsations are superimposed on the mean system pressure. Both the **flow
and pressure pulsations** easily travel to all part of the circuit and
affect the performance of the components like control valve and
actuators in the system and make the component vibrate, sometimes even
resonate. This vibration of system components adds to the noise
generated by the flow pulsations. The transmission of FBN in the circuit
is discussed under transmission below.

A typical axial piston pump with 9 pistons running at 1000 rpm can
produce a sound pressure level of more than 70 dBs.

## Structureborne Noise (SBN)

In swash plate type pumps, the main sources of the structureborne noise
are the fluctuating forces and moments of the swash plate. These
fluctuating forces arise as a result of the varying pressure inside the
displacement chamber. As the displacing elements move from suction
stroke to discharge stroke, the pressure varies accordingly from few
bars to few hundred bars. These pressure changes are reflected on the
displacement elements (in this case, pistons) as forces and these force
are exerted on the swash plate causing the swash plate to vibrate. This
vibration of the swash plate is the main cause of **structureborne
noise**. There are other components in the system which also vibrate and
lead to structureborne noise, but the swash is the major contributor.

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![](Pump_noise.png "Pump_noise.png")

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**Fig. 1 shows an exploded view of axial piston pump. Also the flow
pulsations and the oscillating forces on the swash plate, which cause
FBN and SBN respectively are shown for one revolution of the pump.**

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## Transmission

### FBN

The transmission of FBN is a complex phenomenon. Over the past few
decades, considerable amount of research had gone into mathematical
modeling of pressure and flow transient in the circuit. This involves
the solution of wave equations, with piping treated as a distributed
parameter system known as a transmission line \[1\] & \[3\].

Lets consider a simple pump-pipe-loading valve circuit as shown in Fig.
2. The pressure and flow ripple at any location in the pipe can be
described by the relations:

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$\frac {}{} P = Ae^{-i k x} + Be^{+i k x}$ \...\...\...(1)

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$Q = \frac {1}{Z_{0}}(Ae^{-i k x} - Be^{+i k x})$\.....(2)

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where $\frac {}{} A$ and $\frac {}{} B$ are frequency dependent complex
coefficients which are directly proportional to pump (source) flow
ripple, but also functions of the source impedance $\frac {}{} Z_{s}$,
characteristic impedance of the pipe $\frac {}{} Z_{0}$ and the
termination impedance $\frac {}{} Z_{t}$. These impedances ,usually vary
as the system operating pressure and flow rate changes, can be
determined experimentally.

For complex systems with several system components, the pressure and
flow ripples are estimated using the transformation matrix approach. For
this, the system components can be treated as lumped impedances (a
throttle valve or accumulator), or distributed impedances (flexible hose
or silencer). Various software packages are available today to predict
the pressure pulsations.

### SBN

The transmission of SBN follows the classic source-path-noise model. The
vibrations of the swash plate, the main cause of SBN, are transferred to
the pump casing which encloses all the rotating group in the pump
including displacement chambers (also known as cylinder block), pistons,
and the swash plate. The pump case, apart from vibrating itself,
transfers the vibration down to the mount on which the pump is mounted.
The mount then passes the vibrations down to the main mounted structure
or the vehicle. Thus the SBN is transferred from the swash plate to the
main structure or vehicle via pumpcasing and mount.

Some of the machine structures, along the path of transmission, are good
at transmitting this vibrational energy and they even resonate and
reinforce it. By converting only a fraction of 1% of the pump
structureborne noise into sound, a member in the transmission path could
radiate more ABN than the pump itself \[4\].

## Airborne noise (ABN)

Both FBN and SBN , impart high fatigue loads on the system components
and make them vibrate. All of these vibrations are radiated as
**airborne noise** and can be heard by a human operator. Also, the flow
and pressure pulsations make the system components such as a control
valve to resonate. This vibration of the particular component again
radiates airborne noise.

## Noise reduction

The reduction of the noise radiated from the hydraulic system can be
approached in two ways.

\(i\) **Reduction at Source** - which is the reduction of noise at the
pump. A large amount of open literature are available on the reduction
techniques with some techniques focusing on reducing FBN at source and
others focusing on SBN. Reduction in FBN and SBN at the source has a
large influence on the ABN that is radiated. Even though, a lot of
progress had been made in reducing the FBN and SBN separately, the
problem of noise in hydraulic systems is not fully solved and lot need
to be done. The reason is that the FBN and SBN are interrelated, in a
sense that, if one tried to reduce the FBN at the pump, it tends to
affect the SBN characteristics. Currently, one of the main researches in
noise reduction in pumps, is a systematic approach in understanding the
coupling between FBN and SBN and targeting them simultaneously instead
of treating them as two separate sources. Such an unified approach,
demands not only well trained researchers but also sophisticated
computer based mathematical model of the pump which can accurately
output the necessary results for optimization of pump design. The
amplitude of fluid pulsations can be reduced, at the source, with the
use of an hydraulic attenuator(5).

\(ii\) **Reduction at Component level** - which focuses on the reduction
of noise from individual component like hose, control valve, pump mounts
and fixtures. This can be accomplished by a suitable design modification
of the component so that it radiates least amount of noise. Optimization
using computer based models can be one of the ways.

## Hydraulic System noise

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![](Noise.png "Noise.png")

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**Fig.3 Domain of hydraulic system noise generation and transmission
(Figure recreated from \[1\])**

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## References

1\. *Designing Quieter Hydraulic Systems - Some Recent Developments and
Contributions*, Kevin Edge, 1999, Fluid Power: Forth JHPS International
Symposium.

2\. *Fundamentals of Acoustics* L.E. Kinsler, A.R. Frey, A.B.Coppens,
J.V. Sanders. Fourth Edition. John Wiley & Sons Inc.

3\. *Reduction of Axial Piston Pump Pressure Ripple* A.M. Harrison. PhD
thesis, University of Bath. 1997

4\. *Noise Control of Hydraulic Machinery* Stan Skaistis, 1988. MARCEL
DEKKER , INC.

5 Hydraulic Power System Analysis, A. Akers, M. Gassman, & R. Smith,
Taylor & Francis, New York, 2006,