# Engineering Acoustics/Piezoelectric Transducers

# Introduction

Piezoelectricity from the Greek word \"piezo\" means pressure
electricity. Certain crystalline substances generate electric charges
under mechanical stress and conversely experience a mechanical strain in
the presence of an electric field. The piezoelectric effect describes a
situation where the transducing material senses input mechanical
vibrations and produces a charge at the frequency of the vibration. An
AC voltage causes the piezoelectric material to vibrate in an
oscillatory fashion at the same frequency as the input current.

Quartz is the best known single crystal material with piezoelectric
properties. Strong piezoelectric effects can be induced in materials
with an ABO3, Perovskite crystalline structure. \'A\' denotes a large
divalent metal ion such as lead and \'B\' denotes a smaller tetravalent
ion such as titanium or zirconium.

For any crystal to exhibit the piezoelectric effect, its structure must
have no center of symmetry. Either a tensile or compressive stress
applied to the crystal alters the separation between positive and
negative charge sights in the cell causing a net polarization at the
surface of the crystal. The polarization varies directly with the
applied stress and is direction dependent so that compressive and
tensile stresses will result in electric fields of opposite voltages.

# Vibrations & Displacements

Piezoelectric ceramics have non-centrosymmetric unit cells below the
Curie temperature and centrosymmetric unit cells above the Curie
temperature. Non-centrosymmetric structures provide a net electric
dipole moment. The dipoles are randomly oriented until a strong DC
electric field is applied causing permanent polarization and thus
piezoelectric properties.

A polarized ceramic may be subjected to stress causing the crystal
lattice to distort changing the total dipole moment of the ceramic. The
change in dipole moment due to an applied stress causes a net electric
field which varies linearly with stress.

# Dynamic Performance

The dynamic performance of a piezoelectric material relates to how it
behaves under alternating stresses near the mechanical resonance. The
parallel combination of C2 with L1, C1, and R1 in the equivalent circuit
below control the transducers reactance which is a function of
frequency.

## Equivalent Electric Circuit

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

## Frequency Response

The graph below shows the impedance of a piezoelectric transducer as a
function of frequency. The minimum value at fm corresponds to the
resonance while the maximum value at fn corresponds to anti-resonance.

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

# Resonant Devices

Non resonant devices may be modeled by a capacitor representing the
capacitance of the piezoelectric with an impedance modeling the
mechanically vibrating system as a shunt in the circuit. The impedance
may be modeled as a capacitor in the non-resonant case which allows the
circuit to reduce to a single capacitor replacing the parallel
combination.

For resonant devices the impedance becomes a resistance or static
capacitance at resonance. This is an undesirable effect. In mechanically
driven systems this effect acts as a load on the transducer and
decreases the electrical output. In electrically driven systems this
effect shunts the driver requiring a larger input current. The adverse
effect of the static capacitance experienced at resonant operation may
be counteracted by using a shunt or series inductor resonating with the
static capacitance at the operating frequency.

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

# Applications

## Mechanical Measurement

Because of the dielectric leakage current of piezoelectrics they are
poorly suited for applications where force or pressure have a slow rate
of change. They are, however, very well suited for highly dynamic
measurements that might be needed in blast gauges and accelerometers.

## Ultrasonic

High intensity ultrasound applications utilize half wavelength
transducers with resonant frequencies between 18 kHz and 45 kHz. Large
blocks of transducer material is needed to generate high intensities
which makes manufacturing difficult and is economically impractical.
Also, since half wavelength transducers have the highest stress
amplitude in the center, the end sections act as inert masses. The end
sections are often replaced with metal plates possessing a much higher
mechanical quality factor; giving the composite transducer a higher
mechanical quality factor than a single-piece transducer.

The overall electro-acoustic efficiency is:

`             Qm0 = unloaded mechanical quality factor`\
`             QE  = electric quality factor`\
`             QL  = quality factor due to the acoustic load alone`

The second term on the right hand side is the dielectric loss and the
third term is the mechanical loss.

Efficiency is maximized when:

then:

The maximum ultrasonic efficiency is described by:

Applications of ultrasonic transducers include:

` Welding of plastics`\
` Atomization of liquids`\
` Ultrasonic drilling`\
` Ultrasonic cleaning`\
` Ultrasound`\
` Non destructive testing`\
` etc.`

# More Information and Source of Information

MorganElectroCeramics\
Resources for Piezoelectric
Transducers


# Engineering Acoustics/Piezoelectric Acoustic Sensor

## Introduction

Piezoelectric Acoustic Wave technologies have been used for over 60
years. They have many applications for pressure, chemical concentration,
temperature or mass sensors. Their detection mechanism is based on
acoustic wave propagation. An acoustic wave is excited and propagates
through or on the surface of the material. Changes to the
characteristics of the propagation path affect the velocity and/or
amplitude of the wave. Changes in velocity/amplitude can be monitored by
measuring the natural frequency or phase characteristics of the sensor,
which can then be correlated to the corresponding physical or chemical
quantity being measured. [^1] Acoustic waves sensors use piezoelectric
materials to generate and detect acoustic waves. Piezoelectric materials
provide the transduction between electrical and mechanical response
conversion of electrical signal into mechanical acoustic waves and vice
versa. Conventional piezoelectric materials includes quartz, LiNbO3, AlN
and LiTaO3.

## Acoustic Wave Propagation Modes

Piezoelectric acoustic wave devices are described by the mode of wave
propagation through or on a piezoelectric substrate. If the wave
propagates on the surface of the substrate, it is known as a surface
wave; and if wave propagating through the substrate is called a bulk
wave.

### Mechanical waves in sensor devices

Mechanical waves for sensor applications are of two different types:
shear waves and compressional waves. Shear waves (also called S wave)
have particle displacements that are normal to the direction of wave
propagation, as for surface water waves. Compressional waves (also
called P wave) are waves in which the displacement of the particle is
along the same direction as the propagation direction of the wave[^2].

!Shear
wave\|400px
!Compressional
wave\|400px

## Acoustic Wave Technology

Surface acoustic wave (SAW) and bulk acoustic wave (BAW) are two most
commonly used technologies in sensor applications.

### Surface Acoustic Wave

The operation frequency of the SAW device ranges from the MHz to GHz
range, mainly depending on the interdigital transducer's design and
piezoelectric material[^3]:

```{=html}
<center>
```
$f_{res} = \frac{V_{R}}{\lambda}$

```{=html}
</center>
```
where $V_{R}$ is Rayleigh wave velocity determined by material
properties and λ is the wavelength defined as the periodicity of the
IDT. The figure below is SAW delay line configuration, which consists of
two IDTs, one of them acting as the transmitter to generate acoustic
waves, and the other as a receiver, the path between the IDTs is known
as the delay-line. When an electric signal is applied on the
interdigitated electrodes (IDT) with alternate polarity, as shown in
Figure, an alternating regions of tensile and compressive strain between
two fingers of the electrodes due to piezoelectric effect of material. A
mechanical wave is generated at the surface. The mechanical wave
propagates in both directions from the input IDT, only half of the
energy of the mechanical wave propagates across the delay line in the
direction of the output IDT. [^4] The delay-line is sensing area,
usually, the sensor material is deposited on the delay-line for chemical
sensor to absorb the target analytics.

!Surface Acoustic Wave Sensor Interdigitated Transducer
Diagram\|700px
!Surface acoustic wave resulting from opposing polarity of the
electrodes of the IDT

The animation below is time-domain simulation for the 2D structure of
SAW device using COMSOL. The x,y-axis represent the position of model.
The small rectangles on top are electrodes.

!FEM model with SAW\|

#### Sensor Response

The surface acoustic wave is sensitive to changes in the surface
properties of the medium in the delay-line, these changes modulate the
velocity and amplitude of the wave.

The surface wave velocity can be perturbed by various factors, each of
which represents a possible sensor response[^5]

```{=html}
<center>
```
$\frac{\delta v}{v_{0}}= \frac{1}{v_{0}}(\frac{\delta v}{\delta m}\Delta m + \frac{\delta v}{\delta c}\Delta c + \frac{\delta v}{\delta T}\Delta T + ...)$

```{=html}
</center>
```
-   where $v_{0}$ is unperturbed wave velocity, $m$ is mass, $T$ is
    temperature and c is stiffness.

Therefore, this kind of devices can be used in mass, pressure and
temperature sensing applications.

##### Mass sensor

One of the most used surface acoustic waves (SAW) sensor are mass
sensor.

Example of application: Gas Sensor, Bio-sensor

The sensor material is deposited along the propagation path between the
two IDTs. After exposure to a target analytes (e.g. target gas), the
active sensing material of the sensor adsorbs the analytes molecules
only, which causes the mass of the sensing material to increase and the
surface acoustic wave speed to decrease on the propagation path due to
mass loading. This causes a change in the delay time[^6],

```{=html}
<center>
```
$\tau = \frac{L_{path}}{V_{R}}$

```{=html}
</center>
```
where $L_{path}$ is the length of propagation path. By tracking the
delay time change at the receiver IDT, one can infer the concentration
of the target analyte.

```{=html}
<center>
```
$\Delta \tau = \frac{L_{path}}{V_{R}} - \frac{L_{path}}{V_{R^{'}}} \propto concentration$

```{=html}
</center>
```
##### Equivalent circuit

Mason's Crossed-Field model was used to develop the equivalent
electrical circuit for an one period of IDT fingers[^7]. Frequency
dependent resistance blocks were used. Resistance is minimum for the
center frequency of the SAW device, and very high for remaining
frequencies. Thus, the input energy propagates only at the frequencies
in near the resonant frequency. The equivalent circuit below is
implemented using ADS.

### Bulk Acoustic Wave

A bulk acoustic wave is a wave that travels through a piezoelectric
material, as in a quartz delay line. It is also known as a volume
acoustic wave. In some materials, the wave velocity is greater for bulk
acoustic waves than surface acoustic wave because SAW is composed of a
longitudinal and a shear wave. The wave velocity is lower than both of
them. Bulk acoustic waves contain either longitudinal or shear waves
only, and thus propagate faster.

#### Quartz Crystal Microbalance (QCM) technology

QCM is the oldest and simplest acoustic wave device for mass sensors. It
consists of a thin disk of AT-cut quartz with parallel circular
electrodes patterned on both sides. The application of a voltage between
these electrodes results in a shear deformation of the crystal[^8].
!Quartz resonators with front and back
electrodes

The working principle is based on mass-loading, which is similar to SAW
sensor. Bulk adsorption of target analyte onto the coated crystal causes
an increase in effective mass, which reduces the resonant frequency of
the crystal, in direct proportion to the concentration of target
analyte. For ideal sensing material, this sorption process is fully
reversible with no long-term drift effect, giving a highly reliable and
repeatable measurement[^9].

The relation between the frequency shift and the mass-loading can be
obtained from a model developed by Prof. Dr. Günter Sauerbrey from
Tiefenort, Germany, in 1959:

$\Delta f = -\frac{2f_{0}^2}{A \sqrt{\rho_{q}\mu_{q}}}\Delta m$[^10]

-   $f_{0}$ - resonant frequency depends on the wave velocity (v) and
    the piezoelectric material thickness, $f_{0} = \frac{v}{2d}$
-   $\Delta f$ - frequency change
-   $\Delta m$ - mass change
-   $A$ - active area
-   $\rho_{q}$ - density of piezoelectric material
-   $\mu_{q}$ -shear modulus of piezoelectric material

#### Thin-film Bulk Acoustic Resonator (FBAR) technology

FBAR is special case
of QCM with piezoelectric films thicknesses ranging from only several
micrometers down to tenth of micrometers using MEMS technology. They
resonate in the frequency range up to 10 GHz. Their mass sensitivity is
proportional to their resonance frequency.
FBAR can achieve 3X
mass sensitivity compared to QCM. !Thin-film bulk acoustic
resonator(FBAR)_.png "Thin-film bulk acoustic resonator(FBAR)")

## Reference

[^1]: Hoang T 2009 Design and realization of SAW pressure sensor using
    aluminium nitride Dissertation University Joseph Fourier, France

[^2]: (60), H., (69), J., & (68), R. (n.d.). Mechanical waves and shear
    wave induction in soft tissuesteemCreated with Sketch. Retrieved
    April 13, 2018, from
    <https://steemit.com/ultrasonography/@hagbardceline/mechanical-waves-and-shear-wave-induction-in-soft-tissue>

[^3]: H. Wohltjen, "Mechanism of operation and design considerations for
    surface acoustic wave device vapour sensors," Sensors and Actuators,
    vol. 5, no. 4, pp. 307 -- 325, 1984.

[^4]: Kirschner J 2010 Surface acoustic wave sensors (SAWS): design for
    application (www.jaredkirschner.com/
    uploads/9/6/1/0/9610588/saws.pdf)

[^5]: Ricco, A.j., et al. "Surface acoustic wave gas sensor based on
    film conductivity changes." Sensors and Actuators, vol. 8, no. 4,
    1985, pp. 319--333., <doi:10.1016/0250-6874(85)80031-7>.

[^6]: H. Wohltjen, "Mechanism of operation and design considerations for
    surface acoustic wave device vapour sensors," Sensors and Actuators,
    vol. 5, no. 4, pp. 307 -- 325, 1984.

[^7]: Trang Hoang. Design and realization of SAW pressure sensor using
    Aluminum Nitride. Acoustics \[physics.class-ph\]. Université
    Joseph-Fourier - Grenoble I, 2009. English. `<tel-00540305>`{=html}

[^8]: Hoang T 2009 Design and realization of SAW pressure sensor using
    aluminium nitride Dissertation University Joseph Fourier, France

[^9]: <http://www.michell.com/us/technology/quartz-crystal-microbalance.htm>

[^10]:


# Engineering Acoustics/Microphone Technique

## 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.

## Working Distance

### Close Miking

When miking at a distance of 1 inch to about 3 feet 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.

#### Leakage

Leakage 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.

#### 3 to 1 Rule

The 3:1 distance rule is a general rule of thumb for close miking. To
prevent phase anomalies and leakage, the microphones should be placed at
least three times as far from each other as the distance between the
instrument and the microphone.

!3:1 Rule

### 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 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 amplifier, 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.

### 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 windblasts.

### 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.

Some people prefer to use a contact microphone, attached (usually) by a
fairly weak temporary adhesive, however this will give a rather
different sound to a conventional microphone. The primary advantage is
that the contact microphone performance is unchanged as the guitar is
moved around during a performance, whereas with a conventional
microphone on a stand, the distance between microphone and guitar would
be subject to continual variation. Placement of a contact microphone can
be adjusted by trial and error to get a variety of sounds. The same
technique works quite well on other stringed instruments such as
violins.

### 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.

### 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. Several sources also recommend placing the
microphone slightly above the mouth.

### 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 that 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>.

Back to the main
page


# Engineering Acoustics/Sealed Box Subwoofer Design

## Introduction

A sealed or closed box baffle is the most basic but often the cleanest
sounding subwoofer box design. The subwoofer 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 subwoofer 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 subwoofer
box and driver combination.

## Closed Baffle Circuit

The sealed box enclosure for subwoofers can be modeled as a lumped
element system if the dimensions of the box are significantly shorter
than the shortest wavelength reproduced by the subwoofer. Most subwoofer
applications are crossed over around 80 to 100 Hz. A 100 Hz wave in air
has a wavelength of about 11 feet. Subwoofers 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 subwoofer enclosure system.

```{=html}
<center>
```
![](Circuit_schema.jpg "Circuit_schema.jpg")\

```{=html}
</center>
```
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 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 the setup for finding resonance:

```{=html}
<center>
```
\

```{=html}
</center>
```
Where voltage V1 is held constant and the variable frequency source is
vaied
until V2 is a maximum. The frequency where V2 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:

```{=html}
<center>
```
![](Effective_area.jpg "Effective_area.jpg")\

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

C~AS~ = C~MS~S~D~^2^

From the driver acoustic compliance, the box acoustic compliance is
found. This is where the final application of the subwoofer 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 more flattened response is
desire 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 compliance ratio

```{=html}
<center>
```
![](Compliance.jpg "Compliance.jpg")\

```{=html}
</center>
```
## 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~γ

#### 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. The face of the box
which the driver is set in should not be a square. If the driver were
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 either be mounted offset on a
square face, or the face should be rectangular, with the driver closer
to one edge.

The ratios between internal height, width and depth should never be
integer (2:1, 3:1 etc.), as this would encourage the formation of
standing waves inside the box. Some have suggested the use of the
Golden ratio and others the
third root of 2, both of
which are close to each other and close to the
IEC-recommended
ratios for room dimensions (which conform to the same acoustical
requirements). In practice most manufacturers formulate their boxes
based on aesthetic and cost considerations, while ensuring, through
testing, that no major box resonances appear. In high-quality units this
entails the extensive use of rigid in-box bracing, sound absorption
material, sophisticated alloys or polymers, complex geometrical shapes,
including curves, etc.


# Engineering Acoustics/New Acoustic Filter For Ultrasonics Media

## 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
attentuate 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\...

```{=html}
<center>
```
$\hat{R} = \frac{pressure\ of\ reflected\ portion\ of\ incident\ wave}{pressure\ of\ incident\ wave} = \frac{\rho c - Z_{in}}{\rho c + Z_{in}} \,$

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

```{=html}
<center>
```
$\hat{T} = \frac{pressure\ of\ transmitted\ portion\ of\ incident\ wave}{pressure\ of\ incident\ wave} = 1 + \hat{R} \,$,

```{=html}
</center>
```
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,
$Z_{in} \,$ 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)\...

```{=html}
<center>
```
$\frac{Z_a}{Z_0} = \frac{\left( \frac{Z_b}{Z_0} \right) + j\ \tan(kd)}{1 + j\ \left( \frac{Z_b}{Z_0} \right) \tan(kd)} \,$

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

------------------------------------------------------------------------

Back to main page


# Engineering Acoustics/Acoustic Micro Pumps

## Application to Micro Scale Pipes

Acoustic Streaming is ideal for microfluidic systems because it arises
from viscous forces which are the dominant forces in low Reynolds flows
and which usually hamper microfluidic systems. Also, streaming force
scales favorably as the size of the channel, conveying a fluid through
which an acoustic wave propagates, decreases.[^1] Because of acoustic
attenuation via viscous losses, a gradient in the Reynolds stresses is
manifest as a body force that drives acoustic streaming as well as
streaming from Lagrangian components of the flow.[^2] For more
information on the basic theory of acoustic streaming please see
Engineering Acoustics/Acoustic
streaming. When
applied to microchannels, the principles of acoustic streaming must
include bulk viscous effects (dominant far from the boundary layer,
though driven by boundary layer streaming), investigated in the classic
solution developed extensively by Nyborg[^3] in 1953 as well as
streaming inside the boundary layer. In a micromachined channel, the
dimensions of the channels are on the order of boundary layer thickness,
so both the inner and outer boundary layer streaming must be evaluated
to have a precise prediction for flow rates in acoustic streaming
micropumps.

The derivation that follows is for a circular channel of constant cross
section assuming that the incident acoustic wave is planar and bound
within the channel filled with a viscous fluid.[^4] The acoustic wave
has a known amplitude and fills the entire cross-section and there are
no reflections of the acoustic wave. The walls of the channel are also
assumed to be rigid. This is important, because rigid boundary
interaction results in boundary layer streaming that dominates the flow
profile for channels on the order of or smaller than the boundary layer
associated with viscous flow in a pipe. This derivation follows from the
streaming equations developed by Nyborg who starts with the compressible
continuity equation for a Newtonian fluid and the Navier-Stokes and
dynamic equations to get an expression for the net force per unit
volume. Eckart[^5] uses the method of successive approximations with the
pressure, velocity, and density expressed as the sum of first and second
order terms. Since the first order terms account for the oscillating
portion of the variables, the time average is zero. The second order
terms arise from streaming and are time independent contributions to
velocity, density, and pressure. These non-linear effects due to viscous
attenuation of the acoustic radiation in the fluid are responsible for a
constant streaming velocity\[1\].

Then, the expansion (through the method of approximations) of the
variables are substituted into the standard force balance equations
describing a fluid resulting in two equations\[5\] where:

$(1)\  -F=-\nabla p_2 + \left(\beta_\mu + \frac{4}{3}\mu \right)\nabla \nabla u_2 - \mu \nabla\times\nabla\times u_2$

$(2)\  -F\equiv \rho_0 |\left(u_1\cdot\nabla u_1\right) + u_1\left(\nabla\cdot u_1\right)|$

where the signifier $|expression|$ denotes time average, $F$ is the body
force density, $\beta_\mu$ is the bulk viscosity, $p_2$ is the second
order pressure, $\mu$ is the dynamic viscosity, $\rho_0$ is the density,
$u_2$ is the streaming velocity, and $u_1$ is the acoustic velocity. The
acoustic velocity, represented two dimensionally in axial and radial
directions respectively, is described by:

$(3)\  u_{1x}=V_ae^{-(\alpha+ik)x} \left(1-e^{-(1+i)\zeta z}\right) e^{i\omega t}$

$(4)\  u_{1z}=\frac{-V_ae^{-(\alpha+ik)x} \left(\alpha+ik\right) \left(1-e^{-(1+i)\zeta z}\right) e^{i\omega t}}{\left(1+i\right)\zeta}$

where

$\zeta=\frac{\omega\rho_0}{2\mu}$

where $V_a$ is the acoustic velocity at the source,
$k=\frac{\omega}{c_0}$is the wave number, $c_0$ is the velocity of sound
in the fluid, and $\alpha$ is the acoustic absorption coefficient. The
$\zeta$ term describes the viscous penetration depth, or how large the
boundary layer is. The components of the acoustic velocity given in
Equation (3) and (4) can be substituted into Equation (2) to solve for
the first-order body force. This gives the one-dimensional body force
per unit volume in axial and radial components, respectively:

$(5)\  F_{xv}=\rho_0 V_a^2 e^{-2\alpha x}$

$(6)\  F_{xb}=\frac{1}{2}\rho_0 V_a^2 e^{-2\alpha x}\left[ke^{-\zeta z}\left(cos(\zeta z)+sin(\zeta z)-e^{-\zeta z}\right)+\alpha e^{-\zeta z}\left(e^{-\zeta z}-3(cos(\zeta z)+sin(\zeta z)\right)\right].$

$F_{xv}$ and $F_{xb}$ are expressions for the body force due to viscous
losses and due to the acoustic wave touching the rigid boundary\[5\].
With no-slip boundary conditions imposed on Equation (1), with Equations
(5) and (6) inserted, the streaming velocity $u_2$ can be found. The
differential pressure is assumed to be zero and static head can be
derived by evaluating Equation (1) with a boundary condition of zero net
flow through any fluid element. The solution of Equation(1) for the
streaming velocity profile in two terms relating to the viscous effects
(outer boundary layer streaming) and the boundary layer effects (inner
boundary layer streaming), respectively, results in:

$(7)\  u_{2v}=\frac{\alpha\rho_0 V_a^2 h^2}{2\mu}\left(\frac{z}{h}\right)\left(1-\frac{z}{h}\right)$

$(8)\  u_{2b}=\frac{V_a^2}{4c_0}\left[1+2e^{-\zeta z} \left(sin(\zeta z)-cos(\zeta z)\right)+e^{-2\zeta z} \right]$

These two expressions are summed when calculating the velocity profile
across the diameter of the pipe. With no-slip conditions, the outer
boundary layer streaming contribution to the acoustic streaming velocity
decreases as the diameter decreases with a with a profile similar to
Hele-Shaw flow in infinitely wide rectangular channels \[7\]. Figure 1
shows this diameter scaling effect in water with an acoustic velocity
$A=.1m/s$ and a driving frequency of 2 MHz. !Figure 1: Scaling of the
velocity profile with different channel dimensions in
microns.{width="380"}
Many groups such as Rife *et al.* \[7\], are underestimating the
possibilities that acoustic streaming has to offer in channels less than
$10\mu m$ because the inner boundary layer streaming velocity is
ignored. The boundary layer effects are present regardless of diameter.
In water, the acoustic boundary layer is about 1 micron, therefore, for
pipes with diameters on the order ten microns or less, there is a marked
increase in the streaming velocity.[^6] From the velocity profile of the
inner boundary layer streaming in Figure 2, the contribution of the
boundary layer factors in favorably as the diameter of the channel
decreases. Note that the magnitude of the inner boundary layer streaming
is not affected by the diameter and that the percentage of the channel
experiencing the boundary streaming decreases as channel diameter
increases. !Figure 2: Percentage of channel diameter feeling influence
of inner boundary layer streaming for different channel
dimensions.{width="380"}
Then the total flow velocity profile, with both the viscous and boundary
layer effects in Figure 3, takes on a flow profile that becomes more
plug-like as the diameter of the channel decreases. !Figure 3:
Normalized total acoustic streaming velocity profile showing effect of
channel
diameter.{width="380"}
Driving frequency does have an effect on the velocity profile for a
channel of constant diameter that experiences a sizable contribution
from the boundary layer. The frequency dependence on the inner boundary
layer contribution is evident for a $10\mu meter$ channel with typical
paramaters for water and an acoustic velocity $A=.1m/s$ in Figure 4.
Note that the viscous contribution to acoustic streaming is also shown,
but does not exhibit a frequency dependence. For small channels (less
than 10 microns), the inner boundary layer streaming affects a more
sizable portion of the channel at lower frequencies. !Figure 4: The
inner boundary layer streaming is shown to take up a larger percentage
of the channel dimensions as the driving frequency decreases, while the
viscous contribution is
unnaffected.{width="380"}
The total acoustic streaming flow profile is then given in Figure 5.
From this plot, matching the driving frequency to channel geometry is
important to achieve the maximum flow velocity for micro-nano fluidic
devices. !Figure 5: Total acoustic streaming velocity profile with
effect of driving frequency. The match of 2Mhz and 10 micron channel
yields the greatest volume
flow.{width="380"}

## Actuation in Microfluidics

In microfluidic systems, a piezoelectric actuator can be used to impart
an acoustic wave field in a liquid. This effect is even imparted through
the walls of the device. The advantage is that the actuator does not
need be in contact with the working fluid.[^7] Since the streaming
effect occurs normal to the resonator, there may be difficulties in
coupling an actuator with typical micromachining techniques which
generally yield 2-D layouts of microfluidic networks. The solutions
developed for acoustic streaming assume that the acoustic wave is planar
with respect to the channel axis. Therefore a configuration that results
in the most predictable flow is one in which the acoustic wave source
(piezoelectric bulk acoustic resonator) is placed such that the channel
is axially oriented to the normal of the actuator surface.[^8] Figure 6
shows such a configuration from a view looking down onto the device.
![Figure 6: Micro pump utilizing acoustic streaming based on one
developed by Rife *et al.* \[7\]. The BAW piezo actuators are in black
and the arrows indicate the direction of
flow.](Acoustic_micro_pump1.JPG "Figure 6: Micro pump utilizing acoustic streaming based on one developed by Rife et al. [7]. The BAW piezo actuators are in black and the arrows indicate the direction of flow."){width="380"}
The piezo actuators are in black. This cartoon of a micromachined device
is based on one created by Rife *et al.* \[7\]. The dimensions of their
device where on the order of 1.6 mm square (much greater than the size
of the boundary layer), which makes their predictions using classical
solutions by Nyborg that do not include inner boundary layer streaming
valid, as can be seen in Figure 3 where channels much larger than the
boundary layer size are relatively uninfluenced by that part of the
acoustic streaming. However, employing this configuration in the context
of microfabrication techniques is difficult for very small channels.
Rife *et al.* \[7\], managed to place piezoelectric actuators oriented
perpendicular to the opened ends of channels milled into a block of PMMA
manually, albeit their channel dimensions are much larger than that at
which the boundary layer effects dominate or contribute significantly.
For smaller channels, the only option is to put the actuators on the
underside or top of a micro machined fluidic circuit \[8\]. This
configuration, shown in Figure 7 , results in acoustic wave reflections.
Reflections or standing waves will complicate the streaming analysis.
![Figure 7: Planar micromachined micro pump based off one created by
Hashimoto *et al.* \[8\]. Every other BAW is actuated to drive flow in
one direction. The black rectangles represent activated BAW piezo
devices. By shutting them off and actuating the brown ones, the flow can
be
reversed.](Acoustic_micro_pump_2.jpg "Figure 7: Planar micromachined micro pump based off one created by Hashimoto et al. [8]. Every other BAW is actuated to drive flow in one direction. The black rectangles represent activated BAW piezo devices. By shutting them off and actuating the brown ones, the flow can be reversed."){width="380"}

Another option for instigating acoustic streaming results from the
attenuation of surface acoustic waves (SAW) in contact with a fluid
medium \[6\]. In this case, the transverse component of a Rayleigh wave
(or a Lamb wave) propagating along a surface in contact with a fluid is
effectively transferred into a compression wave in the fluid. The energy
in the SAW is dissipated by the fluid and little disturbance is felt in
the substrate far from the SAW source (interdigital piezo actuators).
Figure 8 is a cartoon of the principle. ![Figure 8: Conversion of SAW,
in a solid substrate in contact with a liquid, to planar acoustic
pressure waves
\[6\].](SAW_interaction_with_liquid.jpg "Figure 8: Conversion of SAW, in a solid substrate in contact with a liquid, to planar acoustic pressure waves [6]."){width="380"}
This is the case so long as the velocity of the SAW is greater than the
acoustic velocity of the liquid. The compression wave radiating from the
surface leaves at the Rayleigh angle given by:

$(9)\ \phi_R = arcsin(\frac{V_a}{V_R})$

where $V_R$ is the velocity of the surface acoustic wave. Given that
angle then, theoretically, two actuators producing SAWs could be placed
opposite each other to produce a standing wave field in the fluid across
the channel and a traveling planar wave parallel to the channel axis.
Figure 9 shows how this could be done. !Figure 9: Parallel SAW acoustic
streaming
micro-pump{width="380"}
Finally, a very interesting pump that uses acoustic standing waves and a
diffuser nozzle is shown in Figure 10, which has been developed by
Nabavi and Mongeau.[^9] !Figure 10: Valveless acoustic standing wave
micro
pump.{width="380"}
While this pump does not use the same acoustic streaming principles, it
is included because it uses acoustic waves to generate a flow. The
standing wave, induced by relatively small movements of the piston, has
a maximum pressure at the anti-node and a min at the node. Positioning
the inlet and the outlet at these two locations allows fluid to enter
the chamber immediately after the part of the cycle where the pressure
at the anti-node has overcome the discharge pressure at the diffuser
nozzle. Most importantly, the diffuser nozzle outlet has an asymmetric
resistance. After the fluid is ejected and the pressure in the chamber
is temporarily lower than the ambient pressure the fluid does not flow
right back in the outlet but enters at the pressure node, where pressure
is less. This clever design allows for a valveless pumping apparatus.
The forward and back flow resistance of the diffuser nozzle are not the
same, so a net mass flow out of the resonance chamber is observed.

Another interesting pump for precise positioning of fluid droplets in
microchannels that uses similar acoustic standing waves in a resonance
chamber has been developed by Langelier *et al.*.[^10] Instead of a
piezoelectric membrane generating the acoustic standing waves in the
resonance chamber, the resonance chamber is filled with air and
connected to a larger container that has a speaker at one end. Multiple
quarter wavelength resonance chambers are tuned to specific frequencies,
each with a different length and width. Different pipes connected to
each resonance chamber can then be activated with one source, each one
independently depending on which frequencies the speaker is emitting.
Just like the acoustic standing wave pump of Nabavi and Mongeau, an
outlet is located at the point of peak pressure amplitude, which in this
case is at the end of the resonance chamber. With a rectification
structure, an oscillating flux of fluid out of the resonance chamber is
converted into a pulsed flow in the microfluidic channel.

## References

[^1]: K. D. Frampton, et al., \"The scaling of acoustic streaming for
    application in micro-fluidic devices,\" Applied Acoustics, vol. 64,
    pp. 681-692, 2003.

[^2]: J. Lighthill, \"ACOUSTIC STREAMING,\" Journal of Sound and
    Vibration, vol. 61, pp. 391-418, 1978.

[^3]: W. L. Nyborg, \"ACOUSTIC STREAMING DUE TO ATTENUATED PLANE
    WAVES,\" Journal of the Acoustical Society of America, vol. 25, pp.
    68-75, 1953.

[^4]: K. D. Frampton, et al., \"Acoustic streaming in micro-scale
    cylindrical channels,\" Applied Acoustics, vol. 65, pp. 1121-1129,
    Nov 2004.

[^5]: C. Eckart, \"VORTICES AND STREAMS CAUSED BY SOUND WAVES,\"
    Physical Review, vol. 73, pp. 68-76, 1948.

[^6]: G. Lindner, \"Sensors and actuators based on surface acoustic
    waves propagating along solid-liquid interfaces,\" Journal of
    Physics D-Applied Physics, vol. 41, 2008.

[^7]: J. C. Rife, et al., \"Miniature valveless ultrasonic pumps and
    mixers,\" Sensors and Actuators a-Physical, vol. 86, pp. 135-140,
    Oct 2000.

[^8]: K. Hashimoto, et al., \"Micro-actuators employing acoustic
    streaming caused by high-frequency ultrasonic waves,\" Transducers
    97 - 1997 International Conference on Solid-State Sensors and
    Actuators, Digest of Technical Papers, Vols 1 and 2, pp. 805-808,
    1997.

[^9]: Nabavi, M. and L. Mongeau (2009). \"Numerical analysis of high
    frequency pulsating flows through a diffuser-nozzle element in
    valveless acoustic micropumps.\" Microfluidics and Nanofluidics
    7(5): 669-681.

[^10]: S. M. Langelier, et al., \"Acoustically driven programmable
    liquid motion using resonance cavities,\" Proceedings of the
    National Academy of Sciences of the United States of America, vol.
    106, pp. 12617-12622, 2009.


# Engineering Acoustics/Sonic Supercharging of 2 Stroke Engines

## Sonic Supercharging of 2 Stroke Engines

This page of the Engineering Acoustics Wikibook discusses the merits and
design of Tuned Pipes for 2 Stroke Engines. For introductory material on
2 Stroke Engines please see the following links:

Wikipedia 2 Stroke
Engines

HowStuffWorks 2 Stroke
Engines

## Introduction

For a 2 stroke engine the tuned pipe is the section of the exhaust
system that begins at the exhaust port and ends at the end of the
converging section. A tuned pipe is made of between 3 and 4
characteristic sections depending on the desired effect. The figure
below depicts cross sections for 3 typical configurations of tuned pipes
as well as a straight pipe:

!Typical pipes

The purpose of straight and tuned pipes is to utilize the pressure waves
originating from the exhaust port to assist the breathing of the engine.
This is achieved by designing the pipe in such a way that positive and
negative reflected waves arrive back at the exhaust port at an instant
when a low or high pressure is desired. This is beneficial for two
stroke engines because unlike four stroke engines, they do not have
dedicated intake and exhaust strokes and valves.

The following picture labels the various elements of a two-stroke engine
which are referred to in this Wikibook page.

!Two-stroke engine
elements

Furthermore; abbreviations will help as well:

-   In order of accensding crank angle
    -   TDC - Top Dead Center, 0 deg
    -   EPO - Exhaust Port Open
    -   TPO - Transfer Port Open
    -   BDC - Bottom Dead Ceneter, 180 deg
    -   TPC - Transfer Port Close
    -   EPC - Exhaust Port Close

For introductory material on 2 Stroke Engines please see the following
links:

Wikipedia 2 Stroke
Engines

HowStuffWorks 2 Stroke
Engines

## Straight Pipe

The goal of a tuned straight pipe in this application is to use the
reflected negative pressure waves from the open end of the pipe to help
draw out the exhaust gases. By selecting the appropriate length of the
pipe, the reflected rarefaction wave arrives at the exhaust port just as
the transfer port opens thus assisting the flow of fresh mixture into
the cylinder, and exhaust gases out of the cylinder. The figure below
illustrates this action. In the figure, even though the piston has
reached bottom dead center (BDC), fresh mixture continues to enter the
cylinder because the rarefaction wave causes P2 to be smaller than P1. A
key point to note is that the velocity with which the pressure and
rarefaction waves travel down and up the exhaust pipe is for the most
part independent of the engine operating frequency (RPM). Due to this
fact, the conclusion must be made that for a given pipe length there is
an optimal RPM for which the waves will arrive producing the greatest
benefit for the breathing of the engine. At this optimal RPM, the engine
breathes significantly better and hence produces a noticeable increase
in output power. This effect is quantified by calculating the ratio of
fresh mixture to exhaust gases within the cylinder as the compression
stage begins at EPC. If the rarefaction wave is very large, it is
possible that fresh mixture is pulled into the exhaust pipe while both
transfer and exhaust ports are open. This phenomenon is known as short
circuiting the engine and produces undesired effects such as a decrease
in fuel economy and an increase in release of volatile organic
compounds. These negative effects can be mitigated by designing the
exhaust system such that either no fresh mixture is pulled into the
exhaust pipe (i.e. perfectly tuned straight pipe) or further utilizing
the exhaust pressure wave to inhibit short circuiting. For performance
two-stroke engines, the second solution is most often employed by means
of a tuned exhaust pipe known as a tune pipe.

!Schematic of two-stroke engine with straight
pipe

## Tune Pipe

With a converging-belly section-diverging type tune pipe the goal is to
have the diverging section create a returning rarefaction wave and the
converging section create a returning pressure wave. The belly section
acts as an appropriate time delay between the returning waves such that
the pressure wave arrives at the exhaust port after the transfer port
has closed. This pressure wave pushes the excess fresh mixture in the
exhaust pipe from a short circuit, back into the cylinder. Here the
short circuited fresh mixture is actually desired since this allows the
returning pressure wave to \"super charge\" the cylinder giving it more
fresh mixture than if the cylinder were filled at ambient pressure. This
is a similar result to turbo-charging or super-charging a four-stroke
engine. If the mixture contained within the cylinder before combustion
occurs were allowed to expand to ambient pressure, its volume would be
larger than the displacement of the engine. This phenomenon is
quantified as volumetric efficiency; it is calculated as the ratio of
the ambient pressure volume of the fresh charge, divided by the
displacement volume of the engine. The operation of a two-stroke engine
equipped with a properly tuned pipe is shown in the animation below, for
a step by step description of the process, please follow the link below
the animation.

There are exhaust manufacturers now that are mating up tuned pipes to
tuned (ported) engines to get the best possible \'supercharging effect\'
at given rpm\'s. In the past a tuned pipe would have been tested on a
stock engine but the length and shape of the pipe will differ on a
\'tuned\' engine because of the way it is able to rev higher .

Two-stroke engine
operation

## Tune Pipe Design Geometry

The most basic form of a tune pipe is shown in the figure below with
corresponding wave equations.

!geometry and equations for simple
pipe

This pipe consists of an expansion chamber which serves to create both
the returning rarefaction and pressure waves. From reference \[1\], we
know that wave speed in the pipe is effectively independent of engine
RPM and largely dependent on temperature of the gases in the pipe. This
means that a tune pipe with basic geometry operates optimally for only
one specific RPM, as the engine RPM deviates from this optimal value the
timing of the arrival of the returning waves is less optimal for the
volumetric efficiency. The relation between the volumetric efficiency
and the engine RPM is characterized qualitatively by the following
graph:

!Qualitative relation of volumetric efficiency to
RPM

Although the basic tune pipe performs the desired task of increasing the
volumetric efficiency, the narrow RPM band width for which increased
power is available reduces the practicality of the basic pipe since
engines are typically required to operate within a wide range of RPM.
One way to broaden the effective RPM band width of a pipe is to taper
the pipe at sections of increasing and decreasing cross section. To
understand how this works, we can represent a tapered section as many
small step increases/decreases in cross section. Each step will produce
transmitted and reflected waves in the same way as the basic geometry;
however, the overall effect is weaker waves with longer wave lengths
arriving back at the exhaust port. Although the waves have smaller peak
amplitudes, the effect on volumetric efficiency is greater due to the
longer interaction times of the waves with the cylinder and crankcase.
If the number of steps increased to n diverging steps and m converging
steps, the equations shown represent the plane waves as well as the
transmission and reflection factors for each change in cross section.

!geometry and equations for stepped
pipe

!reflection and transmission coefficients for
step

The graph below shows qualitatively how pressure at the exhaust port
varies with crank angle for both basic and tapered pipe geometry.

!qualitative different between EPP for basic and tapered
pipes

The important differences to notice in the graph are the relative
magnitudes and durations of the positive and negative pressure waves
arriving back at the exhaust port at TPO and EPC. In this graph, if we
pin down the waves with respect to time along the horizontal axis and
then we increase or decrease the RPM, the effect will be that the
positions of the port timing will no longer match up with the same
positions of the waves. This is due to the fact that, as mentioned
above, the wave speed is independent of RPM. In more detail if we
increase the RPM it would have the effect of shrinking the port timing
scale while keeping EPO in the same position. If we increase the RPM,
the port timing scale expands with EPO remaining in the same position.

Looking at things the other way around, if we change some aspects of the
pipe geometry we can see how they change wave propagation in the pipe
and hence operation of the pipe with respect to the engine.

-   Length of the pipe between the exhaust port and the diverging
    section (L1) - this length set by the difference in crank angle
    between EPO and TPO and the desired effective RPM range of the pipe.
    Making the section longer would fit a lower RPM range or a greater
    difference between the crank angle of EPO and TPO.
-   Length of the belly section - this length is set by the difference
    in crank angle between TPO and EPC and the desired effective RPM
    range of the engine. This length and L1 are interdependent since the
    crank angles are also interdependent, (e.g. EPC=(0-EPO)).
-   The angle of the diverging/converging sections - changing this angle
    from steep angled cone (interior angle \> 90 degrees) to a shallow
    angle cone (interior angle \< 90 degrees) has the effect of
    broadening out the wave length. This increases the effective band
    RPM band width of the pipe since there greater flexibility of crank
    angle for which an appropriate pressure will at the exhaust port.
    This also has the effect of decreasing the maximum attainable
    volumetric efficiency of the pipe since the peak pressure amplitude
    is diminished by spreading out the waves energy over a longer
    wavelength. Note that if the diameter of section L1 and the belly
    section are kept constant, changing the angle and changing the
    length of the diverging/converging section is geometrically the
    same.
-   The ratio of the cross section of section L1 and the belly section -
    the ratio is largely dependent on the desired angle and length of
    the diverging/converging sections and the minimum diameter desired
    to avoid impeding the flow of the exhaust gases.

## Further Investigation

For further investigate of the operation of two-stroke engines with
tuned exhaust pipes it is most appropriate to analyze actual test data.
For this we can go to the TFX website where they demonstrate their
testing and data analysis software, or read the paper referenced below
titled \"Exhaust Gas Flow Behavior in a Two-Stroke Engine.

For the class presentation I would like to do both given sufficient
time.

## References

1.  Exhaust Gas Flow Behavior in a Two-Stroke Engine; Y. Ikeda T.
    Takahashi, T. Ito, T. Nakajima; Kobe Univ.; SAE 1993
2.  Power Tuning - Two Stroke
    Engines
3.  2 STROKE WIZARD - Tuned Pipe Design
    Software
4.  Ian Williams Tuning MOTA Tune Pipe Design
    Software
5.  Les phénomènes d\'ondes dans les moteurs; M. Borel; Edition TECHNIP;
    2000
6.  Engineering Acoustics Course Notes; McGill University MECH 500; L.
    Mongeau; 2008
7.  Engineering Acoustics; L. Kinsler, A. Frey; 4th Edition; 2000


# Engineering Acoustics/Thermoacoustics

One ordinarily thinks of a sound wave as consisting only of coupled
pressure and position oscillations. In fact, temperature oscillations
accompany the pressure oscillations and when there are spatial gradients
in the temperature oscillations, oscillating heat flow occurs. The
combination of these oscillations produces a rich variety of
"thermoacoustic" effects. In everyday life, the thermal effects of sound
are too small to be easily noticed; for example, the amplitude of the
temperature oscillation in conversational levels of sound is only about
0.0001 °C. However, in an extremely intense sound wave in a pressurized
gas, these thermoacoustic effects can be harnessed to create powerful
heat engines and refrigerators. Whereas typical engines and
refrigerators rely on crankshaft-coupled pistons or rotating turbines,
thermoacoustic engines and refrigerators have no moving parts (or at
most only flexing parts without the need for sliding seals). This
simplicity, coupled with reliability and relatively low cost, has
highlighted the potential of thermoacoustic devices for practical use.
As a result, thermoacoustics is maturing quickly from a topic of basic
scientific research through the stages of applied research and on to
important practical applications\[1\]. Recently, thermoacoustic
phenomena have been employed in the medical field for imaging of
tissues.

## History

The history of thermoacoustic engines is long but sparsely populated. A
review of Putnam and Dennis\[2\] describes experiments of Byron
Higgins\[3\] in 1777 in which acoustic oscillations in a large pipe were
excited by suitable placement of a hydrogen flame inside. The Rijke
tube\[4\], an early extension of Higgins\' work, is well known to modern
acousticians. Higgins\' research eventually evolved into the modern
science of pulse combustions\[5\] whose applications have included the
German V-I rocket (the \"buzz bomb\") used in World War II and the
residential pulse combustion furnace introduced by Lennox, Inc., in
1982. The Sondhaus tube is the earliest thermoacoustic engine that is a
direct antecedent of the thermoacoustic prime movers. Over 100 years
ago, glass blowers noticed that when a hot glass bulb was attached to a
cool glass tubular stem, the stem tip sometimes emitted sound, and
Sondhauss quantitatively investigated the relation between the pitch of
the sound and the dimensions of the apparatus.

The English physicist, Lord Rayleigh explained the Sondhauss tube
qualitatively in 1896: \"In almost all cases where heat is communicated
to a body, expansion ensues and this expansion may be made to do
mechanical work. If the phases of the forces thus operative be
favorable, a vibration may be maintained. For the sake of simplicity, a
tube, hot at the closed end and getting gradually cooler towards the
open end, may be considered. At a quarter of a period before the phase
of greatest condensation, the air is moving inwards, i.e., towards the
closed end, and therefore is passing from colder to hotter parts of the
tube; but in fact the adjustment of temperature takes time, and thus the
temperature of the air deviates from that of the neighboring parts of
the tube, inclining towards the temperature of that part of the tube
from which the air has just come. From this it follows that at the phase
of greatest condensation heat is received by the air, and at the phase
of greatest rarefaction heat is given up from it, and thus there is a
tendency to maintain the vibrations.\"

The history of imposing acoustic oscillations on a gas to cause heat
pumping and refrigeration effects is even briefer and more recent than
the history of thermoacoustic prime movers. In a device called a
pulse-tube refrigerator, Gifford and Longsworth produced significant
refrigeration by applying a very low-frequency high-amplitude pressure
oscillation to the gas in a tube. As they explained the phenomenon, \"If
any closed chamber is pressurized and depressurized by delivery and
exhaustion of gas from one point on its surface and the flow is
essentially smooth, heat pumping will occur away from the point on its
surface,\" because of the temperature changes that accompany the
pressure changes in the gas, and their time-phasing relative to the
oscillatory gas flow.

## Principle of operation

When a sound wave is sent down a half-wavelength tube with a vibrating
diaphragm or a loudspeaker, the pressure pulsations make the gas inside
slosh back and forth. This forms regions where compression and heating
take place, plus other areas characterized by gas expansion and cooling.
!Working
principle{width="500"}!Working
principle{width="500"}

A thermoacoustic refrigerator is a resonator cavity that contains a
stack of thermal storage elements (connected to hot and cold heat
exchangers) positioned so the back-and-forth gas motion occurs within
the stack. The oscillating gas parcels pick up heat from the stack and
deposit it to the stack at a different location. The device \"acts like
a bucket brigade\" to remove heat from the cold heat exchanger and
deposit it at the hot heat exchanger, thus forming the basis of a
refrigeration unit.
1
The governing mathematical equations of the thermoacoustic phenomenon
are given below.

!Working principle{width="700"}

### Standing-wave systems

The thermoacoustic engine (TAE) is a device that converts heat energy
into work in the form of acoustic energy. A thermoacoustic engine is
operating using the effects that arise from the resonance of a
standing-wave in a gas.  A standing-wave thermoacoustic engine typically
has a thermoacoustic element called the "stack". A stack is a solid
component with pores that allow the operating gas fluid to oscillate
while in contact with the solid walls. The oscillation of the gas is
accompanied with the change of its temperature. Due to the introduction
of solid walls into the oscillating gas, the plate modifies the
original, unperturbed temperature oscillations in both magnitude and
phase for the gas about a thermal penetration depth δ=√(2k/ω) away from
the plate, where k is the thermal diffusivity of the gas and ω=2πf is
the angular frequency of the wave. Thermal penetration depth is defined
as the distance that heat can diffuse though the gas during a time 1/ω.
In air oscillating at 1000 Hz, the thermal penetration depth is about
0.1 mm. Standing-wave TAE must be supplied with the necessary heat to
maintain the temperature gradient on the stack. This is done by two heat
exchangers on both sides of the stack.[^1]

If we put a thin horizontal plate in the sound field the thermal
interaction between the oscillating gas and the plate leads to
thermoacoustic effects. If the thermal conductivity of the plate
material would be zero the temperature in the plate would exactly match
the temperature profiles as in Fig. 1b. Consider the blue line in Fig.
1b as the temperature profile of a plate at that position. The
temperature gradient in the plate would be equal to the so-called
critical temperature gradient. If we would fix the temperature at the
left side of the plate at ambient temperature *T*~a~ (e.g. using a heat
exchanger) then the temperature at the right would be below *T*~a~. In
other words: we have produced a cooler. This is the basis of
thermoacoustic cooling as shown in Fig. 2b which represents a
thermoacoustic refrigerator. It has a loudspeaker at the left. The
system corresponds with the left half of Fig. 1b with the stack in the
position of the blue line. Cooling is produced at temperature *T*~L~.

It is also possible to fix the temperature of the right side of the
plate at *T*~a~ and heat up the left side so that the temperature
gradient in the plate would be larger than the critical temperature
gradient. In that case we have made an engine (prime mover) which can
e.g. produce sound as in Fig. 2a. This is a so-called thermoacoustic
prime mover. Stacks can be made of stainless steel plates but the device
works also very well with loosely packed stainless steel wool or
screens. It is heated at the left, e.g., by a propane flame and heat is
released to ambient temperature by a heat exchanger. If the temperature
at the left side is high enough the system starts to produces a loud
sound.

Thermoacoustic engines still suffer from some limitations, including
that:

-   The device usually has low power to volume ratio.
-   Very high densities of operating fluids are required to obtain high
    power densities
-   The commercially-available linear alternators used to convert
    acoustic energy into electricity currently have low efficiencies
    compared to rotary electric generators
-   Only expensive specially-made alternators can give satisfactory
    performance.
-   TAE uses gases at high pressures to provide reasonable power
    densities which imposes sealing challenges particularly if the
    mixture has light gases like helium.
-   The heat exchanging process in TAE is critical to maintain the power
    conversion process. The hot heat exchanger has to transfer heat to
    the stack and the cold heat exchanger has to sustain the temperature
    gradient across the stack. Yet, the available space for it is
    constrained with the small size and the blockage it adds to the path
    of the wave. The heat exchange process in oscillating media is still
    under extensive research.
-   The acoustic waves inside a thermoacoustic engines operated at large
    pressure ratios suffer many kinds of non-linearities such as
    turbulence which dissipates energy due to viscous effects,  harmonic
    generation of different frequencies that carries acoustic power in
    frequencies other than the fundamental frequency.

The performance of thermoacoustic engines usually is characterized
through several indicators as follows:[^2]

-   The first and second law efficiencies.
-   The onset temperature difference, defined as the minimum temperature
    difference across the sides of the stack at which the dynamic
    pressure is generated.
-   The frequency of the resultant pressure wave, since this frequency
    should match the resonance frequency required by the load device,
    either a thermoacoustic refrigerator/heat pump or a linear
    alternator.
-   The degree of harmonic distortion, indicating the ratio of higher
    harmonics to the fundamental mode in the resulting dynamic pressure
    wave.
-   The variation of the resultant wave frequency with the TAE operating
    temperature

### Travelling-wave systems

Figure 3 is a schematic drawing of a travelling-wave thermoacoustic
engine. It consists of a resonator tube and a loop which contains a
regenerator, three heat exchangers, and a bypass loop. A regenerator is
a porous medium with a high heat capacity. As the gas flows back and
forth through the regenerator it periodically stores and takes up heat
from the regenerator material. In contrast to the stack, the pores in
the regenerator are much smaller than the thermal penetration depth, so
the thermal contact between gas and material is very good. Ideally the
energy flow in the regenerator is zero, so the main energy flow in the
loop is from the hot heat exchanger via the pulse tube and the bypass
loop to the heat exchanger at the other side of the regenerator (main
heat exchanger). The energy in the loop is transported via a travelling
wave as in Fig. 1c, hence the name travelling-wave systems. The ratio of
the volume flows at the ends of the regenerator is *T*~H~/*T*~a~, so the
regenerator acts as a volume-flow amplifier. Just like in the case of
the standing-wave system the machine "spontaneously" produces sound if
the temperature *T*~H~ is high enough. The resulting pressure
oscillations can be used in a variety of ways such as in producing
electricity, cooling, and heat pumping.

## Thermoacoustic machines

There are two basic kinds of thermoacoustic machines:

1.  Thermoacoustic prime movers
2.  Thermoacoustic refrigerator

!A standing wave demonstration
refrigerator{width="550"}

## Basic components of a thermoacoustic system

A thermoacoustic machine generally consists of:

1.  Acoustic driver
2.  Stack or regenerator
3.  Heat exchanger
4.  Resonator

## Acoustic driver

Electrodynamic drivers are used in a class of electrically driven
thermoacoustic refrigeration systems. The mechanical and electrical
characteristics of the driver, in conjunction with the acoustic load
impedance at the driver piston, determine the electroacoustic efficiency
of the actuator. The electroacoustic efficiency is, of course, a key
factor in the overall efficiency of the cooling system. For this reason,
it is useful to develop models that allow the efficiency of any such
driver to be predicted for varying operating conditions and loads. A
detailed description of linear models of loudspeakers using equivalent
electrical circuits is readily available.\[2\] !Equivalent electrical
circuit of an electroacoustic
driver
Several methods based on such linear models have been proposed in order
to determine the model parameters experimentally.

## Stack

In the thermoacoustic refrigerator the stack is the main component where
the thermoacoustic phenomenon takes place. Below shown are two stacks of
different materials used in a standing wave thermoacoustic refrigerator.

!A ceramic stack{width="400"} !A
parallel plate
stack{width="400"}

## Heat exchanger

The heat exchangers employed in a thermoacoustic refrigerator influence
the acoustic field created in the resonator. There are many design
constraints such as porosity of the heat exchanger and high heat
transfer coefficient for efficiency. Due to these constraints, special
kind of heat exchangers are used. One typical micro channel aluminum
heat exchanger is shown below.

!Heat exchanger{width="400"}

## Resonator

This the part of refrigerator which is only there for maintaining the
acoustic wave. Because it is a dead volume which causes heat loss and
adds bulk, quarter wavelength resonators are preferred over half
wavelength.

## References

-   Greg Swift *et al.*,\"Thermoacoustics for Liquefaction of Natural
    Gas\",LNG Technology.
-   L. L. Beranek, *Acoustics*, McGraw--Hill, New York.
-   R. W. Wakeland, ''Use of electrodynamic drivers in thermoacoustic
    refrigerators,''J. Acoust. Soc. Am. 107, 827--832,2000.

[^1]: M. Emam, Experimental Investigations on a Standing-Wave
    Thermoacoustic Engine, M.Sc. Thesis, Cairo University, Egypt
    (2013).

[^2]: G.W. Swift, A unifying perspective for someengines and
    refrigerators, Acoustical Society of America, Melville, (2002).


# Engineering Acoustics/Acoustic streaming

## Definition

Streaming is a term to describe a steady time-averaged mass-flux density
or velocity induced by oscillating acoustic waves in a fluid.[^1] Large
amplitude sound propagation in a fluid may result in a steady motion of
the fluid. This nonlinear phenomenon is known as acoustic streaming and
may be theoretically described by quadratic convective terms in the
governing equations of the fluid flow.

## Applications

Acoustic streaming may be effective in enhancement of convective heat
transfer, ultrasonic cleaning of contaminated surfaces, localized
micro-mixing, development of micro-actuators such as micro-manipulators
for small particles and micro-pumps for liquids.Engineering
Acoustics/Acoustic Micro
Pumps

## Different types of acoustic streaming

There exist several ways to classify streaming.

*A first classification of streaming is based on the mechanism by which
streaming is generated.*[^2]

1.  **Boundary-layer driven streaming:** Flow driven by viscous stresses
    on boundaries and caused by boundary layer effects between a solid
    and a fluid. Boundary-layer driven streaming consists of two types
    of streaming which occur always together: outer and inner
    boundary-layer streaming.
    1.  **Outer boundary-layer streaming:** Rayleigh analysed acoustic
        streaming when a standing wave is present between parallel
        plates and explained that the air motion is caused by a
        nonlinear second order effect. Rayleigh focussed his
        investigations on mean flows outside the boundary layer and his
        approach became since then the analytical tool for the study of
        acoustic streaming.[^3]
    2.  **Inner boundary-layer streaming:** The study of inner boundary
        layer streaming was developed by Schlichting, who investigated
        an incompressible oscillating flow over a flat plate and
        calculated the two-dimensional streaming field inside the
        boundary layer. Figure 1 shows the inner and outer streaming in
        a channel. The length of such a cell is a quarter of the wave
        length.!Figure 1: Schematic diagram of inner and outer
        streaming
        cells{width="380"}
2.  **Jet driven streaming:** Periodic suction and ejection of a viscous
    fluid through an orifice or a change in cross section. The mechanism
    relies on the fact that a viscous fluid behaves quite differently
    during the suction and ejection periods. During the suction period
    the flow comes from all kind of directions, whilst during the
    ejection period a jet is produced. In Figure 2, the outflow and
    inflow patterns at the transition between a small tube and open
    space are shown. These two outflows can be regarded as the
    superposition of a broadly distributed oscillating flow and a
    time-averaged toroidal circulation.!Figure 2: Outflow and inflow
    patterns{width="480"}
3.  **Gedeon streaming:** Associated with travelling wave, as opposed to
    a standing wave as for the previous example. In boundary layer and
    jet driven streaming, there is no net mass transport. In travelling
    wave streaming, a non-zero net mass transport occurs due to the
    phase between the acoustic velocity and density. Travelling wave
    streaming in Stirling thermoacoustic engines and refrigerators is
    called Gedeon streaming or DC flow.
4.  **Eckart streaming:** Eckart streaming or \'quartz wind\' is
    generated by the dissipation of acoustic energy in the fluid.
    Although Eckart was not the first to observe \"quartz wind\", he was
    the first one who gave a mathematical analyses for it in 1948.[^4] A
    recent paper gives orders of magnitudes of velocities observed in
    this type of acoustic streaming [^5]

Figure 3 shows the schematic of Gedeon, Rayleigh and Jet-driven
streaming.

!Figure 3: Gedeon, Rayleigh and jet-driven
streaming{width="480"}

*The second classification is based on the relative magnitude of the
acoustic wavelength to the characteristic length of induced vortical
structures.*

1.  **Fine scale:** For the inner boundary-layer streaming (Schlichting
    streaming), the boundary-layer thickness is equal to the width of
    the vortices. So, it is part of the fine scale classification.
2.  **Comparable scale:** For outer boundary-layer streaming (Rayleigh
    streaming) and jet driven streaming, the wavelength and the vortex
    size are comparable.
3.  **Large scale:** Eckart streaming belongs to the large scale
    classification because the vortex length scale exceeds the acoustic
    wavelength.

*The third classification is based on the magnitude of the streaming
velocity.*

1.  **Slow streaming:** Slow streaming is when the streaming velocity is
    considerably smaller than the magnitude of the fluid velocity. In
    fact, streaming can be characterized by an appropriate Reynolds
    number, Re~NL~, which compares inertia and viscosity and determines
    the degree to which the streaming velocity field is distorted. The
    case Re~NL~\<\<1 corresponds to the slow streaming.[^6]
2.  **Fast streaming:** Fast streaming is when the streaming velocity
    and fluid velocity are of the same magnitude. The case Re~NL~\>\>1
    is referred to fast streaming or nonlinear streaming. Most types of
    acoustic streaming are slow, only the jet driven streaming is
    considered as fast.

## Axial and transverse components of the streaming velocity

Here, the outer boundary-layer streaming velocity characteristics
obtained from the analytical solution of the linear wave equation are
presented. The amplitude of the axial component of the acoustic velocity
field in the linear case is given as,

$(1)\   u=u_{max}sin(2\pi x/\lambda)$

where *u~max~=P~0~/ρ~0~c~0~*. The axial component *(u~st~)* and the
transverse component *(v~st~)* of the streaming velocity field are,

$(2)\  u_{st}=\frac{3}{8}\frac{u_{max}^2}{c}(1-\frac{2y^2}{(H/2)^2})sin(\pi x/l)$

$(3)\  v_{st}=-\frac{3}{8}\frac{u_{max}^2}{c}\frac{2\pi y}{\lambda}(1-\frac{2y^2}{(H/2)^2})cos(\pi x/l)$

where (*-H/2\<y\<H/2*), *H* is the height of the tube and *l=λ/4* [^7]

## References

[^1]: see video on <http://media.efluids.com/galleries/all?medium=749>

[^2]: Greg Swift,\"Thermoacoustics: A unifying perspective for some
    engines and refrigerators\", Condensed Matter and thermal Physics
    Group,Los Alamos National Laboratory, Forth edition, 1999.

[^3]: S. Boluriaan, P. J. Morris,\"Acoustic streaming: from Rayleigh to
    today\", International Journal of Aeroacoustics, 2 (3-4): 255-292,
    2003.

[^4]: O. V. Rudenko, S. I. Soluyan,\"Theoretical foundations of
    nonlinear acoustics\", Consultants Bureau, New York and London,
    1977.

[^5]:
        .

[^6]: S. Moreau, H. Bailliet, J. Valiere,\"Measurements of inner and
    outer streaming vortices in a standing waveguide using laser doppler
    velocimetry\", Journal of Acoustical Society of America, 123 (2):
    640-647, 2008.

[^7]: M. W. Thompson, A. A. Atchley,\"Simultaneous measurement of
    acoustic and streaming velocities in a standing wave using lase
    doppler anemometry\", Journal of Acoustical Society of America,
    117:1828-1838, 2005.