# Engineering Acoustics/Acoustic Levitation

## Definition

Acoustic levitation employs sound radiation to lift objects. It mostly
deals with non-linear phenomenon (since the resulting force on the
object is due to non linear properties of wave motion).

## Motivation behind developing an acoustic reactor

The force generated due to acoustic radiation pressure is generally much
larger than force of electromagnetic radiation pressure which makes the
study of these forces interesting and noteworthy.

Secondly, this phenomenon will allow successful containerless
experiments. The importance of such studies is illustrated by the
following:

Kinetic studies can be classified into two categories:

1.  The first includes material fixed to the walls.
2.  The second includes the flow of particles into and from an apparatus

The drawback of existing methods is that only one type of particle can
be used. Consequently, the behavior reported isn\'t accurate (since the
walls in the first case and the surrounding particles in the second case
can have an effect on the behavior under study).

This elimination of walls can provide further insight by discarding
supports in addition to reducing the interactions with other particles
(e.g.: by handling a single bubble).

One way to achieve this airborne application is by employing a
fascinating application of acoustics, namely acoustic levitation which
involves levitating objects using sound radiation.

Applications of this phenomenon and the corresponding technology can
include material processing in space without using any containers. This
may be particularly useful in the study of materials that are extremely
corrosive.

Moreover, sonoluminescence and acoustic
cavitation encounter this acoustic force.

Other applications can include measuring densities and analyzing fluid
dynamics in which surface tension plays an important role. Lastly,
acoustic positioning is another potential application.

-   Discovery
    News
    lists an interesting application of acoustic levitation.
-   Acoustic Levitation on
    Mars illustrates an
    adventurous application of this technology.

## Components of an Acoustic Reactor

!Figure 1: Schematic of the set
up{width="300"}
A simple acoustic reactor requires a:

-   A transducer to generate the desired sound waves. These transducers
    usually generate intense sounds, with sound pressure levels greater
    than 150 dB.
-   A reflector

In order to focus the sound, transducers and reflectors in general have
concave surfaces. The reflection of longitudinal sound waves off the
reflector leads to interference between the compressions and
rarefactions. Perfect interference will result in a standing acoustic
wave, i.e., a wave that will appear to have the same position at any
time.

With this simple arrangement of transducer and reflector, one can
achieve stable levitation but cannot steer the sample. To do so, Weber,
Rey, Neuefeind and Benmore have described an arrangement in their paper
that describes the use of two transducers. These transducers adjust the
location by altering the acoustic phases (which is carried out
electronically).

## Single Bubble Sonoluminescence

This phenomena occurs when a single bubble encounters a non linear
dynamic, namely, rapid compression of bubble preceded by an expansion
which takes place slowly. When the bubble is compressed rapidly, it can
get so hot that it emits a flash of light

!Figure 2: Sonoluminescence -
mechanism{width="500"}

## Theory

(source: Theory of long wavelength acoustic radiation pressure by
Löfstedt and Putterman)

Starting with the integral form of conservation of momentum,

$$\frac{\partial \rho v_i}{\partial t}+ \frac{\partial \Pi_{ij}}{\partial r_j}=0$$

$$\int_{v}\frac{\partial \rho v_i}{\partial t}+ \int_{S_o}\Pi_{ij} dS_j+\int_{S_K}\Pi_{ij} dS_j=0$$

where

`<big>`{=html}$\Pi_{ij}$`</big>`{=html} is the stress tensor,

`<big>`{=html}$\rho, v$`</big>`{=html} are the local fluid density and
velocity,

`<big>`{=html}$S_o$`</big>`{=html} the surface of the object (at time
t),

`<big>`{=html}$S_K$`</big>`{=html} a surface far from the object, and

V the volume bounded by these surfaces.

Using the relation,

$$\int_v \frac{\partial \rho v_i}{\partial t} dr + \frac{d}{dt}\int\rho v_i dr - \int_{S_o} \rho v_i v_j dS_j = 0$$

$$\frac{d}{dt} \int \rho v_i dr + \int_{S_o} (\Pi_{ij} - \rho v_i v_j) dS_j + \int_{S_R} \Pi_{ij} dS_j = 0$$

Time average of this equation gives an expression for the force on a
moving sphere

$$\langle F_i \rangle  =  \langle \int_{S_o} (\Pi_{ij} - \rho v_i v_j) dS_j \rangle  =  - \int_{S_R} \langle \Pi_{ij} \rangle dS_j = 0$$

Assuming an ideal fluid,

The Galilean invariant contribution to the stress tensor is

:   `<big>`{=html}`<big>`{=html}`<math>`{=html}

\\Pi\_{ij}-\\rho v_i v_j = p\\delta\_{ij}
`</math>`{=html}`</big>`{=html}`</big>`{=html}

and

$$p = - \rho_{eq} \frac{\partial \phi}{\partial t} - \rho_{eq} v^2/2 +  \frac{\rho_{eq}}{2c^2} \left({\frac{\partial \phi}{\partial t}}\right)^2$$

here, `<big>`{=html}$v=\nabla \phi$`</big>`{=html}

`<big>`{=html}$\rho_{eq}$`</big>`{=html} represents the equilibrium
density

`<big>`{=html}$c$`</big>`{=html} denotes the speed of sound

Acoustic radiation force on an object in an ideal fluid is

```{=html}
<center>
```
$\langle F_i \rangle = - \int_{S_R} {[- \frac{\rho \langle v^2 \rangle}{2} +  \frac{\rho}{2c^2} \langle{\frac{\partial \phi}{\partial t}}^2\rangle + \rho \langle v_i v_j \rangle] dS_j}$

```{=html}
</center>
```
### Multipole Expansion of acoustic radiation force

Consider the linear wave equation,

$$\frac{\partial^2 \phi}{\partial t^2} - c^2 \Delta^2 \phi = 0$$

where

:   `<big>`{=html}`<big>`{=html}$\phi = \phi_i + \phi_s$`</big>`{=html}\
    `</big>`{=html}

here `<big>`{=html}$\phi_i$`</big>`{=html} is given by transducer,

`<big>`{=html}$\phi_s$`</big>`{=html} is given by the corresponding
boundary condition at the object where s stands for \'scattered\'

$\phi_s = Re \sum_{n=0}^\infty B_n h_n (k r)P_n (cos\theta) e^{-i\omega t}$

`<big>`{=html}$h_n$`</big>`{=html} here are the outgoing spherical
Hankel functions

`<big>`{=html}$P_n$`</big>`{=html} here are the Legendre polynomials

`<big>`{=html}`<big>`{=html}$\omega = k c$`</big>`{=html}`</big>`{=html}

k: wave number of the sound field imposed on the object

As r approaches infinity,

$\lim_{r \to \infty} \phi_s \to Re \frac{e^{i (kr - \omega t)}}{kr} \sum_{n} (-i)^{n+1} B_n P_n (cos\theta)$

For standing waves,

:   `<big>`{=html}`<big>`{=html}`<math>`{=html}

\\phi_i = Re A sin(kz) e\^{-i\\omega t}
`</math>`{=html}`</big>`{=html}`</big>`{=html}

Thus by computing $\phi$ and using the result in the expression for the
radiation force we get

```{=html}
<center>
```
`<big>`{=html}`<big>`{=html}$F_z = 2\pi\rho A[coskz_o(Im B_o)-sinkz_o(Im B_1)$`</big>`{=html}`</big>`{=html}
\...\...\...\...\...\...\...\...\...\...\...\...\...\..... (1)

```{=html}
</center>
```
### Radiation Force on a spherical object

Consider a spherical body with radius $R_o$ and density ρ~o~

The wave equation inside the sphere is given by:
$:\frac{\partial^2 \Phi}{\partial t^2}-
c_o^2{\nabla^2 \Phi_o} - \zeta_o \rho_o \frac{\partial \nabla^2 \Phi_o}{\partial t} = 0$

where

`<big>`{=html}$\Phi_o$`</big>`{=html} is the velocity potential

`<big>`{=html}$c_o$`</big>`{=html} is the speed of sound in the sphere

`<big>`{=html}$\zeta_o$`</big>`{=html} characterizes the damping in the
sphere

(Effects due to thermal conductivity and shear viscosity are neglected)

The solution to this equation is given by:

$:\Phi_o = Re \sum_{n=0}^\infty C_n j_n (k_o r)P_n (cos\theta) e^{-i\omega t}$

where `<big>`{=html}$j_n$`</big>`{=html} are spherical Bessel functions

`<big>`{=html}$k_o$`</big>`{=html} is complex (because of dissipation)

`<big>`{=html}$k_o = k_o +i \alpha_o$`</big>`{=html}

where

`<big>`{=html}$k_o = \omega/c_o$`</big>`{=html}

`<big>`{=html}$\alpha_o = k_o^2 \zeta_o/2c_o \rho_o$`</big>`{=html}

Here,`<big>`{=html}$\alpha_o$`</big>`{=html} is the attenuation
coefficient of sound in the sphere

The boundary conditions are given by

`<big>`{=html}`<big>`{=html}$\rho \Phi (R_o) = \rho_o \Phi_o (R_o)$`</big>`{=html}`</big>`{=html}

At r = `<big>`{=html}$R_o$`</big>`{=html}

$\frac{\partial \Phi}{\partial t} = \frac{\partial \Phi_o}{\partial t}$

To satisfy these conditions, the incident wave is expanded using
spherical harmonics

$sin(kz) = sin(k z_o) \sum_{n=0}^\infty a_{2n} (kr) P_{2l}  cos\theta + cos(k z_o) \sum_{n=0}^\infty a_{2n+1} (kr) P_{2l+1}     cos\theta$

where

$a_{2l} = \frac{4l+1}{2}. \int_{-1}^1 cos krx P_{2l} (x) dx

a_{2l + 1} = \frac{4l+3}{2}. \int_{-1}^1 sin krx P_{2l+1} (x) dx$

Using the above relations, one can compute $a_o$

$a_o = \frac{sin kr}{kr}.$

When `<big>`{=html}$kR_o << 1$`</big>`{=html}

$sin kz = sin kz_o [ 1 - \frac{1}{2}. k^2 r^2 cos^2 \theta + ...] + cos kz_o [kr cos \theta - ...]$

The boundary condition for the case of a standing wave can be deduced as
follows:

For monopole term,

$\rho A sin k z_o - \rho B_o \frac{i e^{ikR_o}}{kR_o} = \rho_o C_o j_o (x_o)$,

and

$- A (\frac{k^2 R_o}{3} sin k z_o + B_o e^{ikR_o}(\frac{1}{R_o} + \frac{i}{(kR_o)^2}) =  C_o k_o j^{'}_o (x_o)$

For dipole term

$\rho A (k R_o) cos k R_o - \rho B_1 e^{ikR_o} (\frac{1}{kR_o} + \frac{i}{(kR_o)^2}) = \rho_o C_1 j_1 (x_o)$,

and

$A k cos k z_o + B_1 e^{ikR_o} (\frac{-i}{R_o} + \frac{2}{kR_o^2}+ \frac{2i}{k^2 R_o^3}) = C_1 k_o j^{'}_1 (x_o)$,

where

`<big>`{=html}$x_o = k_o R_o$`</big>`{=html}

`<big>`{=html}$B_1$`</big>`{=html} and
`<big>`{=html}$B_o$`</big>`{=html} can be obtained as functions of A

Using these relations for `<big>`{=html}$B_o$`</big>`{=html} and
`<big>`{=html}$B_1$`</big>`{=html} in the radiation force expression we
get:

Thus the radiation force on a sphere is given by

`<big>`{=html}`<big>`{=html}$F_{z} = - \pi k^3 R^3 A^2 sin 2kz_o Re [f_o + f_1]$`</big>`{=html}`</big>`{=html}

where
$f_o = \frac{(1/3)(\rho_o/\rho) k^2 R_o^2 b_o (x_o) + 1}{k^2 R_o^2 [1 + (\rho_o/\rho) b_o (x_o)+ ikR_o]}$

$f_1 = \frac{(\rho_o/\rho) b_1 (x_o) - 1}{2(\rho_o/\rho) b_1 (x_o)+ 1}$

with

$b_o (x_o) = \frac{j_o (x_o)}{x_o j^'_o (x_o)}$

$b_1 (x_o) = \frac{j_1 (x_o)}{x_o j^'_1 (x_o)}$

If we neglect damping, assume $x_o << 1$ and assume the sphere is
incompressible (i.e. $c_o$ approaches infinity), then the radiation
force simplifies to:

```{=html}
<center>
```
$F_{z} = - \pi k^3 R_o^3 A^2 sin 2kz_o \rho_o {5 \rho_o-2 \rho\over 3(2\rho_o + \rho)}$

```{=html}
</center>
```
This expression for radiation force (in a standing wave field) was
initially derived by King. Note that the radiation force is directly
proportional to the cube of the radius and directly proportional to the
velocity amplitude.

### Assumptions

1.  kR \<\< 1, i.e., the wavelength of the sound field is much larger
    than the dimension of the sphere.
2.  incompressible object (used by King, although Gorkov has derived
    results that permits finite compressibility of the sphere)

Sphere is suspended when sum of the forces acting on it equals zero,
i.e., when force due to gravity balances the upward levitation force.

As a result, the object is attracted to regions of minimum potential
energy (pressure nodes). Antinodes are regions experiencing high
pressures.

To ensure the generation of a standing wave, the transducer must be
placed at a certain distance from the reflector and a particular
frequency should be used to get satisfactory results. This distance
should be a multiple of half the wavelength of the sound produced to
make sure the nodes and antinodes are stable.

Secondly, the direction of the force exerted by the radiated pressure
due to the sound waves must be parallel to the direction of gravity.

Since the stable areas should be large enough and able to support the
object to be levitated, the object\'s dimensions should lie between one
third and one half of the wavelength. It is important to note that the
higher the frequency, the smaller the dimensions of the object one is
trying to levitate (since wavelength and frequency are inversely
proportional to each other)

The materials of the object is important too, since the density along
with the dimensions will give the value for its mass and determine the
gravitational force and consequently whether the upward force produced
by the pressure radiation is suitable.

Another characteristic important when talking about material properties
is the **Bond number** which is important when dealing with drops of
fluid. It characterizes the surface tension and size of the liquid
relative to the fluid surrounding it. The lower the Bond number, the
greater the chances that the drop will burst.

Finally, to achieve such high pressures (that can cancel the
gravitational force), linear waves are insufficient. Therefore,
non-linear waves play an important role in acoustic levitation. This is
easily one of the reasons why the study of acoustic levitation is
challenging. Nonlinear acoustics is a field that deals with physical
phenomena difficult to comprehend. Based on experimental observations,
heavy spheres incline to velocity antinodes, light particles are closer
to the nodes.

## Other effects on levitation force

Temperature, pressure, fluid medium characteristics (density, particle
velocity) affect the levitation force. It is important to remember that
the medium changes as conditions change. The fluid medium consists of
reactants and products that change with reaction rate.

Thus, consequently the levitation force is affected. To compensate for
medium changes, resonance tracking system can be employed (which helps
to maintain stable levitation of the particle under study)

## Design considerations

The sphere or particle under study should experience a lateral force
which will act as a positioning force (along with the more obvious
vertical levitating force) Rotation of the sphere about its axis will
ensure uniform heating and stability.

## Using non-spherical particles

When levitating non-spherical particles, the largest cross section of
the object will end up aligning itself perpendicular to the axis of the
standing wave.

## Traveling vs Standing waves

King discovered that the radiation pressure exerted by a standing wave
is much larger than the pressure exerted by a traveling wave (which has
the same amplitude as the standing wave)

This is because the pressure exerted by a standing wave is due to the
interference between the incident and scattered waves. Pressure exerted
by a traveling wave is due to contributions from scattered field only.

## References

1.  Theory of long wavelength acoustic radiation pressure by Löfstedt
    and Putterman
2.  Development of an acoustic levitation reactor by Cao Zhuyoua, Liu
    Shuqina, Li Zhimina, Gong Minglia, Ma Yulongb and Wang Chenghaob
3.  HowStuffWorks

## Useful Sites

1.  Spherical Hankel
    Functions
2.  Legendre
    Polynomials
3.  Multipole expansion


# Engineering Acoustics/Biomedical Ultrasound

## Biomedical Ultrasound

This chapter of the Engineering Acoustics Wikibook provides a brief
overview of biomedical ultrasound applications along with some
introductory acoustical analysis for ultrasound beams. As a whole, the
field of Biomedical Ultrasound is one that provides a wealth of topics
for study involving many base disciplines. As such, this limited entry
does not cover all aspects of Biomedical Ultrasound, but instead chooses
to focus on providing readers with an introductory understanding, from
which additional study of the topic is possible. For readers interested
in a more thorough reference on Biomedical Ultrasound the 2007 text by
Cobbold [^1] is suggested.

## Diagnostic Applications

The most well know application of Biomedical Ultrasound is in medical
imaging, also known as ultrasonography. For a list of specific
applications of ultrasonography refer to the corresponding Wikipedia
entry. The following section
provides a qualitative description of the acoustical process used to
generate and capture sound signals used in producing ultrasound images.

An ultrasound transducer emits a short pulse of high frequency sound.
Depending on application the wave frequency ranges between 1 MHz and
15 MHz.[^2] As the emitted sound waves propagate they will be partially
reflected or scattered by any variation in acoustic impedance, *ρc*,
that is encountered. In the context of to biomedical imaging, this
corresponds to anywhere there are density changes in the body: e.g. the
connection of bone to muscle, blood cells in blood plasma, small
structures in organs, etc.[^3]

The behavior of the reflected wave depends largely on the size of the
reflective feature and the wavelength of the emitted sound wave. When
the wavelength is short relative to the reflective structure reflections
will be governed according to the principals of acoustic transmission
and reflection with normal or oblique interfaces.[^4] When the
wavelength is long relative to the structure the principals of acoustic
scattering [^5] are applicable. The latter condition, which occurs for
small reflection sources, sets the requirement for frequencies used in
ultrasound imaging. As discussed by Cobbold,[^6] analysis for a planar
wave incident on a spherical reflection source of effective radius *a*,
shows the acoustic intensity of the scattered wave, *I~s~*, varies
according to:

\
$$I_s \propto  \frac{a^6}{\lambda ^4}$$

This relation shows that when wavelength is long relative to a scatter
sources effective radius, the scattered energy becomes very small, thus
negligible amounts of the incident wave will be reflected back to the
transducer. To reliably capture a feature in an ultrasound image the
emitted wavelength must be smaller than the features of interest. Other
consideration for wavelength are also applicable: due to attenuation of
the propagating wave lower frequencies offer greater imaging depth,
while higher frequencies (with smaller wavelength) offer increase
ability for lateral focusing of the emitted beam (small beam width at
focus, see below).[^7] Table 1 gives the correlation between frequency
and wavelength in water for several frequencies used in ultrasound
imaging (*λ* = *c*/*f*).

\
{\| class=\"wikitable\" style=\"text-align: center;\" align=\"center\"
\|+ Table 1: Medical ultrasound frequencies and corresponding
wavelengths. \|- \| Frequency (MHz) \|\| 1 \|\| 2 \|\| 5 \|\| 8 \|\| 10
\|\| 12 \|\| 15 \|- \| Wavelength (mm) \|\| 1.50 \|\| 0.75 \|\| 0.30
\|\| 0.19 \|\| 0.15 \|\| 0.13 \|\| 0.10 \|}

After the wave burst is transmitted the ultrasound transducer can act as
a receiver, much like a microphone or hydrophone. Waves reflected off
structures and density gradients are returned to the transducer and
recorded. The delay time between the transmitted and received signal is
correlated to the distance of the reflection source, while the intensity
of the received signal is correlated to the reflection sources acoustic
impedance and size.[^8] In instances where doppler ultrasonography is
utilized, the frequency shift between the transmitted and received
signals can be correlated to the velocity of the reflection source.

Modern ultrasonography use arrays of small transducers, each of which
are individually electronically controlled to achieve an effect know as
beamforming. When using this technique,
control of the phase relation between array elements results in control
over the emitted beam\'s direction and focal depth.[^9] To produce a
two-dimensional ultrasound image, the ultrasound beam focal position is
swept through a region, and the recorded reflected waves are correlated
to the particular focal locations. The exact process by which this
general concept is accomplished varies with each ultrasonography
instrument. Figure 1 provides a sample 2D image produced by the sweeping
of the focal location through a 2D plane.

\
!Figure 1: 2D Obstetric ultrasound
image.{width="420"}

## Clinical and Therapeutic Applications

A number of important clinical and therapeutic applications make use of
high intensity, focused ultrasound beams. In many of these applications,
the therapeutic effect is achieved due the heat generation associated
with dissipation of the high intensity acoustic beam. In some
applications, such as
lithotripsy the
therapeutic effect is obtained from acoustic non-linearity, causing
wave deformation
and shock wave
formation.
This effect is discussed in more detail in a section to
follow.

Provided below is a partial list of Therapeutic applications of
ultrasound:

-   Ultrasound is sometimes used to clean teeth in dental
    hygenist.
-   Focused ultrasound may be used to generate highly localized heating
    to treat cysts and tumors (benign or malignant). This is known as
    Focused Ultrasound Surgery (FUS) or High Intensity Focused
    Ultrasound (HIFU). These procedures generally
    use lower frequencies than medical diagnostic ultrasound (from
    250 kHz to 2000 kHz), but significantly higher energies.
-   Focused ultrasound may be used to break up kidney stones by
    lithotripsy.
-   Ultrasound may be used for cataract treatment by
    Phacoemulsification.
-   Additional physiological effects of low-intensity
    ultrasound have
    recently been discovered, such as the ability to stimulate
    bone-growth and the potential to disrupt the blood-brain
    barrier for drug delivery.

## Acoustic Properties of Ultrasound Beams

As a first approximation, an ultrasound beam can be considered as
resulting from flat circular piston oscillating on an infinite baffle.
In practice, such a system would lead to relatively high diffusion of
the sound beam, severe side lobes, and an inability to choose a focal
length of the acoustic energy. In current biomedical applications the
use of phased arrays is a common approach stemming from the more general
field know as beamforming. Despite the limitations of planar
transducers, their relatively simple analysis serves well to illustrate
the basic properties of any formed beam and the challenges of designing
more advanced systems.

The analytical approach utilized for the simple cylindrical transducer
appears in many acoustics reference Texts, such as those by Pierce,[^10]
Kinsler et al.[^11] and Cheeke.[^12] The sound field solution is
obtained by first considering the sound emitted by the harmonic motion
of a single point source (small sphere) vibrating in free space. The
resulting sound pressure field from this point source is:

\
$$\boldsymbol{P} = i\left(\frac{\rho_o c U_o k a^2}{r}  \right )e^{-ikr}$$

$$I(r) = \frac{1}{2}\rho_o c_o U_o^2 \left(\frac{a}{r} \right )^2 \left(ka \right )^2$$

Where ***P***(*r*) is the harmonic pressure amplitude at the radial
distance *r*, *ρ~o~* is the fluid density, *c~o~* is the fluid sound
speed, *U~o~* is the maximum velocity of the spherical source, *a* is
the sphere radius, and *k* = 2*πf*/*c~o~* is the wave number. In the
preceding equations, *i* = -1^1/2^, which incorporates both amplitude an
phase into the harmonic pressure variable.

To apply this result to the ultrasound transducer as a cylindrical
radiator, each differential section of the cylinder surface can be
considered as a separate spherical source. The resulting sound field
from this approximation is the integral sum of each spherical source. In
general the resulting equation cannot be analytically integrated;
however, when considering regions of the field for r \>\> a, where a is
now the cylinder radius, a simple result is found. Forgoing the full
derivation (for reference see Kinsler [^13] or Cheeke [^14]), the
equations for the produced sound field and acoustic intensity are:

\
$$\boldsymbol{P}(r, \theta) = i \left( \tfrac{1}{2}\rho_o c_o U_o \frac{a}{r}ka \right ) H(\theta)e^{-ikr},$$

$$H(\theta) = \frac{2J_1(ka\sin\theta)}{ka \sin \theta},$$

$$I(r,\theta) = \frac{|\boldsymbol{P}(r)|^2}{2\rho_oc_o},$$

where *H*(*θ*) is the directivity function, *J~1~* is the Bessel
Function of the first
kind, and *I*(*r*) is the
acoustic intensity in W/m^2^. Physically the directivity function
represents the pressure amplitude for beam angles not parallel to the
cylinder axis. It is worthwhile to note that roots of the Bessel
function produce certain beam angles with zero amplitude; the regions
between these angles are known as side lobes, with the on axis component
know as the main lobe. Physically, lobes result from the phase
interaction of waves originating from different parts of the cylindrical
transducer, and are in some ways analogous to pressure nodes in simple
harmonic waves.

To illustrate the phenomena of side lobes in ultrasound beams, the
resulting directivity function and acoustic intensity is calculated for
a 1 MHz beam transmitted into water using a 1 cm radius transducer.
Figure 2 plots the Directivity function, while Figure 3 plots the
acoustic intensity relative to the intensity at the transducer surface.

\
! Figure 2: Beam function for a 1 cm radius cylinder radiating 1 MHz
ultrasound into
water.{width="520"}

! Figure 3: Normalized acoustic intensity field for a 1 cm radius
cylinder radiating 1 MHz ultrasound into
water.{width="520"}

For the purposes of diagnostic and therapeutic ultrasound the presence
of side lobes is an undesirable effect. In diagnostic imaging wave
reflection originating from the side lobes can be misinterpreted as
reflections from the main beam, and act to reduce the resulting image
quality. In therapeutic applications, side lobes represent energy
dissipation on regions not intended to be effected. To reduce the
effects of side lobes, ultrasound devices use transducer designs based
on beamforming theory,
making the analysis substantially more complicated than the simple
cylindrical transducer discussed. One technique to reduce side lobes is
the use of a phased array to focus the main at a particular depth, thus
reducing the relative magnitude of side lobes. Another technique known
as acoustic shadowing reduces side lobes by emitting lower amplitude
waves near the edge of the transducer. As will be discussed in a
proceeding section, an emerging technique to enhance focusing and reduce
side lobes is the purposeful consideration of nonlinear acoustic effects
in ultrasound beams.[^15][^16]

## Nonlinear Acoustics in Biomedical Ultrasound

In many fields related to application of acoustic theory, the assumption
of linear wave propagation is sufficient. In Biomedical ultrasound
however, the propagation of sound waves is often accompanied by
progressive wave distortion resulting from nonlinear, finite amplitude,
effects. The nonlinear effect of most interest in many diagnostic
applications is the generations of harmonics in the ultrasound beam. As
a primer to this section, a review of the acoustic parameter of
nonlinearity,
and harmonic
generation is
suggested.

Nonlinearities relevant to Biomedical Ultrasound are relatively weak,
making their effects on the propagating acoustic wave cumulative with
distance. For appreciable harmonic generation to occur four conditions
should be met:

-   Sufficient pressure and velocity amplitude. The waves emitted for
    almost all applications of biomedical ultrasound meet this
    requirement.[^17]
-   Sufficient propagation distance with near planar wave conditions.
    For directional beams, such as those used in ultrasonography, this
    conditions is approximately met within the Rayleigh Distance, *x* =
    1/2 *ka^2^*, on the main lobe.[^18] Furthermore, harmonic generation
    is proportional to the number of wavelengths propagated and not the
    absolute distance. The ultrasonic frequencies utilized have very
    short wavelengths, for example, a 10 MHz wave must propagate over
    500 wavelengths for a focal depth of 10 cm.
-   Sufficient value of the parameter of
    nonlinearity,
    B/A. For the same acoustic intensity, a material with a higher value
    of B/A will produce harmonics more quickly. The value of B/A in
    water is ten times that in air, and B/A for some biological tissues
    can be double that of water.
-   Low acoustic absorption. In many tissues with high values for B/A
    there are also high values for acoustic absorption. As the extent of
    wave dissipation increases with frequency, generated harmonics are
    absorbed more readily than the fundamental frequency. This effect
    reduces the influence of B/A in biological tissues relative to that
    in low-loss fluids.[^19]

Reviewing these conditions it can be seen that in many circumstances,
harmonic generation will be appreciable in biomedical ultrasound. Two
developing applications that make use of this harmonic generation are:

-   Use of harmonic content in recorded ultrasonography signals. As
    acoustic intensity and propagation distance is highest on the main
    lobe, harmonic generation occurs most significantly on the main
    lobe, and is smaller on side lobes. As a result, the beam pattern
    produced by 2nd harmonics is more directional than the beam produced
    by the fundamental frequency. This allows for potential improvement
    in the resulting image.[^20]
-   The analysis of harmonic profiles for tissue characterization using
    the B/A parameter. Referring to The Acoustic Parameter of
    Nonlinearity,
    values of B/A vary for tissues that have otherwise similar acoustic
    impedance. As a result, harmonic content in ultrasound waves has the
    potential to produce images correlated to the B/A parameter of
    tissues. Practical realization of this concept is an area in
    development, as current imaging methods are unable to utilize this
    potential.[^21]

## External links

-   Medical Ultrasonic
    Transducers

## References

[^1]:

[^2]:

[^3]:

[^4]:

[^5]:

[^6]:

[^7]:

[^8]:

[^9]:

[^10]:

[^11]:

[^12]:

[^13]:

[^14]:

[^15]:

[^16]:

[^17]:

[^18]:

[^19]:

[^20]:

[^21]:


# Engineering Acoustics/Human Voice Production

## Physiology of Vocal Fold

Human vocal fold is a set of lip-like tissues located inside the
larynx, and is the source of
sound for a human and many animals. The Larynx is located at the top of
trachea. It is mainly composed of cartilages and muscles, and the
largest cartilage, thyroid, is well known as the \"Adam\'s Apple.\"

The organ has two main functions; to act as the last protector of the
airway, and to act as a sound source for voice. This page focuses on the
latter function. In the following image, the cross section of vocal cord
is shown. This three dimensional geometry is made using CT scan data.

!*Vocal fold cross
section*{width="400"}

Links on Physiology: Discover The
Larynx

## Voice Production

Although the science behind sound production for a vocal fold is
complex, it can be thought of as similar to a brass player\'s lips, or a
whistle made out of grass. Basically, vocal folds (or lips or a pair of
grass) make a constriction to the airflow, and as the air is forced
through the narrow opening, the vocal folds oscillate. This causes a
periodical change in the air pressure, which is perceived as sound.

Vocal Folds Video

When the airflow is introduced to the vocal folds, it forces open the
two vocal folds which are nearly closed initially. Due to the stiffness
of the folds, they will then try to close the opening again. And now the
airflow will try to force the folds open etc\... This creates an
oscillation of the vocal folds, which in turn, as I stated above,
creates sound. However, this is a damped oscillation, meaning it will
eventually achieve an equilibrium position and stop oscillating. So how
are we able to \"sustain\" sound?

As it will be shown later, the answer seems to be in the changing shape
of vocal folds. In the opening and the closing stages of the
oscillation, the vocal folds have different shapes. This affects the
pressure in the opening, and creates the extra pressure needed to push
the vocal folds open and sustain oscillation. This part is explained in
more detail in the \"Model\" section.

This flow-induced oscillation, as with many fluid mechanics problems, is
not an easy problem to model. Numerous attempts to model the oscillation
of vocal folds have been made, ranging from a single mass-spring-damper
system to finite element models. In this page I would like to use my
single-mass model to explain the basic physics behind the oscillation of
a vocal fold.

Information on vocal fold models: National Center for Voice and
Speech

## Models

### Single mass model

image: Singlemass - vocal folds motion model schematic (single
mass-spring-damper
system).png.png "wikilink")

`                                                       Figure 1: Schematics`

The most simple way of simulating the motion of vocal folds is to use a
single mass-spring-damper system as shown above. The mass represents one
vocal fold, and the second vocal fold is assumed to be symmetry about
the axis of symmetry. Position 3 respresents a location immediately past
the exit (end of the mass), and position 2 represents the glottis (the
region between the two vocal folds).

#### The Pressure Force

The major driving force behind the oscillation of vocal folds is the
pressure in the glottis. The Bernoulli\'s equation from fluid mechanics
states that:

$P_1 + \frac{1}{2}\rho U^2 + \rho gh = Constant$ \-\-\-\--EQN 1

Neglecting potential difference and applying EQN 1 to positions 2 and 3
of Figure 1,

$P_2 + \frac{1}{2}\rho U_2^2 = P_3 + \frac{1}{2}\rho U_3^2$ \-\-\-\--EQN
2

Note that the pressure and the velocity at position 3 cannot change.
This makes the right hand side of EQN 2 constant. Observation of EQN 2
reveals that in order to have oscillating pressure at 2, we must have
oscillation velocity at 2. The flow velocity inside the glottis can be
studied through the theories of the orifice flow.

The constriction of airflow at the vocal folds is much like an orifice
flow with one major difference: with vocal folds, the orifice profile is
continuously changing. The orifice profile for the vocal folds can open
or close, as well as change the shape of the opening. In Figure 1, the
profile is converging, but in another stage of oscillation it takes a
diverging shape.

The orifice flow is described by Blevins as:

$U = C\frac{2(P_1 - P_3)}{rho}$ \-\-\-\--EQN 3

Where the constant C is the orifice coefficient, governed by the shape
and the opening size of the orifice. This number is determined
experimentally, and it changes throughout the different stages of
oscillation.

Solving equations 2 and 3, the pressure force throughout the glottal
region can be determined.

#### The Collision Force

As the video of the vocal folds shows, vocal folds can completely close
during oscillation. When this happens, the Bernoulli equation fails.
Instead, the collision force becomes the dominating force. For this
analysis, Hertz collision model was applied.

$F_H = k_H \delta^{3/2} (1 + b_H \delta')$ \-\-\-\--EQN 4

where

$k_H = \frac{4}{3} \frac{E}{1 - \mu_H^2} \sqrt{r}$

Here $\delta$ is the penetration distance of the vocal fold past the
line of symmetry.

#### Simulation of the Model

The pressure and the collision forces were inserted into the equation of
motion, and the result was simulated.

image: matlab.png

`                                              Figure 2: Area Opening and Volumetric Flow Rate`

Figure 2 shows that an oscillating volumetric flow rate was achieved by
passing a constant airflow through the vocal folds. When simulating the
oscillation, it was found that the collision force limits the amplitude
of oscillation rather than drive the oscillation. Which tells us that
the pressure force is what allows the sustained oscillation to occur.

#### The Acoustic Output

This model showed that the changing profile of glottal opening causes an
oscillating volumetric flow rate through the vocal folds. This will in
turn cause an oscillating pressure past the vocal folds. This method of
producing sound is unusual, because in most other means of sound
production, air is compressed periodically by a solid such as a speaker
cone.

Past the vocal folds, the produced sound enters the vocal tract.
Basically this is the cavity in the mouth as well as the nasal cavity.
These cavities act as acoustic filters, modifying the character of the
sound. The acoustics of vocal tract have traditionally been described on
the basis of a source-filter theory.
Whereas the glottis produces a sound of many frequencies, the vocal
tract selects a subset of these frequencies for radiation from the
mouth. These are the characters that define the unique voice each person
produces.

### Two Mass Model

The basic two mass model is shown in Figure.3 and the two mass model of
vocal fold is shown in Figure.4.

!CGtypes\|500px\|Figure3. Two mass
model

!CGtypes\|350px\|Figure4. Smoothed two mass
model

### Three Mass Model

In Figure5, three mass model of vocal fold is shown. !Figure5. Three
mass model{width="400"}

### Rotating Plate Model[^1]

!Rotatingplate{width="400"} The
motion of vocal fold can be described with two degrees of freedom. First
rotation of mass **M2** and displacement **r**. The equation of motion
will be:

$$\ddot{I}+ B \dot{\theta} +k \theta= T$$

$$m\ddot{r}+b(\dot{r}-\dot{r_b})+k_2(r-r_b)=F$$

Where in these equations: *T* is the applied aerodynamic torque

$I_c$ is the moment of inertia for rotation cover

$B$ is the rotational damping

$k$ is the rotational stiffness

$k_2$is the translational stiffness

$b$ is the translational damping

$F$ is the force\
$M_2$is the cover mass\
$r_b$ is the displacement of the body

$r$ is the displacement of the cover

The equation of motion for the body mass can be written as:

$$M_1\ddot{r_b}+b(\dot{r_b}-\dot{r})+k_2(r_b-r)+K_1r_b+B\dot{r_b}=0$$
where:

$K_1$ is the body stiffness

$M_1$ is the body mass

$B$ is the body damping

### Finite Element Models

## Lumped-element flow circuit for the vocal tract

In the image below, the lumped element flow circuit for the vocal tract
airway is shown. The input impedance to the vocal tracts can be shown by
resistive and inertive lumped elements.[^2] According to the shown
circuit we have:

!lumped element circuit for vocal
tract{width="600"}

$$P_L-R_s u-I_s\dot{u} -\frac{1}{2} \rho u^2/a_g^2-R_eu-I_e \dot{u}=0$$
Where

$P_L$ is a steady lung pressure

$R_s$ is subglottal resistance

$I_s$ is subglottal inertance(epilaryngeal) input resistance

$R_e$ is supraglottal(epilaryngeal) input resistance\
$I_e$ is supraglottal(epilaryngeal) input inertance

## References

\[1\] Fundamentals of Acoustics; Kinsler *et al.*, John Wiley & Sons,
2000

\[2\] Acoustics: An introduction to its Physical Principles and
Applications ; Pierce, Allan D., Acoustical Society of America, 1989.

\[3\] Blevins, R.D. (1984). Applied Fluid Dynamics Handbook. Van
Nostrand Reinhold Co. 81--82.

\[4\] Horacek, J., Sidlof, P., Svec, J.G. Self-Oscillations of Human
Vocal Folds. Institute of Thermomechanics, Academy of Sciences of the
Czech Republic

\[5\] Lucero, J.C., Koenig, L.L. (2005). Simulations of temporal
patterns of oral airflow in men and women using a two-mass model of the
vocal folds under dynamic control, Journal of the Acoustical Society of
America 117, 1362--1372.

\[6\] Titze, I.R. (1988). The physics of small-amplitude oscillation of
the vocal folds. Journal of the Acoustical Society of America 83,
1536--1552

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

Back to main page

[^1]: Ingo R. Titze, \"The myoelastic aerodynami theory of phonation\"
    2006

[^2]:


# Engineering Acoustics/Vocal Folds

## Introduction

The mechanism of sound production in speech and singing would be the
results of airflow in the human lung system, and is also connected to
the digestive system. The diaphragm action from the lungs pushes air
through the vocal folds, and produces a periodic train of air pulses.
This pulses train is shaped by the resonances in the vocal tract, and
has various frequencies and loudness. Vocal formants are basic
resonances, and can be changed by the movements of the articulators to
produce different vowel sounds. To produce different vowel sounds, the
vocal mechanism is controlled to produce the resonances of the vocal
tract is produced the vocal formants. The vocal tract can be considered
to be a cavity resonator, and the soft palate position, the area of
opening, the tongue positions an shape, and the position of jaw would
established the shape of this cavity. More specifically, voice
articulation is the movement of the tongue, pharynx palate, jaw or lips
that changes the volume of cavity, area of opening, and port length,
which determine the frequency of the cavity resonance. The voice
mechanism can be modeled as the lung and diaphragm being the power
source, along with the larynx, pharynx, mouth and nose. At the end of
the tubular larynx rest the vocal folds, also known as vocal cords.
During speech and singing, the larynx is connected to pharynx, and is
covered by epiglottis during swallowing. The vocal tract acts as a
resonator.

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

## The vocal folds

Vocal folds are twin enfoldings of mucous membrane that act as a
vibrator during phonation. Phonation is the process by which the energy
from the lungs in the form of air pressure is converted vibration that
is perceivable to the human ear. There are two methods to phonation. One
is by the air pressure setting the elastic vocal folds into vibration,
which is called voicing. The other is air passing through the larynx to
vocal tract, where airstream gets modified as produces transient of
aperiodic sound waves. In aperiodic phonation, the transient or
aperiodic sound waves generates plosive sound, /t/, where sound is
produced by blocking the airstream and suddenly releasing the built-up
air pressure, fricative sound, /sh/, where a continuous noise type
sounds is made by forcing air through a constricted space, affricate
sound, /ch/, which is a combination of plosive and fricative sound, and
a voiced consonant, /d/, which is a plosive sound followed by a voiced
sound. While vocal folds are open for breathing, the folds are close by
the pivoting of the arytenoid cartilages for speech of singing. Positive
air pressure from the lungs forces the vocal folds to open but the high
velocity air produces a lowered pressure due to the Bernoulli equation
which brings them back together. In an adult male, the vocal folds are
17-23 mm long, and it is around 12.5-17 mm in an adult female. Due to
the action of muscles in the larynx, the vocal folds can be stretched 3
to 4 mm. The frequency of the adult male speaking voice is typically 125
Hz, while the frequency of the adult female voice is about 210 Hz.
Children's voice is around 300 Hz. In comparison to a piano keyboard,
the men's voice would be 1 octave lower than a women's voice, and a
child's voice would be 1 octave higher than an adult women's voice. The
front end of the vocal folds is attached to the thyroid cartilage. The
back end is attached to the arytenoid cartilages, which separates the
folds for breathing. ![](Vocalfolds1.png "Vocalfolds1.png")

## Electrical circuit representation of vocal folds

The vibration model of vocal folds, and the acoustic impedance model of
the vocal can be represented as electrical circuit representation. A
gyrator can be used to combine the acoustic impedance model and the
vibration model, so that the velocity(as potential) from the mobility
analogy for the vibration model can be transferred to the velocity(as
current) in the acoustic impedance model, and where pressure would be
force divided by surface area of the vocal folds tube.

## References

1.  <http://hyperphysics.phy-astr.gsu.edu/hbase/Music/voice.html>


# Engineering Acoustics/Resonating feathers of Manakins

# Introduction

During lekking), Male
Club-winged Manakins, *Machaeropterus deliciosus* (Aves: Pipridae) alter
their secondary feathers by
hypertrophy. The oscillation
of the secondary feathers caused by the cause them to collide and
vibrate in order to produce sustained harmonic tones with a fundamental
frequency of 1500 Hz. The male manakin produces a totally unique sound
in order to attract the attention of the female. Instead of the
conventional way of using voice for sound production, he uses his wings
as a musical instrument. \[\[Image:Golden-headed
Manakin.jpg\|center\|thumb\|500x750px\|

```{=html}
<center>
```
Figure 1: Male Golden Headed Manakin

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

# Mechanism of Sonation

\[\[Image:Manakin.jpg\|thumb\|350x700px\|

```{=html}
<center>
```
Figure 2: Phasing of sonation

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

The male manakin modifies the
sixth and seventh secondary feathers, which act as a pair of coupled
resonators. The five other secondary feathers oscillate in phase with
them to result in sonation, which sounds like a ringing *Tick-Tick-Ting*
\[1\]. The series of motions involves two brief mechanical *ticks*, when
the wings are flicked and a sustained mechanical *ting*, when they are
flipped above the back \[2\]. These two sounds are not acoustically
different except for the longer duration of the *ting*. Each of them is
composed of a fundamental frequency of 1.49 kHz and its higher frequency
harmonics.

Tick-Ting video and audio
demonstration

\[\[Image:Manakin_feather.jpg\|thumb\|350x700px\|

```{=html}
<center>
```
Figure 3 :Modified Secondary feathers

```{=html}
<center>
```
\]\] Unlike typical secondaries, secondary feathers 1--5 exhibit
increasingly wider rachi and an
increasingly pronounced transition from continuously tapering to an
abrupt taper around the distal two-third, three-fourth and then
four-fifth for the third, fourth and fifth secondaries, respectively.
Beyond the sudden taper on the fifth secondary, the rachi bend medially.
This 'kink' in the rachis causes it to overlap and contact the rachis of
the adjacent sixth secondary feather, while at rest\[3\]. The sixth and
seventh secondaries exhibit the following distinct modification: the
rachi is thick at the base, and at approximately one-half of its length,
their width doubles and they twist along their long axis so that the
dorsal feather surface is oriented medially. The sixth secondary feather
has ridges, while the fifth feather has a curved tip \[4\]. The
innermost pair of the modified feathers (seventh secondaries) collide
across the back. Immediately following this collision, the wings shiver
laterally and medially, pulling them just millimeters apart.
Approximately 8ms later, they are
adducted to produce another
collision. The sonation tone is produced continuously throughout this
process and the feathers generate vibrations at just the right frequency
\[3\].

# Acoustic analysis

The male Manakin leans forward and flicks his wings together at a
frequency which peaks at 1500 Hz and has unusually high Q-values. This
frequency is faster than that at which a humming bird beats its wings.
The quality factor Q is a measure of the rate at which a system reaches
maximum amplitude. Using the spectral method, Q can be determined as

$Q=\frac{f_0}{BW - 3  dB, SPL}$

where f0 is the natural frequency of the system and BW -- 3 dB,SPL is
the bandwidth at 3 dB SPL below the peak. From experiments, Q factor was
found to be above 10 for all the feathers and was as high as 27. This
implies that the structures can be good biological resonators \[3\].

# Wing modification

The extra support required for such a high speed wing movement comes
from the super-sized wing bones. The ulna of the male manakin is
modified such that it has bumps and grooves to support the wings and its
width is fourfold the usual width. Another surprising modification is
that the manakin has solid wing bones, as against most birds which have
hollow bones which allow them to fly. The wings thus modified are able
to create the perfect pitch. \[\[Image:TANGARA fêmea ( Chiroxiphia
caudata ).jpg\|thumb\|350x700px\|

```{=html}
<center>
```
Figure 4: Female Blue Manakin

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

# References

\[1\] K. S. Bostwick and R. O. Prum, \"Courting Bird Sings with
Stridulating Wing Feathers,\" Science, vol. 309, pp. 736-, July 29, 2005
2005.

\[2\] K. S. Bostwick, \"Display Behaviors, Mechanical Sounds, and
Evolutionary Relationships of the Club-Winged Manakin (Machaeropterus
deliciosus),\" The Auk, vol. 117, pp. 465--478, 2000.

\[3\] K. S. Bostwick, et al., \"Resonating feathers produce courtship
song,\" Proceedings of the Royal Society B: Biological Sciences, vol.
277, pp. 835--841, March 22, 2010 2010.

\[4\]
<http://www.pbs.org/wnet/nature/episodes/what-males-will-do/photo-gallery-manakin-anatomy/955/>

\<\< Back to Main page


# Engineering Acoustics/Echolocation in Bats and Dolphins

Echolocation is a form of acoustics that uses active sonar to locate
objects. Many animals, such as bats and dolphins, use this method to
hunt, to avoid predators, and to navigate by emitting sounds and then
analyzing the reflected waves. Animals with the ability of echolocation
rely on multiple receivers to allow a better perception of the objects'
distance and direction. By noting a difference in sound level and the
delay in arrival time of the reflected sound, the animal determines the
location of the object, as well as its size, its density, and other
features. Humans with visual disabilities are also capable of applying
biosonar to facilitate their navigation. This page will focus mainly on
how echolocation works in bats and dolphins.

## Sound Reflection

!Figure 1: Reflection and Refraction of
Waves{width="300"}

When a wave hits an obstacle, it doesn\'t just stop there but rather, it
gets reflected, diffracted, and refracted. Snell\'s law of reflection
states that:

$$\frac{sin \theta_i}{c_1}=\frac{sin \theta_t}{c_2}=\frac{sin \theta_r}{c_1}$$

where the nomenclature is defined in Figure 1. The law of reflection
states the angle of incidence is equal to the angle of reflection
($\theta_i=\theta_r$) which is clearly shown in the previous equation.

In order to determine the reflection coefficient $R$, which determines
the proportion of the wave is being reflected, the acoustic impedance is
needed and is define as where $c$ is the speed of sound and $\rho$ is
the density of the medium:

:

    :   `<big>`{=html}`<big>`{=html}$Z=\rho c$`</big>`{=html}\
        `</big>`{=html}

For fluids only, the sound reflection coefficient is defined in terms of
the incidence angle and the characteristic impedance of the two media as
\[3\]:

$$R=\frac{\frac{Z_2}{Z_1}-\sqrt{1-[n-1]\tan^2 \theta_1}}{\frac{Z_2}{Z_1}+\sqrt{1-[n-1]\tan^2 \theta_1}}\qquad$$
where $n=\left ( \frac{c_2}{c_1} \right )^2$

As for the case where medium 2 is a solid, the sound reflection
coefficient becomes \[9\]:

$$R=\frac{(r_n-\frac{r_1}{cos \theta_i})+jx_n}{(r_n+\frac{r_1}{cos \theta_i})+jx_n}\qquad$$
where $Z_n=r_n+jx_n$ is the normal specific acoustic impedance.

The law of conservation of energy states that the total amount of energy
in a system is constant therefore if it is not being reflected, it is
either diffracted, or transmitted into the second medium which may be
refracted due to a difference in refraction index. !Figure 2:
Interaural time difference and Interaural Intensity
Level{width="300"}

## Sound Localization

Sound localization denotes the ability to localize the direction and
distance of an object, or \"target\" based on the detected sound and
where it originates from. Auditory systems of humans and animals alike
use the following different cues for sound location: interaural time
differences and interaural level differences between both ears, spectral
information, and pattern matching \[8\].

To locate sound from the lateral plane (left, right, and front), the
binaural signals required are:

-   Interaural time differences: for frequencies below 800 Hz
-   Interaural level differences: for frequencies above 1600 Hz
-   Both: for frequencies between 800 and 1600 Hz

### Interaural Time Differences

!Figure 3: Shadow cast by the
head{width="300"}

Humans and many animals uses both ears to help identify the location of
the sound; this is called binaural hearing. Depending on where the sound
comes from, it will reach either the right or left ear first, therefore
allowing the auditory system to evaluate the arrival times of the sound
at the two reception points. This phase delay is the interaural time
difference. The relationship between the difference between the length
of sound paths at the two ears, $\Delta d$, and their angular position,
$\theta$, may be calculated using the equation \[1\]:

:

    :   `<big>`{=html}`<big>`{=html}$\Delta d=r(\theta + sin \theta)$`</big>`{=html}`</big>`{=html}

where $r$ is the half distance between the ears. This is mainly used as
cue for the azimuthal location. Thus, if the object is directly in front
of the listener, there is no interaural time difference. This cue is
used at low frequencies because the size of the head is less than half
the wavelength of the sound which allows a noticeable detection in phase
delays between both ears. However, when the frequencies are below 80 Hz,
the phase difference becomes so small that locating the direction of the
sound source becomes extremely difficult.

### Interaural level difference

As the frequency increases above 1600 Hz, the dimension of the head is
greater than the wavelength of the sound wave. Phase delays no longer
allow to detect the location of the sound source. Hence, the difference
in sound level intensity is used instead. Sound level is inversely
proportional to the source-receiver distance, given that the closer you
are to the emitting sound, the higher the sound intensity. This is also
influenced greatly by the acoustic shadow cast by the head. As depicted
in Figure 3, the head blocks sound, decreasing the sound intensity
coming from the source\[4\].

## Active Sonar

Active sonar supplies their own source and then wait for echoes of the
target\'s reflected waves. Bats and dolphins use active sonar for
echolocation. The system begins with a signal produced at the
transmitter with a source level (SL). This acoustic wave has an
intensity of I(r) where r is the distance away from the source. Next,
the source signal travels to the target while gathering a transmission
loss (TL). Once arriving at the target, the fraction of the initial
source signal which is denoted by the target strength (TS), is reflected
toward the receiver. On the way to the receiver, another transmission
loss (TL\') is experienced. For a monostatic case where the source and
the receiver are located in the exact position, TL is equal to TL\',
thus, the echo level (EL) is written as \[9\]:

:

    :   `<big>`{=html}`<big>`{=html}$EL=10\log \frac{I(r) \sigma}{I_{ref} 4 \pi}-TL'$`</big>`{=html}`</big>`{=html}

The equation for target strength is \[9\]:

:

    :   `<big>`{=html}`<big>`{=html}$TS=10\log\frac{\sigma}{4\pi}$`</big>`{=html}`</big>`{=html}

### Reverberation

As sound is emitted, other objects in the environment can cause the
signal to be scattered creating different echoes in addition to the echo
produced by the target itself. If we look at underwater for instance,
reverberation can result from bubbles, fish, sea surface and bottom, or
planktons. These background signals mask the echo from the target of
interest, thus, it is necessary to find the reverberation level (RL) in
order to differentiate between the echo level. The equation for RL is as
follow:

$TS_R$ represents the target strength for the reverberating region and
is defined by:

:

    :   `<big>`{=html}`<big>`{=html}$TS_R=S_v+10\log V=S_A+10\log A$`</big>`{=html}`</big>`{=html}

where \"$V$ (or $A$) is the volume (or surface area) at the range of the
target from which scattered sound can arrive at the receiver during the
same time as the echo from the desired target\" \[9\] and $S_v$
(or$S_A$) is the scattering strength for a unit volume (or a unit
surface area).

## Echolocation in Bats

!Figure 5: Echolocation in
bats{width="300"}
Bats produce sounds through their larynx, emitting them from the mouth
or some, via their nose. Their calls consist of various types: broadband
components (varies in frequencies), pure tone signals (constant
frequency), or a mix of both components. The duration of these sounds
varies between 0.3 and 100 ms over a frequency range between 14 to
100 kHz \[7\]. Each species' calls can vary and have been adapted in a
particular way due to their lifestyles and hunting habits.

The broadband components are used for hunting in a closed environment
with background noise. The short calls yield precision in locating the
target. The short rapid calls, also prevent overlapping of waves, thus,
allowing the use of interaural time difference. Pure tones signals are
used while hunting in open environments without much background noise.
The calls are longer in duration allowing bats to locate preys at
greater distances. When searching for preys, bats emit sounds 10 to 20
times per second. As they approach their target, the number of waves
emitted can reach up to 200 times per second. The usual range for
echolocation is around 17 m.

For other mammals, the interaural time difference and the interaural
intensity level are cues used only for lateral detection. However bats
can also use interaural intensity level to establish objects in the
vertical direction. This is only applicable if the signals received are
broadband. Another difference is that bats' ears are capable of moving,
thus, allowing them to change between different acoustic cues.

The sound conducting apparatus in bats is similar to that of most
mammals. However, over years of evolution, they have been adapted to
suit their needs. One of these special characteristics is the large
pinnae which serves the purpose of acoustic antennae and mechanical
amplifiers. The movement of the pinnae permits focusing of the coming
sound wave, amplifying or weakening it.

## Echolocation in Dolphins

!Figure 6: Echolocation in
Dolphins{width="350"}

The basic idea of echolocation is comparable between bats and dolphins,
however, since both animals live in such different environment, there
are specific characteristics that differ amongst them.

Dolphins use the nasal-pharyngeal area to produce various types of
sounds (clicks, burst pulses, and whistles) in order to achieve two main
functions: echolocation and communication. Clicks, slow rate pulses that
lasts 70-250 μs at a frequencies of 40 to 150 Hz, and bursts, pulses
produced at rapid rates, are primarily used for echolocation \[1\].
After the clicks have been produced, the associated sound waves travel
through the melon, the rounded area of the dolphin\'s forehead comprised
of fats. Its function is to act as an acoustic lens to focus the
produced waves into a beam, sending it ahead. At such high frequencies,
the wave doesn\'t travel very far in water, hence, echolocation is most
effective at a distance of 5 to 200 m for preys that are 5 to 15 cm in
length \[6\].

When waves get reflected after hitting an object, unlike bats that has
pinnae to direct the waves to the inner ear, dolphins receive this
signal via the fat-filled cavities of the lower jaw bones (Figure 6).
Due to the high acoustic impedance of water, these soft body tissues
also have a similar impedance allowing sound waves to travel to the
inner ear without being reflected. Sound is then conducted to the middle
ear and inner ear, from which is then transferred to the brain.

## References

\[1\] Au, W. W. L., Popper, A. N., & Fay, R. R. (2000). Hearing by
whales and dolphins. Springer handbook of auditory research, v. 12. New
York: Springer.\
\[2\] Thomas, J. A., Moss, C., & Vater, M. (2004). Echolocation in bats
and dolphins. Chicago: The University of Chicago Press\
\[3\]Wave Reflection. (n.d.). Retrieved October 17, 2010, from
<http://www.sal2000.com/ds/ds3/Acoustics/Wave%20Reflection.htm>\
\[4\] Diracdelta.co.uk, science and engineering encyclopedia. (n.d.).
Retrieved November 6, 2010, from Interaural Level Difference:
<http://www.diracdelta.co.uk/science/source/i/n/interaural%20level%20difference/source.html>\
\[5\] Murat Aytekin, J. Z. (2007, November 27). 154th ASA Meeting, New
Orleans, LA. Retrieved November 10, 2010, from Sound localization by
echolocating bats: Are auditory signals enough?:
<http://www.acoustics.org/press/154th/aytekin.html>\
\[6\] Seaworld. (2002). Retrieved September 20, 2010, from Bottlenose
Dolphins Communication and Echolocation:
<http://www.seaworld.org/infobooks/bottlenose/echodol.html>\
\[7\] Wikipedia. (2010, February 4). Retrieved September 19, 2010, from
Animal Echolocation: <http://en.wikipedia.org/wiki/Animal_echolocation>\
\[8\] Wikipedia. (2009, March 14). Retrieved September 19, 2010, from
Sound Localization: <http://en.wikipedia.org/wiki/Sound_localization>\
\[9\] Kinsler, L. E. (1982). Fundamentals of acoustics. New York:
Wiley.\


# Engineering Acoustics/The Human Ear and Sound Perception

**Summary:** This page will briefly overview the human auditory system,
transducer analogies, and some non linear effects pertaining to specific
characteristics of the auditory system. Results from the discipline of
Psycho-Acoustics will be
presented.

## Introduction

The human ear is a small physical device with disproportionately large
properties. On one hand it can withstand sounds with acoustic pressure
levels close to 1kPa which are pretty much the loudest encountered in
nature and on the other hand it can detect pressure levels that
correspond to displacements of the eardrum of about one tenth the
diameter of the hydrogen atom.[^1] When including the information
processing done in the brain and the physiological response that it
elicits, one can see why the Human Auditory System has been giving
researchers a hard time since the turn of the twentieth century.\
: Some researchers approached the auditory system as a very complicated,
active transducer;one that transmits the wave information first
acoustically, then mechanically, then hydro dynamically, and finally
electro-dynamically to the brain.[^2] Others like the legendary Georg
Von Bekesy, maintained that the
continuously regenerative nature of the living organism should be taken
into account when considering the behavior of the auditory system.[^3]\
: Humans however are no strangers to complicated problems. After all, we
have been to the moon, so what is going on?

## The Problem with humans

In order to explore the behavior of any physical system one needs a set
of variables describing the system.These variables should be well
defined and arise naturally from the physical principles governing the
behavior of the system.The same physical principles also provide any
researcher with well established means of assessing what constitutes a
valid measurement.\
Furthermore,in any well behaved physical system the experimenter has
control over the variables, to such an extent that he or she can hold
most of the variables constant and individually vary a few of them to
evaluate the relationship between them and quantify their dependence.\
Additionally, in any linear system the principle of superposition holds,
so that the overall effect of varying several variables at the same time
equates to a linear combination of the individual contributions observed
from varying every individual variable independently while keeping
everything else constant\
: The above mentioned usually constitute what can be described as a very
happy researcher.However, problems arise when one sets out to evaluate
the human auditory system, because hearing is a sensation and just like
every other sensation it is an esoteric process. To resolve this problem
one has to venture into the realm of psycho-physics and the principles
of psychological measurements. It is known that one cannot directly
measure sensation, but one can measure the response that the sensation
elicits.[^4]\
: With the above approach one can measure quantities such as just
noticeable differences; perceptible excitation; increased nervous
activity etc. However, the validity or relevance of those measurements
cannot readily be confirmed by first principles.[^5] The nature of the
human auditory system is such that one is not able to decouple and
independently vary any of the variables of interest(however those might
be defined) and even if one could, the principle of superposition,in
general, does not apply.

## Non Linearity \| Part1

After acknowledging the difficulties involved in quantifying the
behavior of the auditory system and developing models of hearing one
should take a look into specific sources of non-linearity and the
mechanisms through such a behavior is imposed upon the auditory system.
There is probably no better example of such behavior than what is called
the acoustic, auditory or intra-aural reflex.

### The Acoustic Reflex

The acoustic reflex in man refers to the tendency of the middle ear
muscles controlling the behavior of the ossicles (the little bones in
the middle ear) to tense under an intense acoustic stimulus, thereby
making the inner ear stiffer and in that way limiting the motion of the
stapes (the last bone in the chain). This reduction in the motion of the
stapes equates to a real rather than a perceived reduction in the
amplitude of the vibrations transmitted through the middle ear to the
inner ear. The reflex serves to protect the sensitive inner ear from
damage during exposure to loud sounds.

Unfortunately, although fast, the auditory reflex it is not an
instantaneous reaction. For low frequencies, the response takes from 20
to 40ms to be elicited and therefore offers no protection against loud
impulsive sounds like gunshots and explosions.[^6] With the onset of the
auditory reflex the entire ear exhibits a marked change in acoustic
impedance which was observed by 1934 Geffcken and measured by Bekesy and
other researchers in subsequent years. It is argued, however, that the
onset of the auditory reflex happens for sound of very high intensity
and therefore its effect on perception is limited.[^7] On the other
hand, the same reflex can be voluntarily elicited by, for example,
vocalizing. According to Lawrence A. Kinsler, it seems that the
mechanical characteristics of the ear are mainly responsible for the
response elicited by the auditory system and hence sound perception.[^8]
Whatever the exact nature of the auditory reflex may be, or what the
precise range that it has the most effect however is beyond the scope of
this article.

### Perceived loudness of pure tones

The intensity and loudness of sound are two highly interdependent
quantities. Loudness belongs to the psychological attributes of sound
and intensity is a precisely defined and measurable physical quantity.
Because of their strong similarity, the two quantities were once thought
to be one and the same, since if one increases the intensity of a
particular sound, the sound becomes louder.[^9] In the simplest and
clearest terms: intensity is measured sound level and loudness is
perceived sound level.

The measured sound level is expressed in terms of **intensity** and
**intensity level**, while the perceived sound level in expressed in
terms of **loudness** and **loudness level**.

Sound Intensity is defined as the acoustic power per unit area and it is
measured in Watts per square meter\
$$I=\frac{Power}{Unit Area}\quad\left[\frac{W}{m^2}\right]$$

However, the human ear is capable of detecting sound intensity ranging
from 1x10^−12^Wm^−2^ to 1x10^2^Wm^−2^(above which intensity permanent
damage to the ear will occur). This gives a scale in which the maximum
value is 10 000 000 000 000 times larger than the smaller one.[^10]

In order to provide more insight and get around the cumbersome numbers
we use the **Intensity level** **I~L~** which is defined as the
intensity relative to 10x10^−12^ Wm^−2^, on a logarithmic scale and it
has unit of decibels.\
.

$$I_L=10\log\left(\frac{I}{I_{ref}}\right)\quad[dB]$$

In planar waves in air and standard temperature and mean pressure,
acoustic pressure and Intensity are linked by the following relation:

$$I=\frac{P^2}{\rho c}$$

Where ρ is the air density and c the speed of sound in air. By doing the
following:\
$$I_L=10\log\frac{\left(\frac{P^2}{\rho c}\right)}{\left(\frac{P_{ref}^2}{\rho c}\right)}=10\log{\left(\frac{P}{P_{ref}}\right)}^2=20\log\frac{P}{P_{ref}}=SPL$$

The expression on the right is deemed the **Sound Pressure Level** and
it is identical to the Intensity Level, but in terms of acoustic
pressure.The reference pressure used is 20μPa. It is very close to the
average minimum audible acoustic pressure in air in the absence of any
noise.[^11] It is important to note that the minimum audible pressure is
averaged over multiple subjects, therefore for a given percentage of the
population, negative Sound pressure levels are perceptible i.e. they can
perceive sound pressures smaller than the reference pressure. The chosen
reference pressure level corresponds to the reference Intensity through
the aforementioned relationship, in a way that SPL and I~L~ are
identical.

The qualitative expressions of loud, not very loud, extremely loud, are
used to describe loudness. Although these expressions are adequate in
describing the sensation in any specific individual, they do a very poor
job in quantifying the result. The above qualitative expressions have
been made qualitative for pure tones, i.e. sinusoidal waves, with the
use of **Loudness Level** and **Loudness**.

The **loudness level** of a particular test tone is an indirect measure
of loudness and it is defined as the **Sound Pressure Level(SPL)** of a
1000 Hz pure tone that sounds as loud as the test tone.[^12] The 1000 Hz
tone was chosen arbitrarily and retained as the standard. The Loudness
level is measured in phons. The Loudness Level of the just audible
1000 Hz tone is defined as 3 phons because the minimum perceptible SPL
of a 1 kHz tone is 3 dB. Increments in phons are logarithmic because the
SPL is measured in decibels.

The **loudness level** is very useful in quantifying the sensation,
however it fails to provide information on the relation between sounds
of different loudness levels. In other words it fails to provide insight
on how much louder a sound of e.g. 20 phons is than a sound of 50 phons.
To get around this problem, we use **Loudness** which has units of
sones. **Loudness** is based on the 40 dB, 1000 Hz pure tone which is
defined as to have a loudness of 1 sones. The Loudness scale is derived
by increasing or decreasing the SPL of the 1 kHz tone until it \"sounds
twice as loud as before\" or \"half as loud \" etc. Successive halving
of the loudness creates the rest of the scale. The **Loudness** for the
remaining tones is determined by the same equal loudness judgment that
provides the **Loudness Levels**.[^13]

Loudness and Loudness level are best illustrated and are most useful
when plotted against the SPL of pure tones, in what are called equal
loudness contours or Fletcher & Munson
curves, so named after the
earlier researchers, however the way loudness is measured has been
significantly altered and standardized since the time when such
measurements were first made.\

<File:Lindos1.svg%7CEqual> Loudness contours

## A little bit of bio

### The cochlea

The **cochlea**, or inner ear, constitutes the hydrodynamic part of the
ear. It is a small, hollow, snail shaped member formed from bone and
filled with colorless liquid. It has an uncoiled length of about 35mm
and a cross-sectional area of about 4mm^2^ on the end closest to the
inner ear, that tapers off to about 1mm^2^ at the far end.[^14]\
: It is filled with **two** different fluids separated in **three**
different channels that run together from the base of the stapes to the
apex of the cochlea, however two of the channels are separated by
Raleigh\'s membrane, which is thin and flexible enough to be neglected
from a hydromechanical point of view.[^15] The vibrations are
transmitted directly from the base-plate of the stapes, the last of the
three ossicles, to the contained fluid. The cochlea is divided down the
middle by the **basilar membrane** which is a partly bony and partly
gelatinous membrane. It is on this membrane that the organ of corti and
the infamous **hair cells** reside.

<File:Anatomy> of Human Ear with Cochlear Frequency Mapping.svg\|The
Auditory system and Cochlea <File:Cochlea.png%7CThe> three Fluid filled
cavities

### The basilar membrane

As previously mentioned, the basilar membrane is a flexible gelatinous
membrane that divides the cochlea longitudinally. It is the flexible
part of the cochlear partition (the other being rather bony)and it
contains about 25 000 nerve endings attached to numerous haircells
arranaged on the surface of the membrane. It extends from the base to
just before the apex of the cochlea at which point it terminates at the
helicotrema. This creates two hydromechanically distinct channels, with
the baseplate of the stapes attached to the entrance of the upper
channel at the **oval window**, and a highly flexible membrane called
the **round window** sealing the lower channel. The two channels connect
at the apex through the **helicotrema** which is basically a gap through
the cochlear partition.\
:

<File:Gray928.png%7CDiagrammatic> longitudinal section of the cochlea
showing the location of the Basilar Membrane <File:Two> Views of
Cochlear Mechanics.svg\|Two Views of Cochlear Mechanics

:

The vibrations transmitted to the stapes set up acoustic waves in the
fluid that travel down the upper channel, through the helicotrema and
back up through the lower channel. Since the walls of the cochlea are
relatively rigid and the contained fluids relatively incompressible,
this causes the basilar membrane to flex. In order to equalize the
pressure in the cochlea, the round window \"bulges out\" and in this way
provides pressure release.

:

The basilar membrane starts out narrowly, with a width of about 0.04mm
near the oval window and then widens to about 0.5mm near the
helicotrema. This non uniformity in width, along with the pressure
release provided at the round window cause the basilar membrane to
exhibit maxima of vibration at different locations (distances from the
oval window) along the membrane, depending on the frequency of
vibration. This makes the basilar membrane behave as an acoustic filter
that separates the constituent frequencies of an incoming sound signal
depending on the location of the maxima.\
:Basilar Membrane
Animation

:

<File:Uncoiled> cochlea with basilar membrane.png\|Uncoiled cochlea with
basilar membrane

:

### Whats up with all the hair?

The hair-cells that populate the top surface of the basilar membrane are
the last part in the chain of transformation of the mechanical energy of
the acoustics wave into electrical impulses. These cells are arranged in
an inner row and an outer row in the organ of corti (which runs along
the basilar membrane) and they are surrounded by electrically charged
cells at different potentials(synapses).[^16][^17]\
:

<File:Gray932.png%7CSection> exposing the hair and hair-cells
<File:Cochlea-crosssection.svg%7CCochlea-crosssection> with the
hair-cells visible

:

As was already mentioned, the basilar membrane exhibits various
vibration maxima at different locations when excited by a sound input.
As a result of these excitations a relative motion of the fluid parallel
to the membrane is effected. This motion produces a shear force on
myriads of minuscule hairs protruding from these cells. The disruption
produces an electrochemical cascade on the surrounding electrically
active cells, which results in a signal to be sent to the brain.\
: What is really important is to note that these hair-cells are not
evenly distributed over the surface of the basilar membrane, but rather
they are concentrated in discreet patches. Since different frequencies
make different different parts of the membrane vibrate more than others,
this means that there are ranges of frequencies that we can perceive
better than others, depending on the number density of the hair-cells
surrounding the corresponding region on the basilar membrane. This
introduces discreetness and gives a sort of minimum resolution to our
sense of hearing, thus causing some interesting non linear effects to be
discussed soon.

:

<File:Hair> Cell Patterning Defects in the Cochlea.png\|Arrangement of
the hair cells on the cochlea. Left=healthy;Right=pattern defects

:

Due to the similarity between the behavior of the inner ear and the
behavior of a band-pass filter, the above groups of frequencies have
been named **critical bandwidths**.[^18]

## Non Linearity \| Part 2

Now that a little bit more has been presented about the workings of the
inner ear, more peculiarities of the idiosyncratic auditory system can
be illustrated, starting with a non linear effect that is fairly common
and very noticeable when it occurs. It is the phenomenon of beating.

### Beating Phenomena

Beating phenomena are a characteristic of multiple degree of freedom
systems, where the various degrees of freedom are coupled to some extent
and that receive two harmonic excitations at slightly different
frequencies. The excitations can be summed as follows:[^19]

:   $$x=A_1e^{j(\omega_1 t)+\Phi_1}+A_2e^{j(\omega_2 t)+\Phi_2}$$

The resulting vibration is no longer simple harmonic.

:

The inner ear is a continuous system, with the basilar membrane serving
as a complicated bandpass filter to separate frequencies. When one or
both ears are exposed to sound that consists of two tones with a slight
difference in their frequencies, the non uniform distribution and strong
localization of the hair cells on the surface of the basilar membrane
result in the same group (or critical bandwidth) of hair cells being
excited by both tonal components of the incident sound.\
:

<File:Beating> Frequency.svg\|Beating

As a result, the listener perceives the combination sound as that of a
single frequency tone but with periodically varying intensity. This is
known as beating.\
: The tones remain indistinguishable until the frequency separation
between them , is greater than the a bandwidth. It is really interesting
to note that if the two tones are presented to each ear separately, then
no beating occurs and the ear is able to resolve the difference.[^20]

## References

## Links and Resources

-   Dancing Hair Cell
-   American Society of Acoustics
-   Table of sound
    levels
-   Basilar Membrane
    Animation

[^1]: Acoustics, Leo L. Beranek 1993, Copyright: Acoustical Society of
    America, Ch13 Hearing, Speech Inteligibility and Psychoacoustic
    Criteria

[^2]: Overall Loudness of Steady Sounds According to Theory and
    Experiment, Walton L. Howes, Nasa reference Edition 1001

[^3]: THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 23.NUMBER
    5, SEPTEMBER.1951,DC Potentials and Energy Balance of the Cochlear
    Partition,GEORG V. Bekesy,Psycho-Acousic Laboratory, Harvard
    University, Cambridge, Massachusetts,(Received May 5, 1951)

[^4]: The Measurement of Hearing,Ira J. Hirsh, McGraw Hill Book Company,
    Inc, First Edition,1952

[^5]:

[^6]: Fundamentals of Acoustics,Lawrence E. Kinsler, Alan B.
    Coppens,4rth Edition

[^7]: Acoustic Reflex in Man,Aage R. Moller, J. Acoust. Soc. Am. 34,
    1524 (1962), <DOI:10.1121/1.1918384>

[^8]:

[^9]:

[^10]: <http://www.engineeringtoolbox.com/sound-intensity-d_712.html>

[^11]:

[^12]:

[^13]:

[^14]:

[^15]:

[^16]:

[^17]:

[^18]:

[^19]:

[^20]:


# Engineering Acoustics/Conductive Mechanisms of Hearing

!Thumb\|right\|border\|400px\|Anatomy of the Human
Ear

In this section we will discuss the pathways for conduction of acoustic
sound waves to the inner ear. Two methods of conduction will be covered;
first we will cover conduction to the inner ear through the outer and
middle ear then conduction to the inner ear through bone conduction.
Without these conductive pathways the acoustic waves that transmit sound
would not be able to reach the inner ear, and we would therefore be
unable to hear.

## Outer Ear

The first area where sound enters the ear is in the outer ear which can
be broken down into two parts, the Pinna (visible portion of the ear)
and the ear canal.

### Pinna

It was once thought that the Pinna funneled sound into the ear canal,
but the primary function of the Pinna is to aid in sound source
localization. The various ridges in the Pinna filter the signal at
frequencies over 4000 Hz according to the direction of the sound.The
resulting spectral variations allow our brain to determine the elevation
of a sound and localize sounds in reference to our position.[^1]

### Ear Canal

The ear canal can be modeled as a tube approximately 2.8 cm long that is
closed at one end by the tympanic membrane. Unlike a rigid tube that has
a sharp spike in amplitude at the resonant frequency, the ear canal has
a wide resonance peak from approximately 2--5 kHz.[^2] It is important
to remember the ear canal is not rigid nor is not a straight tube, and
therefore damping is introduced.While damping is introduced to the
system, thereby altering the resonant frequency, the important thing is
that the ear canal can raise the Sound Pressure
Level (SPL) of the ear by up to 15 dB
which will amplify the incoming acoustic signal.[^3]

!Thumb\|center\|border\|400px

## Middle Ear

The primary purpose of the middle ear is to act as an impedance-matching
transformer that will allow for acoustic energy to be efficiently
transferred from the air filled outer ear to the liquid filled cochlea.
Knowing the impedance ratio, r, of the liquid in the cochlea to the air
is 4000:1 we can use the following equation to determine what the energy
transmission coefficient, T, would be without the middle ear

```{=html}
<center>
```
$T = \frac{4r}{ \left( r+1 \right) ^2}$

```{=html}
</center>
```
which gives us a transmission of 0.001 or 0.1%. This transmission value
is equivalent to a SPL drop of around 40 dB.[^4]

To overcome this impedance mismatch,the middle ear employs three
mechanical amplification systems; the areal ratio between the Tympanic
Membrane and the
Stapes, the areal ratio of the
stapes to the Oval Window
the lever effect of the Ossicular
Chain.

### Areal Ratio

While the ossicular chain helps to increase the pressure of the incoming
acoustic signal, the majority of gain in the middle ear is due to the
areal ratio of the tympanic membrane and the stapes. The average area of
the tympanic membrane is 66 $mm^2$ while that of the stapes is 3.2
$mm^2$. The areal ratio between the two areas should therefore be
approximately 20:1, but as mentioned earlier the effective area of the
tympanic membrane is only 65% so, 66 $mm^2$ x 65% = 42.9 $mm^2$. This
results in an areal ratio of 13.4.

!Thumb\|center\|border\|375px

### Tympanic membrane

The tympanic membrane plays an important role in the conduction of
sound. Serving as a
Transducer, the membrane
converts acoustic pressure waves into mechanical motion. Experiments
have shown that the tympanic membrane is not uniform, but has a
spatially variable stiffness. Due to these variations, the effective
area of the membrane is only 65%.[^5] It is important to mention that
the stiffness of the tympanic membrane is also very important to the
efficient transfer of energy as being too stiff would reflect a large
amount of energy back through the ear canal, while being too flacid
would cause the membrane to absorb too much energy.[^6]

Another Important contribution of the Tympanic membrane is the gain
contributed by the curved membrane principle. This model represents the
membrane in two sections with the manubrium at the center. Due to the
curved membrane principle, the force on the manubrium is greater than on
the membrane and thus the strength of the incoming signal is increased
by a factor of 2.[^7] Including these effects, we find the total gain of
the areal ratio and tympanic membrane will be 26.8. Using this pressure
ratio we can calculate the associated gain in decibels

```{=html}
<center>
```
$\text{dB Gain} = 20\,log(26.8/1) = 28.6\,dB\$

```{=html}
</center>
```
Similar to the areal ratio between the tympanic membrane and the stapes,
the areal ratio between the stapes and the oval window of the
Cochlea is 20:1

```{=html}
<center>
```
$\text{dB Gain} = 20\,log(20/1) = 26\,dB\$

```{=html}
</center>
```
Which yields a pressure gain of 26 dB.[^8]

### Ossicular Chain

The ossicular chain functions like a basic lever. The manubrium of the
Malleus is 1.3 times longer than
the long process of the Incus, and
the two ossicles are connect with ligaments so that they can move
together. Their attachment in the middle ear evenly distributes their
mass about an axis of rotation which allows them to be easily set into
motion by incoming acoustic signals. In addition, the ossicles are
damped enough that when the incoming signal is stopped, the malleus and
incus stop as well which is desirable as continued motion of the
ossicles would lead to echoing. The difference in lengths between the
two ossicles also corresponds to a pressure gain at the incus equivalent
to the length ratio of the two ossicles. This can be best shown by
modeling the two bones as a simple lever:

!Thumb\|center\|border\|325px\|Anatomy of the Human
Ear

Knowing this pressure ratio we are able to compute the equivalent gain
in decibels:[^9]

```{=html}
<center>
```
$\text{dB Gain} = 20\,log(1.3) = 2.3\,dB\$

```{=html}
</center>
```
While the pressure gain is small, it still contributes to the conduction
of sound to the inner ear.

### Overall Middle Ear Gain/ Ideal Transformer Prediction

Using the pressure gain from the ossicular chain, the areal ratio of the
tympanic membrane to the stapes, and the areal ratio of the stapes to
the oval window we find the total gain from the middle ear is

```{=html}
<center>
```
$\text{Middle Ear dB Gain} = 2.3\,dB+ 28.6\,dB\ + 26\,dB = 56.9\,dB$

```{=html}
</center>
```
This gain in the middle ear is called \"The Ideal Transformer
Prediction\...compensate\[s\] for the air-to-cochlea impedance
mismatch\" [^10] and has been confirmed in tests performed on cadavers
with the middle ears removed.

## Bone Conduction

Bone conduction is the transfer of sound to the cochlea through the
bones of the skull. In for conduction to occur in the skull, the
air-conduction threshold of sound must be exceeded by 50-60 dB which
often occur through direct vibration to the skull, primarily the
temporal bone. When this threshold is exceeded, the traveling waves
produced in the cochlea resemble those induced by the stapes.[^11] Due
to the impedance mismatch, bone conduction does not play a significant
role in hearing, but is often used for measuring hearing to determine if
there is damage to the middle ear, or to test the viability of the inner
ear.[^12]

One use for bone conduction is with people with a damaged middle ear but
having a normal functioning inner ear. For these people a traditional
hearing aid can not be used, but due to a healthy inner ear bone
conduction hearing aids can be. For these hearing aids a titanium screw
can be implanted into the skull, and an external device housing the
microphone and receiver can conduct acoustic waves to the inner ear
through the temporal bone. While hearing through bone conduction is not
perfect, it does allow for clear and understandable sound
recognition.[^13]

## References

[^1]: Gelfand, Stanley A. Hearing, an Introduction to Psychological and
    Physiological Acoustics. New York: M. Dekker, 1981. Print. p.85-102

[^2]:

[^3]: Kinsler, Lawrence E. Fundamentals of Acoustics. New York: Wiley,
    2000. Print. p.312-15

[^4]:

[^5]: Durrant, John D, and Jean H. Lovrinic. Bases of Hearing Science.
    Baltimore: Williams & Wilkins, 1977. Print.

[^6]: Auditory Function. Emanuel,Maroonroge,Letowski.
    <http://www.usaarl.army.mil/new/publications/HMD_Book09/files/Section%2016%20-%20Chapter%209%20Auditory%20Function.pdf>

[^7]:

[^8]:

[^9]: Hamill, Teri, and Lloyd L. Price. The Hearing Sciences. San Diego:
    Plural Pub, 2008. Print. p.166-69

[^10]:

[^11]:

[^12]: Yost, William A. Fundamentals of Hearing: An Introduction. San
    Diego: Academic Press, 2000. Print. p.72

[^13]:


# Engineering Acoustics/Hearing Protection

## Introduction

The human ear is in constantly exposed to noise. In some situations, the
intensity of this noise can be infuriating, like on a subway, a train,
or a plane. One may want to put on headphones, and crank up the music
volume to overcome the maddening turbojet engine noise or the roar of
the city. In the cabin of an aircraft, the intensity of the noise during
cruise condition is about 85 dB, while reaching over 100 dB during
take-off and landing. A solution to reduce the exposure to high noise
levels is the use of noise-canceling headphones.

## Noise Control Mechanisms

There are two types of noise-canceling headphones. One using passive
elements, the other one making use of active elements.

### Passive Noise Control

Passive noise control elements do not require any source of energy. The
noise reduction comes from the material and the shape of the hearing
device. An example of headphones only using a passive element to block
sound are the earmuffs and ear plugs worn by workers on construction
sites. This type of headphones can reduce the noise level by about 15 to
25 decibels [^1].

The hearing protection device acts as a sound barrier. Sound barriers
are more effective with high frequency noise. Large wavelength noise, or
low frequency sounds, can bend around the device more easily, while high
frequency sound will be diffracted. An important factor in how well the
headphones will block the outside noise is how good the seal created is.
For the same pair of headphones or earphones, the attenuation can be
very good or bad, depending on how they fit on the user.

The phase speed of the device is described by the bulk speed[^2],

$c^2=\frac{(\mathcal{B}+\frac{4}{3}\mathcal{S})}{\rho_0}$

where $\mathcal{B}$ and $\mathcal{S}$ are the bulk and shear moduli of
the solid. $\rho_0$ is its density. The reflection coefficient in a
solid can be described as
$\mathbf{R}=\frac{\frac{r_n-r_1}{\cos{\theta_i}}+jx_n}{\frac{r_n+r_1}{\cos{\theta_i}}+jx_n}$

where $r$ and $n$ denote the specific acoustic resistance and reactance,
respectively. The specific acoustic resistance is

$r=\rho c$

The subscripts $n$ and $1$ denote the property in the normal direction
and of the first medium, respectively. With the reflection coefficient,
the level of attenuation and level of transmission can be found. The
power transmission coefficient is

$T_\pi(\theta)=\frac{1}{1+[(\omega \rho_s / 2r_1)\cos{\theta}]^2}$

### Active Noise Control

What makes noise-canceling headphones such an interesting device is the
use of the active element to eliminate undesirable noise. It is
important to note that a typical active noise-canceling headphone will
also have a passive element that will serve as a first barrier to the
unwanted, high frequency sound waves. Behind the noise barrier, a
combination of microphone, electrical circuit, and speaker is set in
place to destroy some of the noise that made it through the passive
element. The steps to cancel the undesirable noise are fairly
straightforward. First, the microphone detects the noise coming from
outside. Then the electrical circuit reads the information coming from
the microphone and creates a noise signal that has the same frequency
and amplitude as the outside noise, but with a phase of 180° to the
outside source. This signal is sent to the loudspeaker, which will put
out the desired sound[^3]. What happens then is the outside sound waves
get canceled -- or destroyed -- by the sound waves generated by the
speaker. The difficult part is the implementation of the system. The
limiting factor in the efficiency of noise cancelling devices is the
reaction time of the system. \"Coming within 25 degrees of the needed
180-degree phase shift can cut noise by 20 decibels. Headphones that
react more slowly provide less cancellation.\"[^4] claims Mark
Fischetti. Active noise control is effective at low frequencies. For
higher frequency noise, the required response time of the system is too
small for the device to be able to destroy the incoming sound waves.
Such a system is feasible but require complex electronics, difficult to
implement in a relatively small device.

In their studies on different brands of noise-cancelling headphones, L.
Y. L Lang et al. showed that active noise are \"reliable in stationary
noise environments as opposed to those in environment that are highly
transient.\"[^5] In situations with highly transient sound, like in an
airport or inside the cabin while the aircraft is taking off, the active
noise-control system can sometimes become ineffective, or even increase
the sound pressure level. They found that a certain pair of headphones
increased the sound pressure level by 20.4 dB at a frequency of around
600 Hz. In transient noise environment, the headphones can show
fluctuations in the level of attenuation of sound pressure, especially
in the range of 100 Hz to 1000 Hz. The efficiency of the
noise-cancelling headphones can be calculated by finding the insertion
loss. The insertion loss is measured by finding the difference in sound
pressure level in the ear when no headphones are worn and when the
active noise-cancelling device is activated and worn over the ears.

$IL_{T}=L_{0}-L_{C-ON}$

In this equation, $IL_{T}$ is the total insertion loss, $L_{0}$ is the
sound pressure level without the headphones and $L_{C-ON}$represents the
sound pressure level when the ears are covered and the active element is
turned on. The active noise-cancelling headphones have the highest
insertion loss at frequencies below 230 Hz. This is a perfect solution
for usage of headphones while commuting. In an aircraft cabin or a bus
cabin, the sound pressure level is maximum at around 110 Hz. By having
the maximum attenuation at those frequencies, one can protect their ears
from the possibly dangerous noise level found in different
transportation methods.

On top of canceling low frequency noise, noise-cancelling headphones can
improve speech intelligibility. By attenuating low frequency sound
waves, the frequencies located between 500 Hz and 1500 Hz can be
isolated leading to a more intelligible speech. In their 2001 report,
Mongeau, Bernhard and Feist concluded that noise-cancelling headphones
could be a solution to improve communication at noisy toll booth, where
car and truck noise can lead to difficult conversations[^6]. The report
states that a good communication system could raise the speech
intelligibility index (SII) up to 0.75. In regular conditions in a toll
booth, the SII can be as low as 0 when using a regular or raised vocal
effort, or 0.06 when using a loud voice. In perfect conditions, the SII
is equal to 1. Although noise-cancelling devices improve speech
intelligibility, the main issue was the reception of the idea. Having
someone wearing a pair of over-the-ear headphones while serving a client
does not give a good impression as people tend to think the attendant is
also focusing on something else.

Since it is an active system, energy must be added to the system, so the
headphones require an energy source such as a typical battery.

## Types of headphones

There are three different design of headphones. First, the one providing
the least amount of noise attenuation is the supra-aural headphone. This
type of headphone simply sits on the ears. While it is not efficient in
blocking the noise, it provides a more natural sound. The fact that it
is more open to the outside environment allows the headphones to sound
more like a speaker.

The next type is the circumaural headphone. This type of headphones fits
around the user's ears, completely isolating them. They provide good
sound attenuation because they can block sound coming from all
directions. On the other hand, they are usually heavier and not as
comfortable. Due to the size of the circumaural headphones, it is
possible to add an active element to even further decrease outside
noise. This will be discussed in the next section.

Finally, the last type is the in-ear headphone. They are worn directly
in the ear. There are two styles of in-ear headphones. The ear buds are
worn in the opening of the ear, while the canal earphones are worn in
the ear canal. This type provides the best passive noise attenuation as
they can efficiently create a seal and block the outside noise. They
also provide a high-quality sound.

## Conclusion

The noise-canceling headphones are a good example of how theoretical
science can be applied and used to solve an everyday problem. With the
speaker-generated sound wave having the same amplitude and frequency as
the outside sound waves, but with a phase difference of , the two sound
waves cancel each other and the result is a quieter commute, safer for
the ears and the mind.

## Reference

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

[^1]: M. Fischetti, \"Noise-canceling headphones. Reducing a roar,\"
    *Scientific American,* vol. 292, no. 2, pp. 92-3, 2005.

[^2]: L. E. Kinsler, A. R. Frey, A. B. Coppens and J. V. Sanders,
    Fundamentals of Acoustics, United States of America: John Wiley &
    Sons, Inc., 2000.

[^3]: W. Harris, \"How Noise-canceling Headphones Work,\" 15 February
    2007. \[Online\]. Available:
    https://electronics.howstuffworks.com/gadgets/audio-music/noise-canceling-headphone.htm.
    \[Accessed 25 February 2018\].

[^4]:

[^5]: L. Y. L. Ang, Y. K. Koh and H. P. Lee, \"The performance of active
    noise-canceling headphones in different noise,\" *Applied
    Acoustics,* no. 122, pp. 16-22, 2017.

[^6]: L. Mongeau, R. J. Bernhard and J. P. Feist, \"Noise Control and
    Speech Intelligibility Improvement of a Toll Plaza,\" The Institute
    for Safe, Quiet and Durable Highways, West Lafayette, Indiana, 2001.