# Engineering Acoustics/Electro-acoustic analogies

## Electro-acoustical Analogies

### Acoustical Mass

Consider a rigid tube-piston system as following figure.

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

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Piston is moving back and forth sinusoidally with frequency of **f**.
Assuming $f << \frac{c}{l\ or\ \sqrt{S}}$ (where **c** is sound velocity
$c=\sqrt{\gamma R T_0}$), volume of fluid in tube is,

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$\Pi_v=S\ l$

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Then mass (mechanical mass) of fluid in tube is given as,

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$M_M= \Pi_v \rho_0 = \rho_0 S\ l$

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For sinusoidal motion of piston, fluid move as rigid body at same
velocity as piston. Namely, every point in tube moves with the same
velocity.

Applying the Newton\'s second law to the following free body diagram,

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

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$SP'=(\rho_0Sl)\frac{du}{dt}$

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$\hat{P}=\rho_0l(j\omega)\hat{u}=j\omega(\frac{\rho_0l}{S})\hat{U}$

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Where, plug flow assumption is used.

`"Plug flow" assumption:`\
`Frequently in acoustics, the velocity distribution along the normal surface of`\
`fluid flow is assumed uniform. Under this assumption, the acoustic volume velocity U is`\
`simply product of velocity and entire surface. `$U=Su$

#### Acoustical Impedance

Recalling mechanical impedance,

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$\hat{Z}_M=\frac{\hat{F}}{\hat{u}}=j\omega(\rho_0Sl)$

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acoustical impedance (often termed an **acoustic ohm**) is defined as,

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$\hat{Z}_A=\frac{\hat{P}}{\hat{U}}=\frac{Z_M}{S^2}=j\omega(\frac{\rho_0l}{S})\quad  \left[\frac{N s}{m^5}\right]$

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where, acoustical mass is defined.

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$M_A=\frac{\rho_0l}{S}$

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#### Acoustical Mobility

Acoustical mobility is defined as,

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$\hat{\xi}_A=\frac{1}{\hat{Z}_A}=\frac{\hat{U}}{\hat{P}}$

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#### Impedance Analog vs. Mobility Analog

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![](Imp-mov.gif "Imp-mov.gif")

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#### Acoustical Resistance

Acoustical resistance models loss due to viscous effects (friction) and
flow resistance (represented by a screen).

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

***r~A~***is the reciprocal of***R~A~*** and is referred to as
*responsiveness*.

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### Acoustical Generators

The acoustical generator components are pressure, **P** and volume
velocity, **U**, which are analogus to force, **F** and velocity, **u**
of electro-mechanical analogy respectively. Namely, for impedance
analog, pressure is analogous to voltage and volume velocity is analogus
to current, and vice versa for mobility analog. These are arranged in
the following table.

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![](e-a_analogy.gif "e-a_analogy.gif")

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Impedance and Mobility analogs for acoustical generators of constant
pressure and constant volume velocity are as follows:

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

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### Acoustical Compliance

Consider a piston in an enclosure.

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

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When the piston moves, it displaces the fluid inside the enclosure.
Acoustic compliance is the measurement of how \"easy\" it is to displace
the fluid.

Here the volume of the enclosure should be assumed to be small enough
that the fluid pressure remains uniform.

Assume no heat exchange 1.adiabatic 2.gas compressed uniformly , p prime
in cavity everywhere the same.

from thermo equitation ![](Equ1.jpg "Equ1.jpg") it is easy to get the
relation between disturbing pressure and displacement of the piston
![](Equ3.gif "Equ3.gif") where U is volume rate, P is pressure according
to the definition of the impendance and mobility, we can
get![](Equ4.gif "Equ4.gif")

Mobility Analog VS Impedance Analog

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

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### Examples of Electro-Acoustical Analogies

Example 1: Helmholtz Resonator

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![](Example1holm.JPG "Example1holm.JPG")

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Assumptions - (1) Completely sealed cavity with no leaks. (2) Cavity
acts like a rigid body inducing no vibrations.

Solution:

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\- Impedance Analog -

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![](Example2holm1sol.JPG "Example2holm1sol.JPG")

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Example 2: Combination of Side-Branch Cavities

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![](Exam2prob.JPG "Exam2prob.JPG")

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Solution:

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\- Impedance Analog -

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![](Exam2sol.JPG "Exam2sol.JPG")

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