Engineering Acoustics/Electro-acoustic analogies

Electro-acoustical Analogies

Acoustical Mass

Consider a rigid tube-piston system as following figure.

Piston is moving back and forth sinusoidally with frequency of f. Assuming $f<<{\frac {c}{l\ or\ {\sqrt {S}}}}$ (where c is sound velocity $c={\sqrt {\gamma RT_{0}}}$), volume of fluid in tube is,

$\Pi _{v}=S\ l$

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

$M_{M}=\Pi _{v}\rho _{0}=\rho _{0}S\ l$

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,

$SP'=(\rho _{0}Sl){\frac {du}{dt}}$ ${\hat {P}}=\rho _{0}l(j\omega ){\hat {u}}=j\omega ({\frac {\rho _{0}l}{S}}){\hat {U}}$

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,

${\hat {Z}}_{M}={\frac {\hat {F}}{\hat {u}}}=j\omega (\rho _{0}Sl)$

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

${\hat {Z}}_{A}={\frac {\hat {P}}{\hat {U}}}={\frac {Z_{M}}{S^{2}}}=j\omega ({\frac {\rho _{0}l}{S}})\quad \left[{\frac {Ns}{m^{5}}}\right]$

where, acoustical mass is defined.

$M_{A}={\frac {\rho _{0}l}{S}}$

Acoustical Mobility

Acoustical mobility is defined as,

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

Impedance Analog vs. Mobility Analog

Acoustical Resistance

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

File:Ra analogs.png r_A is the reciprocal of R_A and is referred to as responsiveness.

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.

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

File:Acoustic gen.png

Acoustical Compliance

Consider a piston in an enclosure.

File:Enclosed Piston.png

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 File:Equ1.jpg it is easy to get the relation between disturbing pressure and displacement of the piston File:Equ3.gif where U is volume rate, P is pressure according to the definition of the impendance and mobility, we can getFile:Equ4.gif

Mobility Analog VS Impedance Analog

File:Comp.gif

Examples of Electro-Acoustical Analogies

Example 1: Helmholtz Resonator

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

Solution:

- Impedance Analog - File:Example2holm1sol.JPG

Example 2: Combination of Side-Branch Cavities

File:Exam2prob.JPG

Solution:

- Impedance Analog - File:Exam2sol.JPG

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