# Acoustics/Polymer-Film Acoustic Filters

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## Introduction

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

## Changes in Media Properties Due to Sound Wave Characteristics

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

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`     `$Z_0 = \pm \rho_0 c  \,$`  `$(Pa \cdot s/m)$`  `

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where

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`     `$\pm \rho_0  \,$` = (equilibrium) density of medium  `$(kg/m^3)\,$\
`     `$c \,$` = speed of sound in medium  `$(m/s) \,$`  `\
`     `

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

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

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

## Why Coupled Acoustic Media in Acoustic Filters?

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

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

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and

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

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

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

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

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

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

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

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

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

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

## An Application of Coupled Media in Acoustic Filters

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

## References

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

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

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

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

Created by Valdez L. Gant

