# Engineering Acoustics/Anechoic Tiling

## Introduction

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One important application of acoustics research is the testing, design,
and construction of anechoic tiles for underwater acoustic stealth.
Anechoic tiling, first used in the Second World War by German U-boats
(codenamed "Alberich" [^1]), are used to reduce the acoustic signature
of naval vessels. The tiles reduce both reflected active SONAR off the
pressure hull, and reduce internal noise transmitted through the
pressure hull that can be picked up by passive SONAR. In modern times,
anechoic tiles are present on nearly all submarines. The frequency range
of SONAR, and consequently the range of interest of anechoic tiling to
minimize detection, is around 1-30 kHz [^2].

The main sound attenuation mechanism in the tiles comes from resonant
scattering the sound waves due to air cavities in the rubber. The use of
air bubbles to attenuate sound was first published by German
acousticians Erwin Meyer and Eugen Skudrzyk in a report written in the
British occupied zone of Germany in 1946, translated to English in 1950
for an unclassified release by the U.S. Navy [^3]. The air bubbles work
to attenuate sound by acting as resonant oscillators and dissipate
acoustic energy through thermal losses, frictional losses, and other
processes[^4].

Tank measurements of acoustic parameters with the panel surround on both
sides by water, or "free-field", are theoretically simple measurements
that can be conducted in an interior laboratory setting [^5], and will
be described in later sections. Audoly [^6] presents a method to
transfer the results of free-field acoustic properties from tank
acoustic measurements to the acoustic properties of panels with
arbitrary backing, such as the rigid backing of a submarine hull.

## Planar Waves

The transmission and reflection coefficients $\hat{T}$ and $\hat{R}$ are
presented as ratios of the incident, reflected, and transmitted acoustic
pressure magnitudes:

{\\hat{P}\_{incident}},\\quad \\quad \\hat{R} =
\\frac{\\hat{P}\_{reflected}}{\\hat{P}\_{incident}}`</math>`{=html}\|}}

The conservation of acoustic power from transmitted waves through a
panel is as follows: $|\hat{T}|^2+|\hat{R}|^2+|\hat{A}|^2=1$

Where $\hat{A}$ is the acoustic absorption coefficient. From careful
observation of the conservation of energy equation, it is evident that
to minimize $|\hat{T}|$ and $|\hat{R}|$, acoustic energy dissipated
$|\hat{A}|$ should be increased. In materials such as metals in the
frequency ranges pertinent to this study, $|\hat{A}| \approx 0$. In
rubber materials, and especially those with air voids [^7] [^8],
$|\hat{A}|$ can no longer be assumed to be negligible [^9].

The terms \"insertion loss\" ($IL$) or \"echo reduction\" ($ER$) are
used. The insertion loss is the reduction (in decibels) of the acoustic
power of the insertion of a panel, related to the transmission
coefficient: $IL = -10\log|\hat{T}|^2$, and the echo reduction is the
reduction (in decibels) of the acoustic power after a reflection:
$ER = -10\log|\hat{R}|^2$.

### Three-layered media: No panel absorption

For three-layered medium with two infinite fluid layers, each with
impedance $r_{1/3}=\rho_{1/3}c_{1/3}$ on either side of a sample of
thickness $L$ and impedance impedance $r_{2}=\rho_{2}c_{2}$, and
$k=\omega/c_2$, at normal incidence the following equation for the
reflection coefficient $\hat{R}$ may be used [^10]:

For the symmetrical case of the first fluid and third fluid identical
$r_3=r_1$, equation () can be reduced
into the following simplified formulations [^11]:

Where $m=\frac{r_{1}}{r_2}$. By inspection of equations
(), resonances of minima of $\hat{R}$
and maxima of $\hat{T}$ occur at $kL = n\pi$. These lossless $IL$ and
$ER$ are plotted over $kL$ in the following figures in the next section
(for aluminum panels suspended in water) as the black lines.

### Three-layered media: With panel absorption

For panel media with acoustic attenuation, the following formulas in
() can be used to describe the insertion
loss and echo reduction, where $\alpha$ is the attenuation constant in
$dB/m$, and $r=\frac{\alpha}{\omega c_2}$[^12]. For the case of no
acoustic attenuation $\alpha=0$, equations
() are recovered.

A sketch of the effect of absorption in panel materials on $IL$ and $ER$
is shown in the following figures:


For a general formulation of n-layer solid panels with absorption,
methods are described in references [^13] [^14].

### Determination of *α(ω)* from experimental data

A 2nd order approximation for $\alpha$ is used (Equation
()) as it is not constant over frequency
[^15], as shown for the case of aluminum and nitrile rubber in the
figure below.

It is possible to experimentally determine $\alpha (\omega)$ from tank
acoustic tests. First the magnitude of the absorption coefficient is
determined from conservation of energy and measured values of
$|\hat{T}|$ and $|\hat{R}|$ [^16]:

The coefficient $a$ in equation () is
estimated using the following formula, then $b$ is then fit to data
[^17].

## Experimental Investigations

To characterize $|\hat{T}|$ and$|\hat{R}|$ of panel materials, free
field acoustic measurements are performed: In a water-filled tank, a
parametric array source $(a)$ produces an highly directional discrete
acoustic wave [^18] with the far-field directionality function
$D(\theta)$ [^19]. For demonstration, the far field directionality of an
underwater array source is included in the insertion loss sample
measurement figure below. The shape of the discrete wave is shown in
[^20].

### Insertion Loss

Using a hydrophone $(c)$, one recording is made with the sample $(b)$ in
place to record transmitted pressure $P_t$ and one measurement without
the sample in place to record incident pressure $P_i$. The measurement
configurations are shown in the following figures.

### Echo Reduction

For reflection experiments, the reflected pressure $P_r$ off the sample
is measured, and for the incident pressure $P_i$ measurement a foam
reflector is used. A foam reflector has a high acoustic impedance
mismatch with the water and reflects sound efficiently.


### Pressure Measurements

Shown in the following figures, the measured pressure signals over time
are are recorded and processed with a Fourier transform to determine
pressure over frequency. The coefficients $|\hat{R}|$ and $|\hat{T}|$
over frequency are simply the ratio of the resultant pressure spectra.


### Results

Insertion loss and echo reduction plots for, aluminum test samples, such
as those measured in [^21], are shown in the following figures. Aluminum
samples are conducted as it is a material with well known acoustic
properties, and negligibly low absorption over the frequencies studied.
The experimental setup described shows good agreement with theory [^22]
[^23].


### Other Considerations

For applications of anechoic tiling on submarines, the conditions
surrounding the submarine change drastically with submerged depth.
Varying pressure, salinity, temperature all affect acoustic properties
of the rubber, the surrounding water, and the anechoic tile in general.
Environmental tank such as the ones described in reference [^24] can be
used to simulate ocean conditions. The wavelength of the sound produced
by the parametric array is limited by the physical size of the tank.
Achieving lower frequency measurements necessitates the use of larger
tanks for experiments, or semi-anechoic siding on the tank walls [^25].

## References

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

By: Geoffrey Chase

[^1]: James L Lastinger and Gerald A Sabin. Underwater sound absorbers:
    A review of published research with an annotated
    bibliography .
    Technical report, NAVAL RESEARCH LAB ORLANDO FL UNDERWATER SOUND
    REFERENCE DIV, 1970.

[^2]: N. Friedman and United States Naval Institute. The Naval
    Institute guide to world naval weapons
    systems. The Naval
    Institute Guide To\... Series. Naval Institute Press, 1989.

[^3]: Walter Kuhl. Sound absorption and sound absorbers in water
    (Dynamic properties of rubber and rubberlike substances in the
    acoustic frequency
    region). Dept. of
    the Navy, Bureau of Ships, 1950.

[^4]:

[^5]: V.F. Humphrey. The measurement of acoustic properties of limited
    size panels by use of a parametric
    source90403-1). Journal of
    Sound and Vibration, 98(1):67 -- 81, 1985.

[^6]: Christian Audoly. Determination of efficiency of anechoic or
    decoupling hull coatings using water tank acoustic
    measurements. In
    Societe Francaise d'Acoustique, editor, Acoustics 2012, Nantes,
    France, April 2012.

[^7]:

[^8]:

[^9]: E. Eugene Mikeska and John A. Behrens. Evaluation of transducer
    window materials. The Journal of
    the Acoustical Society of America, 59(6):1294--1298, 1976.

[^10]: Lawrence E Kinsler, Austin R Frey, Alan B Coppens, and James V
    Sanders. Fundamentals of
    Acoustics. 4th Edition, pp. 560.
    ISBN 0-471-84789-5. Wiley-VCH, December 1999., page 560, 1999.

[^11]: Robert J Bobber. Underwater electroacoustic
    measurements.
    Technical report, NAVAL RESEARCH LAB ORLANDO FL UNDERWATER SOUND
    REFERENCE DIV, 1970.

[^12]:

[^13]: A. K. Mal, C.-C. Yin, and Y. Bar-Cohen. The Influence of
    Material Dissipation and Imperfect Bonding on Acoustic Wave
    Reflection from Layered
    Solids,
    pages 927--934. Springer US, Boston, MA, 1988.

[^14]: Bernard Hosten and Michel Castaings. Transfer matrix of
    multilayered absorbing and anisotropic media. Measurements and
    simulations of ultrasonic wave propagation through composite
    materials. The Journal of the
    Acoustical Society of America, 94(3):1488--1495, 1993.

[^15]:

[^16]:

[^17]:

[^18]:

[^19]: P. D. Thorne. A broad-band acoustic source for underwater
    laboratory applications.
    IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency
    Control, 34(5):515--523, Sept 1987.

[^20]:

[^21]:

[^22]:

[^23]: Victor F Humphrey, Stephen P Robinson, John D Smith, Michael J
    Martin, Graham A Beamiss, Gary Hayman, and Nicholas L Carroll.
    Acoustic characterization of panel materials under simulated ocean
    conditions using a parametric array
    source. The Journal of the
    Acoustical Society of America, 124(2):803--814, 2008.

[^24]:

[^25]:
