Copyright by Ashley Jean Hicks 2015

Copyright

by

Ashley Jean Hicks

2015

The Thesis Committee for Ashley Jean HicksCertifies that this is the approved version of the following thesis:

Design and Testing of Sub-wavelength Panels for

Underwater Acoustic Isolation

APPROVED BY

SUPERVISING COMMITTEE:

Preston S. Wilson, Supervisor

Michael R. Haberman, Co-Supervisor



by

Ashley Jean Hicks, B.S.

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE IN ENGINEERING

THE UNIVERSITY OF TEXAS AT AUSTIN

August 2015

Dedicated with love to David Johnson (1945–1998),

Anna Jean Oakleaf (1925–1998), and Charles Oakleaf (1913–2013).

Acknowledgments

First, many thanks are due to my advisors, Dr. Preston S. Wilson and

Dr. Michael R. Haberman. Their guidance is much appreciated and their

patience even more so. Additionally, I would like to acknowledge the Office

of Naval Research, who provided the Applied Research Laboratories and The

University of Texas at Austin with the funding necessary to pay the bills while

I, like a child in a great sandbox, explored scientific possibility. Special thanks

to Dr. William V. Slaton, mentor and friend, without whom I would have

never been where I am today. To my family, who supported my graduate

school decisions and helped in anyway they could, thank you. To my friends,

especially Jerrod Ward, who answered late night pleas and cries of frustration

with laughter and more than a few beers, thank you. I couldn’t have done it

without you all, and I am eternally grateful. Live well, pray often, and travel

far.

v



Ashley Jean Hicks, M.S.E.

The University of Texas at Austin, 2015

Supervisors: Preston S. WilsonMichael R. Haberman

Underwater sound isolation is an important area of research for both

environmental and military applications. This work explores present research

in airborne thin panel acoustic metamaterials and underwater acoustic isola-

tion using encapsulated bubbles. These ideas are combined in the design of

sub-wavelength, free-standing underwater panels for acoustic isolation. This

work investigates the resonance of cylindrical air cavities in water with flat

aspect ratios. The spherical resonance approximation proposed by Minnaert

for gas bubbles in water is shown to provide a good approximation of the res-

onance frequency of cylindrical inclusions. Panels with cylindrical inclusions

are developed and tested in the 500 Hz to 1.3 kHz range and the 2 kHz to

5 kHz range. Panels with a void fraction of 1.3% and non-dimensional kT

value of 0.02 to 0.07 show a frequency averaged insertion loss of 3 dB to 8 dB.

Additionally, it is shown that an increase in void fraction yields an increase in

panel isolation capability. It is the hope that this and future work in the area

vi

of sub-wavelength panels will improve the underwater environment for marine

life and underwater naval applications.

vii

Table of Contents

Acknowledgments v

Abstract vi

List of Tables xi

List of Figures xii

Chapter 1. Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Bubbles as Isolators . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Acoustic Metamaterials . . . . . . . . . . . . . . . . . . 5

1.2.3 Development of Void-Filled Panel Isolators . . . . . . . 8

1.3 Objective and Thesis Roadmap . . . . . . . . . . . . . . . . . 9

Chapter 2. Modeling and Measurement of a Unit Cell 12

2.1 The Minnaert Resonance of Bubbles . . . . . . . . . . . . . . . 12

2.1.1 Minnaert Resonance . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Approximation of a Sphere as an Equal-Volume Cylinder 18

2.2 Measurements of Cylindrical Inclusions . . . . . . . . . . . . . 19

2.2.1 Unit Cell Construction . . . . . . . . . . . . . . . . . . 19

2.2.2 Experimental Apparatus and Data Analysis . . . . . . . 22

2.2.3 Results of Unit Cell Experiments . . . . . . . . . . . . . 24

2.3 Finite Element Analysis of Cylindrical Inclusion . . . . . . . . 29

2.3.1 Development of Model . . . . . . . . . . . . . . . . . . . 29

2.3.2 Model Analysis and Results . . . . . . . . . . . . . . . . 32

2.4 Model Comparisons . . . . . . . . . . . . . . . . . . . . . . . . 37

viii

Chapter 3. Measurement of Panel Performance 40

3.1 Panel Construction . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 High Frequency (2 kHz to 5 kHz) Tank Experiments . . 42

3.2.2 Low Frequency (500 Hz to 1.3 kHz) Tank Experiments . 45

3.2.3 Post-Processing of Data . . . . . . . . . . . . . . . . . . 47

3.3 High Frequency (2 kHz to 5 kHz) Panels . . . . . . . . . . . . 48

3.3.1 Effect of Voids . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.2 Effect of Void Fraction . . . . . . . . . . . . . . . . . . . 50

3.3.3 Effect of Using 3-D Printed Components . . . . . . . . . 54

3.4 Low Frequency (500 Hz to 1.3 kHz) Panels . . . . . . . . . . . 62

3.4.1 Effect of Voids . . . . . . . . . . . . . . . . . . . . . . . 63

3.4.2 Mean Insertion Loss . . . . . . . . . . . . . . . . . . . . 68

3.5 Finite Element Analysis of 2 kHz to 5 kHz Panel . . . . . . . . 69

3.5.1 Development of Model . . . . . . . . . . . . . . . . . . . 69

3.5.2 Model Analysis and Results . . . . . . . . . . . . . . . . 71

Chapter 4. Conclusions 74

4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Appendices 80

Appendix A. MATLAB Code for Data Analysis 81

A.1 Unit Cell - Experimental Results . . . . . . . . . . . . . . . . . 81

A.2 Unit Cell - Finite Element Analysis Results . . . . . . . . . . . 82

A.3 Panel Isolation - Experimental Results . . . . . . . . . . . . . 82

A.3.1 Transfer Function Graphs . . . . . . . . . . . . . . . . . 83

A.3.2 Insertion Loss Calculation . . . . . . . . . . . . . . . . . 83

Appendix B. Analytical Model of Unit Cell 84

B.1 Equivalent Circuit Impedance . . . . . . . . . . . . . . . . . . 84

B.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 89

B.3 Effect of Membranes and Aspect Ratios . . . . . . . . . . . . . 92

B.4 MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . 96

ix

Bibliography 98

Vita 106

x

List of Tables

2.1 Dimensions and expected Minnaert resonance, denoted as f0,of cylindrical inclusions in POM unit cells with a thickness of t= 6.35 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Material properties used in COMSOL Multiphysics finite ele-ment analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

B.1 Resonance results from the input impedance as determined bythe circuit model above. Inclusion dimensions are taken fromChapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

B.2 Resonance results from the input impedance as determined bythe circuit model without the membrane present. . . . . . . . 92

B.3 Resonance results from the input impedance as determined bythe circuit model of a unit cell with a modified aspect ratio forthe thickness of the Delrin, both with and without a membranepresent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B.4 Resonance results from the input impedance as determined bythe circuit model of a unit cell with a modified aspect ratio forthe thickness of the Delrin and the thickness of the membrane. 95

xi

List of Figures

1.1 Encapsulated bubble array surrounding a cylindrical sound source.Figure adapted from Ref. [14]. . . . . . . . . . . . . . . . . . . 4

1.2 Concept of a thin-panel isolator with cylindrical air inclusionsencapsulated by two non-perforated elastic layers. . . . . . . . 8

2.1 A bubble of radius R0 undergoing low-amplitude linear oscilla-tions of amplitude Rε0 surrounded by spherical shells of liquid.Figure adapted from Ref. [8]. . . . . . . . . . . . . . . . . . . 14

2.2 A spherical bubble with radius R0 and an equal volume cylin-drical inclusion of radius a and thickness t. . . . . . . . . . . . 18

2.3 A unit cell made of POM plastic with thickness t has a cylin-drical inclusion with radius a in the center. . . . . . . . . . . . 20

2.4 A sheet of POM plastic of thickness t sandwiched between twoneoprene rubber layers which trap air inside of a cylindricalinclusion of radius a creating a unit cell of thickness T. . . . . 21

2.5 Experimental setup to measure the resonance frequency of aunit cell in a cylindrical tank with diameter D and height H. . 22

2.6 Frequency response of the steel tank. . . . . . . . . . . . . . . 24

2.7 Results of the 100 Hz unit cell measurement showing (a) Ptotal

and Ptank and (b) Pbubble. . . . . . . . . . . . . . . . . . . . . . 25







2.11 Schematics of COMSOL geometries used to model the system. 31

2.12 Velocity response of the unit cell designed with a Minnaert fre-quency of 100 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . 33


xii



2.16 Comparison of the Minnaert model of resonance frequency withexperimental data and FEA results for the unit cell. . . . . . . 38

3.1 Schematic showing the air filled inclusion, a panel with air filledinclusions, and a five-sided box created from the isolating panels. 41

3.2 An acrylic tank with 0.37 m deep water was used for high fre-quency experiments. . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Location of the source (S ) and receiver (R) in the high frequencytank experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 A photograph of the void-filled cube in place during an experi-ment, showing the cube suspended around the source from theshaker plate (A) and the cube’s upward-facing side (B) on theair-water interface. . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Tank described in Section 2.2 with shaker and cube suspendedfrom a plate above the tank. . . . . . . . . . . . . . . . . . . . 46

3.6 Low frequency cube during an experiment showing the hook andplate (A) used to suspend the shaker and the cube suspendedon mono-filament such that the upward-facing side (B) is at theair-water interface. . . . . . . . . . . . . . . . . . . . . . . . . 47

3.7 Cubes created from high frequency panels. From (L) to (R):Plain case, perforated case, void-filled case. . . . . . . . . . . . 49

3.8 Isolation effect of panels with and without air-filled inclusionson the frequency response of the tank. . . . . . . . . . . . . . 50

3.9 Isolation effect of panels with varying void fraction on the fre-quency response of the tank. . . . . . . . . . . . . . . . . . . . 52

3.10 Mean insertion loss across specified frequency bands for panelswith varying void fraction. . . . . . . . . . . . . . . . . . . . . 53

3.11 Mean insertion loss of the panel as a function of void fractionacross the entirety of the frequency range. . . . . . . . . . . . 54

3.12 Example of a solid cylinder created on a 3D printer. The interiorhexagonal structure is called in-fill. Image courtesy of MakerBot. 55

3.13 Two 3D printed pieces with high in-fill (a) and low in-fill (b)percentage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.14 Isolation effect of panels with varying in-fill percentages. . . . 58

xiii

3.15 Mean insertion loss across specified frequency bands for a panelmade with PLA plastic center layer of varying in-fill percentage. 59

3.16 Effect of in-fill percentage of a PLA panel on insertion loss forspecified frequency bands. . . . . . . . . . . . . . . . . . . . . 60

3.17 Mean insertion loss of the panel as a function of void fractionfor Delrin plastic and PLA plastic center layers. . . . . . . . . 61

3.18 Cubes created from low frequency panels. From (L) to (R):Plain case, perforated case, void-filled case. . . . . . . . . . . . 63

3.19 Isolation effect for panels designed to operate from 500 Hz to1.3 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.20 Isolation effect of the low frequency panel from 150 Hz to 350 Hz. 64




3.24 Isolation effect of the low frequency panel from 800 Hz to 1 kHz. 67

3.25 Isolation effect of the low frequency panel from 1 kHz to 1.3 kHz. 67

3.26 Mean insertion loss as a function of frequency for panels de-signed to operate from 500 Hz to 1.3 kHz. . . . . . . . . . . . 68

3.27 3D COMSOL geometry used to model the thin-panel system. . 70

3.28 Results finite element analysis of radiated sound power in thepanel system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

B.1 Schematic of the unit cell described in Chapter 2. . . . . . . . 84

B.2 Completed equivalent circuit of the unit cell. . . . . . . . . . . 85

B.3 Simplified circuit model with impedances listed instead of indi-vidual components. . . . . . . . . . . . . . . . . . . . . . . . . 86

B.4 Final simplified circuit model used to determine the input impedance. 88

B.5 Results of circuit model analysis for the 100 Hz unit cell. . . . 89

B.6 Velocity information for the 100 Hz unit cell. . . . . . . . . . . 90

xiv

Chapter 1

Introduction

1.1 Motivation

The isolation of underwater acoustic noise is of growing concern world-

wide. The military has obvious interest in isolating sounds from underwater

vehicles and SONAR devices. Environmental concerns keep scientists search-

ing for new methods of acoustic abatement. Many sources of noise in the

ocean are anthropogenic and have the possibility of interfering with the bi-

ological systems in the environment [1]. Low frequency noises are the most

pervasive as high frequency noise sources tend to decay more quickly in the

underwater environment.

Prominent sources of underwater sound are industrial, like oil drilling

and pile driving [2]. Pile driving hammers generate impulsive noise in primarily

the 20 Hz to 1 kHz band [3]. Marine wind turbines radiate a continuous noise

in the 60 Hz to 300 Hz band [2]. Popper has published several works [1, 4]

that give a review of and estimate the impact anthropogenic noise has on the

surrounding environment and marine life. Developing methods of underwater

isolation at these frequencies would be of great benefit to the environment.

There is also a military interest in isolating the motor noise of ships

1

and submarines. The development of isolation materials for the appropriate

frequency ranges could help increase the acoustic stealth of underwater mil-

itary vehicles. Additionally, there is a commercial gain in the development

of materials that isolate motorized watercraft noise. A standard small boat

engine will produce a continuous noise in the 1 kHz to 5 kHz range [2]. Out-

board motor noise disturbs and deters collections of fish in the underwater

environment [5]. Motorized watercraft have also been shown to significantly

effect the aquatic environment [6]. The development of an isolation material

that reduces outboard motor noise would be very beneficial to fishermen and

recreational water users alike.

All of the above examples illustrate the need for a simple, effective

underwater acoustic isolator. This thesis therefore seeks to design an effective

underwater acoustic isolator of minimal thickness. A deeply sub-wavelength

panel has the additional benefit of isolating acoustic noise while minimizing

its obtrusiveness in the environment as well as minimizing its manufacturing

and installation costs.

1.2 Background

This work seeks to develop an underwater acoustic isolating panel by

combining two different branches of current acoustic research: the use of bub-

bles as isolators and attenuators and the development of acoustic metama-

terials with resonant inclusions. Additional influence was drawn from the

Alberich anti-reflection submarine coatings developed in World War II and

2

resonant bubbles in elastic media. This section will provide a brief review of

current and past work on these topics and discuss how previous work informed

the research presented in this thesis.

1.2.1 Bubbles as Isolators

It has long been known that the presence of bubbles in an underwater

environment drastically alters the acoustical properties of the environment.

Several models exist to describe the behavior of individual bubbles in a liquid

environment [7, 8] and the acoustic behavior of bubbly liquids [9, 10]. The pres-

ence of a resonant bubble in a system can assist in removing energy from the

system, thus acting as an attenuator for an acoustic wave. Waves propagat-

ing through bubbly liquid experience dispersion (frequency dependent sound

speed) and attenuation, both of which are maximized near the individual bub-

ble resonance frequency. This dispersive behavior has been quantified in prop-

agation models for free bubbles [9] and for bubbles with solid elastic shells

[11]. A propagation model that captures the effect of encapsulated bubbles on

overall acoustic wave dispersion and attenuation has also been developed [12].

This review will focus on two types of underwater isolators that use bub-

bles to attenuate sound. The first is the encapsulated bubble array developed

by Lee et al. [12–15] at The University of Texas at Austin. The encapsulated

bubbles in these systems act as simple harmonic isolators. Each individual

cell was tethered to a wire-frame or rope structure which could be made to

surround the noise source.

3

In [13], Lee et al. investigate the isolation ability of bubbles created

from latex balloons. They found that an encapsulated bubble with a radius of

approximately 5 cm successfully attenuated frequencies ranging from 50 Hz to

100 Hz. In that experiment, a single line of encapsulated bubbles were placed

inside of a cylindrical tube and the resulting change in resonance patterns

permitted the determination of the dispersion effect. In [12] and [14], encap-

sulated bubbles were attached to a frame surrounding the source as shown

in Fig. 1.1. These bubble arrays show a maximum reduction of 26 dB for

frequencies below 200 Hz.

Figure 1.1: Encapsulated bubble array surrounding a cylindrical sound source.Figure adapted from Ref. [14].

4

Lee et al. also investigated the attenuation capabilities of a freely ris-

ing bubble cloud [12]. Bubble clouds were generated by attaching perforated

hoses to a steel frame approximately 0.5 m below the sound source. Air flow

through the hoses created a freely rising cloud consisting of bubbles with an

approximate radius of 0.5 cm. The volume fraction of bubbles in the bubble

cloud, also known as the void fraction, was between 0.005 and 0.025. For fre-

quencies between 350 Hz to 1 kHz, the freely rising cloud allowed a reduction

in noise level of up to 40 dB [12].

In similar work, Wursig et al. reported the effectiveness of a bubble

curtain in reducing the noise generated by pile driving [16]. In an in situ

experiment, Wursig et al. reported a broadband reduction of 3 dB to 5 dB

and a octave band reduction of up to 10 dB in the 400 Hz to 800 Hz band and

20 dB in the 1.6 kHz to 6.4 kHz band. A bubble cloud experiment conducted

by Domenico showed that bubble clouds attenuated sound in an underwater

environment and the frequency band of attenuation was dependent on the size

and resonance frequency of the bubbles in the cloud [17].

1.2.2 Acoustic Metamaterials

Another emergent research area of acoustics is the field of acoustic

metamaterials. An acoustic metamaterial is best defined as an artificial com-

posite containing sub-wavelength structures whose dynamic effective proper-

ties are unattainable using conventional materials or composites [18]. An

AMM topic germane to the content of this work is the recent development

5

of a deeply sub-wavelength panel for sound isolation in air [19].

The panels developed in [19] had thickness T of 15 mm and a density

of 3 kg/m3 or less. In that work, each panel was constructed from an elastic

membrane stretched across a rigid plastic grid. Each unit cell in the grid had

a small weight attached to the center, creating a simple harmonic oscillator.

The panel of unit cells exhibited an internal sound transmission loss of 19.5 dB

at 200 Hz. This corresponded to a non-dimensional kT value of 0.05, where k

is the wavenumber in the host medium. Additionally, Yang et al. reported that

this panel could be stacked to obtain total reflection across a broad frequency

range [19]. A stacked panel system with thickness of 60 mm and weight of

15 kg/m3 exhibited a transmission loss of greater than 40 dB over the 50 Hz

to 1 kHz range. The kT value across the frequency range for the stacked panel

was 0.05 to 1.09.

Xiao et al. also developed a thin metamaterial-based plate of sub-

wavelength resonator arrays [18]. Their work reported that thinner plates

allowed the resonators in the panel to have a larger effect. Through a series of

numerical models, [18] showed plates with a thickness of approximately 2 mm

exhibited a transmission loss of 5 dB over the 100 Hz to 200 Hz range. This

corresponded to a kT value of 0.004 to 0.008. Additionally, Xiao et al. showed

these panels can be tuned to a specific frequency band by loading the resonant

cavities with weights.

Naify et al. experimentally measured the transmission loss of a locally

resonant metamaterial [20]. The metamaterial was a thin membrane stretched

6

over a support structure with centrally located masses. They investigated the

relationship between transmission loss and metamaterial properties. The goal

for these panel-type metamaterials was to improve on the transmission loss

given by the mass law. Ultimately, the resonant metamaterial showed a five

fold increase in transmission loss over the mass law prediction in the 100 Hz to

1 kHz range. A numerical model for characterizing thin plate acoustic metama-

terials was developed in [21]. Additionally, Li et al. highlighted the possibility

of using thin-plate acoustic metamaterials for isolating low frequency noises

and limiting underwater SONAR detection.

There are also metamaterials being developed which do not rely on

resonate effects to attenuate or isolate sound. Assour et al. reported on the

development of a 2D panel made of tungsten and silicone rubber [22]. In ex-

perimental measurements, this panel showed wave attenuation in the 650 Hz to

3.5 kHz range, tunable by adjusting the size of tungsten inserts and the thick-

ness of the silicon rubber. Christensen et al. reported a tunable metamaterial

made of a perforated material, called a double fishnet by the authors. The

material was fabricated by stacking two metal plates perforated with square

holes. Experiments showed that this double fishnet material attenuated across

a frequency range dependent upon the gap between the two fishnet layers [23].

Additionally, Klatt and Haberman developed a composite material model that

can be used to design negative stiffness metamaterials from buckled elements

[24]. These elements have the ability to be used for energy absorption over a

wide frequency bandwidth.

7

1.2.3 Development of Void-Filled Panel Isolators

One possible way to combine the insights from the research presented

in Sections 1.2.1 and 1.2.2 would be to consider the construction of a panel

with several air filled inclusions encapsulated in the center layer. Further, one

might expect these panels to have an isolation effect similar to bubbly media

[9] and tethered encapsulated bubble arrays [12–15]. Xiao et al. reported that

resonant behavior dominates for acoustically thin panels [18], specifying that

effects of localized panel resonances will clearly dominate when kT values

are on the order of 10−2. This work introduces a tri-layer thin panel with

cylindrical air inclusions, as shown in Fig. 1.2

Figure 1.2: Concept of a thin-panel isolator with cylindrical air inclusionsencapsulated by two non-perforated elastic layers.

A similar structure designed to attenuate sound in the ultrasonic range

was developed and modeled by Leroy et al. [25, 26]. In [25], Leroy et al. de-

signed and experimentally verified a material consisting of small cylindrical

inclusions completely surrounded by soft elastic silicon. The material was ex-

panded upon in an analytical model presented in [26]. The panel presented

in this work contains cylindrical inclusions surrounded on the radial side by

8

stiff elastic material and on the top and bottom by a thin layer of soft elastic

material. Outside of that thin elastic layer, the panel is in direct contact with

water. The primary behavior of the bubble in the panel should be similar to

the behavior of an encapsulated bubble in a water environment [12–15] rather

than a cylindrical bubble in a solid elastic environment [25, 26].

These panels also resemble a reflection reduction coating developed

for German U-boats in World War II [27]. These coatings, commonly called

Alberich coatings, had a tri-layer structure as shown in Fig. 1.2. However, the

backing layer (usually the hull of a U-boat) was used as structural support.

The acoustic effects of generalized Alberich coatings are reported in [27–32].

Unlike the materials requiring a structural backing layer, the panels proposed

here are free standing. To the author’s best knowledge, there has been no

previously published work on sub-wavelength free standing panel isolators.

The development of such a panel is the primary purpose of the thesis outlined

below.

1.3 Objective and Thesis Roadmap

Having discussed previous work in the fields of thin panel acoustic meta-

materials and underwater acoustic isolation, and having developed the concept

of a free standing sub-wavelength panel with cylindrical inclusions, this thesis

has three main objectives:

1. Present and develop simple models that can be used to design

9

sub-wavelength panels that have cylindrical air-filled inclusions.

2. Demonstrate the underwater noise abatement efficacy of thin-

panels containing cylindrical air-filled inclusions.

3. Experimentally quantify the effect of hole size, void fraction,

and fill fraction on panel isolation ability.

The first objective is met in Chapter 2. Section 2.1 describes the res-

onance of a spherical bubble and explains the process of approximating a

spherical bubble as a cylindrical inclusion. The approximation is tested exper-

imentally in Section 2.2, which presents the development and testing of several

unit cells to determine the resonance frequency of cylindrical inclusions, and

computationally in Section 2.3, which determines the resonance frequency of

a unit cell via a finite element analysis of the system. In Section 2.4, the

validity of approximation of a cylindrical inclusion as a spherical bubble is

determined by comparing the experimental data, the finite element analysis,

and the Minnaert resonance frequency.

The second and third objectives are addressed in Chapter 3. Chapter 3

explores the development and construction of sub-wavelength panels in the

2 kHz to 5 kHz range and 500 Hz to 1.3kHz range. The effect of various

panel properties including void fraction and center material on the isolation

ability of panels designed to isolate in the 2 kHz to 5 kHz range is explored

in Section 3.3. Section 3.4 explores the isolation ability of panels designed

to work in the 500 Hz to 1.3 kHz range. Additionally, Section 3.5 develops

10

a model for finite element analysis of the panel system to compliment the

experimental results.

Finally, Chapter 4 reiterates the work presented in Chapter 2 and Chap-

ter 3 and draws conclusions about the future use of sub-wavelength panels

in underwater acoustic isolation. Though the research presented here repre-

sents a significant amount of work on this novel material, there remain several

questions and possible points of exploration. Future work is highlighted in

Section 4.2.

11

Chapter 2

Modeling and Measurement of a Unit Cell

In order to facilitate the design of a complex system, it is important to

model and measure the system’s fundamental components. A major compo-

nent of the panel system described in the previous chapter is the cylindrical

air-filled inclusion. This chapter focuses on the development and sizing of

cylindrical air-filled inclusions for use in a sub-wavelength panel.

2.1 The Minnaert Resonance of Bubbles

The resonance frequency of a spherical bubble in water is widely stud-

ied. A cylindrical air-filled inclusion like the one proposed in this work will

have a different resonance frequency. As this work is dependent on the res-

onance effect of the cylindrical air-filled inclusion, it is important for design

purposes to have accurate estimates of the resonance frequency of the inclu-

sion. The goal here is a simple and convenient zeroth-order model of resonance

frequency to guide the initial design of panels. To date, several papers have

been published on the resonant behavior of non-spherical bubbles [33–37]. A

relevant recent example is provided by Calvo et al. who describe the resonance

of a single cylinder embedded in a soft elastic medium [34]. The cylindrical

12

inclusions proposed by the present work are seated in a stiff elastic medium

with a very thin soft elastic shell surrounded by water and thus are not well

described by Calvo’s model. Additionally, the cylindrical inclusions are not

exactly represented by the models presented by Geng and Oguz [35, 36], which

describe bubbles that completely fill a portion of a long cylinder of water.

Other models may more closely describe the resonant behavior of the system,

but they are numerically inconvenient [33, 37]. It will be shown that the Min-

naert model for a spherical bubble in water [7] is an effective zeroth-order

model for the cylindrical inclusions in sub-wavelength panels.

2.1.1 Minnaert Resonance

The low-amplitude oscillation of a spherical gas bubble in water was

first explored by Minnaert in 1933 [7]. Minnaert’s work treats the gas bubble

in a liquid environment as a mass on a spring and from there derives the

resonance frequency of the oscillation. This derivation, which assumes no

dissipative losses in the system and low-amplitude, uniform oscillations, is

outlined below using the method and nomenclature given in [8].

13

Figure 2.1: A bubble of radius R0 undergoing low-amplitude linear oscillationsof amplitude Rε0 surrounded by spherical shells of liquid. Figure adapted fromRef. [8].

To begin, consider an oscillating bubble with mean radius of R0, as

shown in Fig. 2.1. For single frequency excitation, the bubble’s wall motion

can be described as

Rε = −Rε0eiω0t. (2.1)

The bubble radius at any point in time is

R = R0 +Rε(t) = R0 −Rε0eiω0t. (2.2)

Here a negative sign is used to indicate that an increase in pressure causes

14

a decrease in overall bubble radius. In an infinite environment, the bubble

is surrounded by spherical shells of water with radius r and thickness ∆r.

Each spherical shell has a mass of 4πr2ρ∆r. The kinetic energy is found by

integrating over all of shells from the bubble wall to infinity,

ΦK =1

2

∫ ∞R

(4πr2ρdr)r2. (2.3)

The mass of liquid moving through a spherical surface around the bub-

ble at any point in time dt is given by 4πr2rρdt. If the liquid is approximated

as incompressible, then by the conservation of mass the flow at the shell can

be equated to the flow at the bubble surface,

r

R=R2

r2.

Equation (2.3) yields

ΦK = 2πR3ρR2. (2.4)

Equation (2.4) has a maximum value when the bubble is at equilibrium radius

R0. Under this condition, R = iω0Rε0eiω0t, and thus the squared magnitude

of the radial velocity is given by∣∣∣R∣∣∣2 = (ω0Rε0)2.

Radiation mass is the effective mass felt by an acoustically radiating

system, and it is dependent on the geometry of the system and the frequency

of oscillation. Normally, other inertial effects would be present in an oscil-

lating system, but for simplicity Minnaert considers only the inertial effect of

15

the oscillation. As such, the radiation mass is defined mradRF = 4πR3

0ρ. The

maximum kinetic energy can then be written

ΦK,max =1

2mrad

RF(Rε0ω0)2. (2.5)

In a loss-less simple harmonic oscillator, the maximum kinetic energy is equal

to the maximum value of potential energy [38]. Using the concept of work [8],

the maximum potential energy can be written

ΦP,max = −∫ Vmin

V0

(pg − p0)dV = −∫ R0−Rε0

R0

(pg − p0)4πr2dr. (2.6)

Assuming the gas behaves poly-tropically, that is pgVκ is a constant

where κ is the poly-tropic index, the pressure and volume at equilibrium are

equal to the pressure and volume when the bubble is at a minimum volume

giving the following expression

pg(R)3κ = p0R3κ0 . (2.7)

Utilizing the radius relationship Rε = R−R0, the pressures can be related to

the equilibrium radius and the magnitude of the radial oscillations as

pgp0

=

(1 +

Rε

R0

)−3κ. (2.8)

For small radial oscillations, Eq. (2.8) can be expanded with the binomial

expansion to yield the following linear relationship between the relevant pres-

sures, equilibrium radius, and radial motion,

p0 − pg =3κRεp0R0

. (2.9)

16

Equation (2.9) is substituted into Eq. (2.6). Using the first order approxima-

tion Rε = R - R0, the potential energy becomes

ΦP,max =

∫ Rε0

0

3κp0Rε

R0

4πR20dRε = 6πκp0R0R

2ε0. (2.10)

Equation (2.10) and Eq. (2.5) are combined to yield the resonance frequency

of the oscillating bubble,

ω0 =1

R0

√3κp0ρ

. (2.11)

In an adiabatic system with negligible surface tension, the Minnaert

resonance frequency is dependent only on the density ρ of the material, the

static pressure p0, and the adiabatic constant, or ratio of specific heats, γ,

which replaces κ in an adiabatic system. The Minnaert resonance frequency

can be written as

ωM =1

R0

√3γp0ρ

. (2.12)

As a simple rule, for a small air bubble in water under one atmosphere of

pressure, Eq. (2.12) reduces to

f0R0 ≈ 3.26 m/s. (2.13)

This form of the Minnaert resonance frequency is easy to remember

and apply. The Minnaert resonance, Eq. (2.13), is the most common reference

point for estimating the resonance frequencies of air bubbles in water, and as

shown in the subsequent sections is remarkably useful for the design of unit

cells of the present system.

17

2.1.2 Approximation of a Sphere as an Equal-Volume Cylinder

Figure 2.2: A spherical bubble with radius R0 and an equal volume cylindricalinclusion of radius a and thickness t.

As a zeroth-order model of the resonance frequency, the cylindrical air

cavity of the present work is considered as an equal volume spherical air bubble.

Equation (2.13) is used to determine the radius of a sphere with a resonance

in the desired frequency range. Assuming that the shapes have equal volume,

the sphere with radius R0 is approximated as a cylinder with radius a and

thickness t, as shown in Fig. 2.2. The volume of the spherical bubble is

Vsphere =4

3πR3

0. (2.14)

The cylindrical inclusion has volume

Vcylinder = πa2t. (2.15)

Equating Eq. (2.15) and Eq. (2.14) enforces the equivalent volume assumption

and yields

a2t =4

3R3

0. (2.16)

18

The radius of the cylindrical inclusion is

a =

√4R3

0

3t. (2.17)

Equation (2.17) requires the specification of height in the calculation

of the radius of the inclusion. For the case of the unit cells described in

Section 2.2 and Section 2.3, the value t was dictated by the thickness of the

material used in the panels, and the radius was determined using Eq. (2.17) to

tune the resonance frequency. It will be shown that panels with cylindrical air

inclusions with an aspect ratio near a/t ≈ 2.4 will have a resonance frequency

near the Minnaert resonance frequency across a broad range of frequencies.

2.2 Measurements of Cylindrical Inclusions

The effectiveness of estimating the resonance frequency of a cylindrical

inclusion using the Minnaert resonance frequency for an equal-volume spher-

ical bubble was tested by constructing a selection of individual cylindrical

inclusions, called unit cells, and measuring their resonance frequency. These

results were also compared to the results of a finite element analysis (FEA) of

the system performed in COMSOL Multiphysics and discussed in Section 2.3.

2.2.1 Unit Cell Construction

To create a unit cell, a single cylindrical hole of radius a was drilled

through of a sheet of a polyoxymethylene (POM) plastic. POM is sold under

various trade-names, including Delrin and Celcon. Delrin, manufactured and

19

sold by DuPont, was used in this work. The overall size of the sheet was

approximately twice the diameter of the cylindrical inclusion, as shown in

Fig. 2.3. Air was trapped inside of the inclusion by sandwiching the POM

sheet between two thin layers of neoprene rubber, as shown in Fig. 2.4. The

rubber was attached to the POM using DAP Weldwood, a neoprene-based

contact cement. The neoprene sheets were attached under enough tension to

remove any air pockets and hold the sheet flat. No additional tension was

applied to the membrane for the creation of these unit cells.

Figure 2.3: A unit cell made of POM plastic with thickness t has a cylindricalinclusion with radius a in the center.

20

Figure 2.4: A sheet of POM plastic of thickness t sandwiched between twoneoprene rubber layers which trap air inside of a cylindrical inclusion of radiusa creating a unit cell of thickness T.

Following the process described in Section 2.1, unit cells were created

with ten different hole radii. The POM height, t, for each unit cell was

6.35 mm. The thickness of the neoprene rubber was 0.8 mm, leading to a

total unit cell thickness, T, of 7.95 mm. The dimensions of each unit cell are

given in Table 2.1. Note that unit cells with a Minnaert resonance frequency

greater than 500 Hz deviated from the dimensions given in Fig. 2.3. The size

of these unit cells was determined to be too small, so each unit cell above

500 Hz was given a side length of 0.051 m, a dimension chosen arbitrarily so

that the unit cell was of workable size.

f0 (Hz) a (m) Cell Size (m)100 0.0853 0.3556200 0.0301 0.1524300 0.0164 0.1016400 0.0107 0.051500 0.0076 0.051600 0.0058 0.051700 0.0046 0.051800 0.0038 0.051900 0.0032 0.0511000 0.0023 0.051

Table 2.1: Dimensions and expected Minnaert resonance, denoted as f0, ofcylindrical inclusions in POM unit cells with a thickness of t = 6.35 mm.

21

2.2.2 Experimental Apparatus and Data Analysis

The resonance frequencies of the unit cells were measured using a sub-

traction method commonly used with encapsulated bubbles in a large, closed

tank designed to provide a flat response at low frequencies [15]. The apparatus

is shown in Fig. 2.5. The tank was a cylindrically shaped with solid steel walls,

with outer diameter D = 1.03 m and height H = 1.03 m. The wall thickness

was 1.27 cm. The tank was completely filled with filtered and degassed water

[39]. It was closed with a tightly fitting lid of 5.08 cm thickness in such a way

that no air was present inside of the tank. The unit cell was suspended on

mono-filament line so that the unit cell was centered in the tank.

Figure 2.5: Experimental setup to measure the resonance frequency of a unitcell in a cylindrical tank with diameter D and height H.

The source used in these experiments was a stainless steel piston with

diameter of 8 cm attached to a stinger with length of 27 cm which fit through

22

a pre-drilled hole in the lid of the tank. The stinger was driven by a Lab-

Works ET-126HF electromagnetic shaker. The driving signal was band-limited

pseudo-random signal with frequency components between 10 Hz and 1.3 kHz.

The signal was generated by an Agilent 89410A vector signal analyzer (VSA).

The shaker was driven through a power amplifier, and the voltage provided

by the VSA applied to the shaker was held constant throughout the experi-

ment. The pressure signal of the acoustic field inside of the tank was measured

using a Reson TC4013 hydrophone conditioned by a Reson VP2000 voltage

preamplifier. The hydrophone was mounted through a hole in the tank wall a

distance of 6 cm below the bottom of the lid.

A common method for measuring the resonance frequency of a single

encapsulated bubble [15] is to consider the total complex pressure field of a

tank with a bubble as a superposition of the field of the empty tank and the

field created by the bubble,

Ptotal = Ptank + Pbubble. (2.18)

A rearrangement of Eq. (2.18) gives the scattered field of the bubble. The

relative amplitude of the pressure response of the bubble is given in dB as

Pbubble = 20 log10(Ptotal − Ptank). (2.19)

The same subtraction technique was used to determine the resonance

frequency of the unit cell. All data was collected from the VSA and processed

in MATLAB. A sample MATLAB script is provided in Appendix A.

23

2.2.3 Results of Unit Cell Experiments

Figure 2.6: Frequency response of the steel tank.

Though the measurements described above were conducted on all ten

unit cells, only four yielded useful and interpretable results. This is due in

part to two issues. Firstly, the tank setup was designed to work best at low

frequencies (below the first tank resonance which occurs at just above 300 Hz).

Fig. 2.6 shows a sample empty tank response from 10 Hz to 1 kHz. As the

frequency increases past an obvious tank resonance at approximately 350 Hz,

the response becomes more modal. It was difficult to clearly extract a single

bubble resonance out of these variations. Additionally, the higher frequency

inclusions were very small. This increased the difficulty of detecting a reso-

nance peak as the unit cell represented an increasingly small volume fraction

within the tank. As such, the presence of the bubble did not significantly alter

24

the pressure field enough to generate a difference that was visible above the

noise of the system.

Figure 2.7: Results of the 100 Hz unit cell measurement showing (a) Ptotal andPtank and (b) Pbubble.

Fig. 2.7 shows the acoustic spectra collected when the 100 Hz unit cell

was in the tank. Fig. 2.7-a shows the raw data before the subtraction technique

was applied. As this data is collected in a frequency range of the tank that is

relatively free from tank resonances and thus the difference between the blank

tank and tank with the inclusion was clear. After applying Eq. (2.19), the

resonance peak is clearly shown in Fig. 2.7-b. The resonance frequency of this

inclusion was found to be 131 Hz.

25


The behavior of the 200 Hz unit cell is shown in Fig. 2.8. This resonance

occurred near a small resonance in the tank (see Fig. 2.6), but a difference

could still be seen when the subtraction method was applied, Fig. 2.8-b. The

resonance of this unit cell was recorded to be 210 Hz, though it could truly lie

anywhere between 205 Hz and 225 Hz.

26


The 500 Hz unit cell is shown in Fig. 2.9. The resonance frequency of

this inclusion was found to occur at 534 Hz.

27

Figure 2.10: Results of the 800 Hz unit cell measurement showing (a) Ptotal

and Ptank and (b) Pbubble.

Finally, the 800 Hz unit cell is shown in Fig. 2.10. Like the 200 Hz unit

cell the inclusion resonance seems to have occurred within a small resonance

of the tank. The resonance of the inclusion was recorded as 822 Hz (the

location of the tallest peak), though it could lie anywhere between 810 Hz

and 825 Hz. These experimental results seem to confirm that the Minnaert

resonance frequency provides a reasonable estimate of the resonance frequency

of the cylindrical unit cells of interest to this work. Now, a finite element

analysis will be used to further confirm these results.

28

2.3 Finite Element Analysis of Cylindrical Inclusion

The resonance frequency of a single cylindrical inclusion can also be

determined using a finite element analysis (FEA). Finite element analysis is a

method of subdividing an entire domain into smaller, discrete, finite elements

thus simplifying the solution of equations that define the physics of the problem

[40]. FEA can be implemented by hand for simple systems [41], but for more

complicated physical systems it is far more convenient to use a computer.

Multiple computer programs exist to aid in FEA of various systems. COMSOL

Multiphysics was used in this work.

This work uses COMSOL Multiphysics 4.3b to COMSOL Multiphysics

5.0, with models created in earlier versions being updated as necessary to

utilize the newest software updates. The simulations described in this sec-

tion were done using the Acoustic-Structure Interaction module of COMSOL

Multiphysics. All models presented below are 2-D, axisymmetric models.

2.3.1 Development of Model

A model of the entire experimental setup described in Section 2.2 is

unnecessarily complicated and computationally demanding. Instead, the unit

cell was approximated as an inclusion in an infinite water domain. The infinite

domain was approximated using a finite spherical water domain with radiation

boundary conditions. An incident pressure wave of unit amplitude was sent

from the radiation boundary. The response of the inclusion was measured

with a velocity probe on the surface boundary of the inclusion. The model

29

was tested and proven to be accurate by successfully predicting the Minnaert

frequency of a free spherical air bubble positioned at the center of the domain.

In all simulations, the thickness of the POM plastic was 6.35 mm, and

the thickness of the neoprene rubber was 0.8 mm, leading to a total unit cell

thickness of 7.95 mm, replicating the real experimental apparatus. Values

for the required input materials were taken from the literature [42–45] and

are given in Table 2.2. The radius of the inclusion was varied in the model

throughout the size range associated with the experimental unit cells.

Material ρ (g/cm3) E (GPa) ν c (m/s) ηsDelrin 1.41 3.2 0.35 — 0.005

Neoprene 1.4 0.01 0.49 — 0.2Water 0.99 — — 1498 —

Air 0.00121 — — 343 —

Table 2.2: Material properties used in COMSOL Multiphysics finite elementanalysis.

To ensure that the situation was modeled appropriately, it was simu-

lated in stages. Fig. 2.11-a shows the initial model, a cylindrical air inclusion

inside of a water domain. The inclusion had radius a that was given by Ta-

ble 2.1. The spherical water domain had a radius of 2a. All domains within

the model were meshed with a free triangular mesh. The largest mesh element

size was at most one sixth the minimum wavelength of the wave water [46, 47].

A velocity probe placed at the center of the inclusion on the boundary between

the air and water domains measured the amplitude of the velocity normal to

the surface.

30

Figure 2.11: Schematics of COMSOL geometries used to model the system.

The next model places a POM plastic disk around the air inclusion.

Fig. 2.11-b shows the air inclusion of radius a situated in the center of a disk

of POM plastic with radius 2a. The disk was centered in a spherical water

domain with radius 3a. An isotropic loss factor, ηs was added to the solid

POM plastic to simulate material losses. This value was chosen to be ηs=0.005.

This ηs value was chosen so that the amplitude corresponds to a material with

very low loss. The meshing conditions were the same as described for the

previous model, and a velocity probe was placed in the same location as in the

cylindrical cavity case.

The final model has a POM plastic disk with an air inclusion sand-

wiched between two neoprene rubber layers as shown in Fig. 2.11-c. As before,

31

the inclusion of radius a was centered inside a POM plastic disk of radius 2a.

The neoprene layers had radius 2a. The entire cell was centered in a spherical

water domain of radius 3a. An isotropic loss factor of ηs=0.005 was assigned

to the POM plastic and a larger isotropic loss factor of ηs=0.2 was assigned

to the neoprene layers. The meshing conditions were the same as described

before, but the neoprene was meshed such that there were three mesh elements

across the thickness of the neoprene layer. A velocity probe was placed in the

center of the neoprene layer at the center of the inclusion.

2.3.2 Model Analysis and Results

For all models, velocity probe data was exported from COMSOL into

MATLAB where the absolute value of the velocity was plotted as a function of

frequency. Each data set was normalized by the maximum amplitude so that

the resonance occurred when the velocity is unity. A sample of the MATLAB

code is given in Appendix A.

32

Figure 2.12: Velocity response of the unit cell designed with a Minnaert fre-quency of 100 Hz.

Figure 2.12 shows the velocity response of the cell designed at 100 Hz

following the process described in Section 2.1.2. The resonance of the system

appears as a sharp peak in velocity. The cylindrical air inclusion (black line in

Fig. 2.12) had a resonance frequency of 134 Hz. When the inclusion was placed

inside of the POM plastic disk (blue line in Fig. 2.12), the resonance shifted

downward to 130 Hz. Finally, the complete unit cell (red line in Fig. 2.12) had

a resonance at 125 Hz. There are several “side lobes” in the resonance pattern

of the 100 Hz unit cell. These “side lobes” have amplitudes less than 20%

of the resonance peak. They are attributed to the presence of modes on the

stretched neoprene layers, which were acting as clamped circular membranes

33

[48, 49]. The presence of membrane modes leads to a leakage of energy in the

system.


The resonance behavior of the 200 Hz unit cell is shown in Fig. 2.13.

The cylindrical air inclusion had a resonance of 235 Hz. Inside of the POM

disk, the resonance shifted downwards to 220 Hz. Finally, the entire unit cell

had a resonance frequency of 205 Hz. Figure 2.13 shows the presence of low

amplitude “side lobes” that are attributed to membrane modes of the neoprene

layers. In this case, the inclusion was smaller in size, and the membrane modes

in the neoprene were not as pronounced.

34


The 500 Hz cell response is shown in Fig. 2.14. The downward shift

in frequency from the stand alone inclusion to the unit cell becomes more

apparent. The resonance frequency of the cylindrical air inclusion was found

to be 524 Hz, and the resonance frequency of the inclusion in the POM disk

was 446 Hz. The full inclusion had a lower resonance frequency of 402 Hz.

Additionally, the full inclusion resonance had a lower Q-factor when compared

to the other cases. At 500 Hz, the air inclusion had radius a of 7.6 mm and

height t of 6.35 mm. The inclusion began to look less like a flattened cylinder

and more like a tube. It is clear that the change in aspect ratio changes the

resonant response of the unit cell.

35


Finally, the 800 Hz cell response is shown in Fig. 2.15. While the

first two models showed a similar downward shift in resonance frequency from

823 Hz to 642 Hz, the full unit cell model showed a significant deviation from

the Minnaert approximation. The resonance frequency of the unit cell was

found to be 1.2 kHz. Additionally, the resonance peak was broad. The 800 Hz

cell had an even more pronounced tubular shape. In fact the aspect ratio of

radius to cylinder height in this case was approximately 1/2.

36

2.4 Model Comparisons

Fig. 2.16 compares the analytical model for an equivalent volume spher-

ical bubble derived in Section 2.1, the experimental results presented in Sec-

tion 2.2, and the FEA results presented in Section 2.3. At lower resonance

frequencies, the experimental and FEA results agree with the Minnaert fre-

quencies. At higher frequencies, the data begins to diverge. The experimental

results are higher than the Minnaert frequencies, and FEA results predict a

significant difference between Minnaert resonance frequency and experimental

resonance frequency. The difference is attributed to the aspect ratio change

at higher frequencies. Inclusions with a smaller radius look less disc-shaped

than the lower frequency inclusions.

37

Figure 2.16: Comparison of the Minnaert model of resonance frequency withexperimental data and FEA results for the unit cell.

Based on the results shown in Fig. 2.16, maintaining a flatter aspect

ratio as given in Eq. (2.20) will lead to a closer correlation with the Minnaert

model. The 100 Hz unit cell has an aspect ratio such that a/t ≈ 10. The

200 Hz unit cell has an aspect ratio such that a/t ≈ 4. From these numbers

and from knowledge of previous work done with encapsulated bubbles, a value

for the aspect ratio is chosen to maintain a pancake inclusion shape yet allow

for inclusions that are reasonably sized. The aspect ratio between radius, a,

and thickness of the plastic center layer, t, for the inclusion should be

a

t≈ 2.4. (2.20)

38

With this knowledge, the approximation outlined in Section 2.1 can

be used to design the radius of inclusions for use in sub-wavelength panels.

The approximate resonance frequency of a cylindrical inclusion is close to the

actual resonance frequency of an equivalent sphere, ensuring that the panel

with resonant inclusions in the desired frequencies range can be easily designed.

This approximation is used for all work presented in Chapter 3.

39

Chapter 3

Measurement of Panel Performance

This chapter focuses on the design, construction, and testing of the

sub-wavelength underwater isolator described in Chapter 1. Section 3.1 will

describe the dimensions of the panels and their construction materials. Sec-

tion 3.2 will overview the experimental methods used to test the thin-panel

isolators, as well as the post-processing done on collected data. Section 3.3

and Section 3.4 will present the results for two different scales of panel design.

Finally, Section 3.5 will discuss a finite element model of the high frequency

panels. Ultimately, the sub-wavelength panel design is shown to be effective

at isolating sound underwater in various frequency ranges.

3.1 Panel Construction

Chapter 2 showed the experimental and FEA results of the resonance

frequency of unit cells. This section will look at the construction of panels

containing several unit cells. For this purpose, panels were constructed using

the materials described previously for the unit cells, polyoxymethylene (POM)

and neoprene rubber. The materials have he material properties provided in

Table 2.2. The neoprene was attached to the POM plastic using a thin layer of

40

DAP Weldwood, a neoprene based contact cement. Air bubbles were removed

from the adhesive layer by pressing them out carefully as the neoprene layer

was placed down.

Figure 3.1: Schematic showing the air filled inclusion, a panel with air filledinclusions, and a five-sided box created from the isolating panels.

The panels have a layered structure as described previously, and as

shown in Fig. 3.1-a again for convenience. The POM layer has thickness t and

the panel has thickness T. Note that to maintain the aspect ratio given by

Eq. (2.20), the values for t and T were specific to the frequency range being

studied; however, the thickness of the neoprene rubber remained a constant

0.8 mm.

The complete panel is shown in Fig. 3.1-b. To create cylindrical air

inclusions holes with specified radius a were drilled through the POM layer.

For the purpose of these experiments, three different cases of panels were

created in order to explicitly measure the influence of different components of

41

the panel on acoustic isolating performance. The first was a plain panel, which

was simply a POM layer of thickness t. The second was a perforated panel,

which was a POM layer of thickness t perforated with a specific number of

holes with radius a to create the desired void fraction. The third was the void-

filled panel, which was the complete three-layered panel shown in Fig. 3.1-b.

In all cases, a five-sided cube-shaped enclosure was fabricated from the panels

so that an underwater sound source could be surrounded on all sides with the

upward-facing side being the air-water interface. Panels were attached to one

another with hot glue, and excess hot glue was removed with a sharp knife.

The cube-shaped enclosure is shown in Fig. 3.1-c.

3.2 Experimental Design

The panel’s effectiveness as an underwater isolator was tested by mea-

suring its effect on the acoustic response of a water-filled test tank. Panels

were designed for two frequency ranges, and hence required two separate appa-

ratus. Section 3.2.1 describes the high frequency apparatus, and Section 3.2.2

describes the low frequency apparatus. Additionally, Section 3.2.3 explains

the technique for post-processing the data presented in the rest of the chapter.

3.2.1 High Frequency (2 kHz to 5 kHz) Tank Experiments

For panels designed to work in the 2 kHz to 5 kHz range, experiments

were conducted inside an acrylic tank with the dimensions shown in Fig. 3.2.

A source and receiver were placed at opposite ends of the tank. The exact

42

locations of the source and receiver are shown in Fig. 3.3. Both the receiver

and source were suspended approximately 4 cm below the air-water interface.

Figure 3.2: An acrylic tank with 0.37 m deep water was used for high frequencyexperiments.

Figure 3.3: Location of the source (S ) and receiver (R) in the high frequencytank experiment.

43

It should be noted that because this experiment focuses on a frequency

response pattern, the position of the source and receiver are arbitrary as long

as the positions meet two conditions. First, neither the source nor the receiver

should be placed in a null of the tank. Placing the receiver in a null was avoided

by moving the receiver around the tank while the source was on and stationary

until the pressure signal was at a maximum. Secondly, care should be taken

to keep the receiver and source stationary during the course of testing. In this

case, the source and receiver were affixed to a scaffolding over the tank with

a series of clamps.

The source used in the high frequency experiments was a stainless steel

piston with diameter of 12.7 mm, driven by a stainless steel stinger with length

of 18 cm that was attached to an LDS V101 electromagnetic shaker. The exci-

tation signal was a continuously repeated periodic chirp from 2 kHz to 5 kHz

generated by an Agilent 89410a Vector Signal Analyzer (VSA) which drove the

shaker through a power amplifier. The voltage level on the power amplifier

was constant throughout all measurements. The receiver used to measure the

acoustic pressure within the tank was a Reson TC4013 hydrophone conditioned

with a Reson VP2000 voltage preamplifier.

To eliminate the formation of air bubbles in the tank, the water was

degassed using a traditional method [39] for approximately three hours prior

to conducting experiments. The thin aluminum tube that acted as a sheath for

the hydrophone wire was back-filled with water from the tank to improve signal

coherence. Each panel cube was made to enclose the source by suspending

44

it from shaker plate with mono-filament fishing line, as shown in Fig. 3.4.

Suspending the cube from the shaker plate allowed for a consistent placement

of the panels. The cube was suspended such that the upward-facing side of

the cube was the air-water interface.

Figure 3.4: A photograph of the void-filled cube in place during an experiment,showing the cube suspended around the source from the shaker plate (A) andthe cube’s upward-facing side (B) on the air-water interface.

3.2.2 Low Frequency (500 Hz to 1.3 kHz) Tank Experiments

For panels designed to work in the 500 Hz to 1.3 kHz range, experi-

ments were conducted using the apparatus described in Section 2.2 with the

lid removed from the tank. To prevent coupling between the low frequency

source and the tank, the source was suspended from a hook above the tank

by a specially designed shaker plate as indicated by the schematic shown in

45

Fig. 3.5. A photograph of the shaker in place is shown in Fig. 3.6. The posi-

tion of the source was arbitrary as long as it was held constant throughout the

experiments and did not sit in a null of the tank’s modal structure. The cube

was suspended on mono-filament line from the shaker plate. The length of the

line was adjusted so that the air-water interface formed the upward-facing side

of the cube, completely enclosing the source.

Figure 3.5: Tank described in Section 2.2 with shaker and cube suspendedfrom a plate above the tank.

The source used in the low frequency experiments was a stainless steel

piston with diameter of 8 cm attached to a stinger with length of 27 cm. The

stinger was attached to a LabWorks ET-126HF electromagnetic shaker. To

46

improve the quality of the signal, the frequency range in question was broken

into six frequency bands. In all cases, the signal was a continuously repeated

periodic chirp generated by an Agilent 89410a Vector Signal Analyzer. The

shaker was driven through a power amplifier, and the voltage at the power

amplifier was kept constant throughout the measurements. The wall mounted

hydrophone was a Reson TC4013 conditioned using a Reson VP2000 voltage

preamplifier.

Figure 3.6: Low frequency cube during an experiment showing the hook andplate (A) used to suspend the shaker and the cube suspended on mono-filamentsuch that the upward-facing side (B) is at the air-water interface.

3.2.3 Post-Processing of Data

For both low and high frequency experiments, the acoustic pressure

response measured by the hydrophone was digitized using the Agilent VSA.

47

The signal analyzer calculated the averaged transfer function (the hydrophone

signal normalized by the excitation signal) using 30 frequency sweeps. Transfer

functions were collected for each of the panel cases (plain, perforated, and void-

filled) at the varying void fractions and material properties discussed below.

Transfer functions were also collected for the empty tank. The empty tank

is referred to in this work as the baseline case. All data was processed using

MATLAB. A sample code for the processing of data can be found in Appendix

A.

The frequency-averaged insertion loss is useful as a single-number met-

ric for comparison of panel noise isolation efficacy, and was calculated by sub-

tracting the value of the transfer function for each case from the value of

the transfer function of the baseline, converting into a dB scale, and taking

the mean across the frequency range. The MATLAB code used to calculate

average insertion loss is found in Appendix A.

3.3 High Frequency (2 kHz to 5 kHz) Panels

This section discusses the effect of including voids in a panel, the effect

of increasing the number of inclusions on a panel, and the effect of changing the

material that the panel was constructed out of on the panel’s isolation effect.

The panels were designed to operate across the 2 kHz to 5 kHz frequency range.

Panels were created and tested as described in Section 3.1. These panels had

POM layer thickness t = 1.9 mm and total thickness T = 3.5 mm. The panels

had side length L = 9.8 cm. Using the method described in Chapter 2, the

48

radius of the inclusions for these panels was a = 4.3 mm. The plain, perforated,

and void-filled panels tested in these experiments are shown in Fig. 3.7.

Figure 3.7: Cubes created from high frequency panels. From (L) to (R): Plaincase, perforated case, void-filled case.

3.3.1 Effect of Voids

The perforated case and the void-filled case had four cylindrical inclu-

sions per panel. The inclusions were placed randomly; however, an effort was

made to ensure that inclusions were placed at least one diameter from their

nearest neighbor or the edge of the panel.

49

Figure 3.8: Isolation effect of panels with and without air-filled inclusions onthe frequency response of the tank.

The measured transfer functions are shown in Fig. 3.8. The baseline

(black line), plain (green line), and perforated (red line) cases all show a similar

received pressure level. The void-filled case, represented by the blue line, shows

a significant decrease in received level across the frequency band. It is clear

that the majority of the isolation effect is due to the presence of enclosed

pockets of air.

3.3.2 Effect of Void Fraction

Attenuation of sound propagating in bubbly water increases with the

increasing void fraction of the air phase [9]. Similarly, a panel with more

50

inclusions should provide more isolation across the frequency band. The void

fraction of the material is defined as the volume of the inclusions divided by

the total volume of a single solid plate,

Void Fraction =ntπa2

TL2. (3.1)

In Eq. (3.1), t is the thickness of the center layer of the panel, T is

the total thickness of the panel including neoprene, L is the side length of the

panel, and n is the number of inclusions present on the panel. Panels with void

fractions of 0.6%, 1.3%, 2.6%, and 3.9% were created. These void fractions

represent panels that had 2, 4, 8, and 12 air filled inclusions respectively.

Again, care was taken to ensure that inclusions were at least one diameter

from their nearest neighbor or the edge of the panel. These conditions became

more difficult to meet at the highest void fraction (3.9%, 12 inclusions).

51

Figure 3.9: Isolation effect of panels with varying void fraction on the frequencyresponse of the tank.

Transfer functions for each of the void fraction cases are presented in

Fig. 3.9. It can be seen that for a majority of the frequency range an increase

in void fraction on the panel leads to an increase in isolation effect. The mean

insertion loss was calculated for six 500 Hz bands across the frequency range,

as shown by Fig. 3.10. The case of the plain panel (no inclusions, 0% void

fraction) was included for comparison.

52

Figure 3.10: Mean insertion loss across specified frequency bands for panelswith varying void fraction.

Figure 3.10 shows that below 3.5 kHz, an increasing void fraction corre-

sponded directly to an increasing insertion loss. Between 3.5 kHz and 4.5 kHz,

the trend shifted. For the remainder of the frequency range the 0.6% and 1.3%

void fraction panels had a higher isolation effect. In cases with a high void

fraction, maintaining the one diameter distance between the inclusion’s neigh-

bor or the edge of the panel became difficult. The inclusions are likely coupling

to one another, causing a shift in resonance frequency and deteriorating the

isolation effect in the desired band [50–52].

53

Figure 3.11: Mean insertion loss of the panel as a function of void fractionacross the entirety of the frequency range.

The mean insertion loss across the entire 2 kHz - 5 kHz range showed

the expected monotonic increase, Fig. 3.11. The exact values for insertion loss

of the 0.6%, 1.3%, 2.6%, and 3.9% panels were 8.25 dB, 9.08 dB, 9.13 dB,

and 10.25 dB respectively. In these frequencies, panels had a non-dimensional

thickness, kT, of 0.02 to 0.07.

3.3.3 Effect of Using 3-D Printed Components

Until this point, panels have been constructed with a center layer of

POM. For these experiments, the POM layer was replaced with a porous poly-

lactide (PLA) polymer created using 3-D printing. PLA is a plastic with prop-

54

erties similar to Delrin. It has a density of 1.25 g/cm3 and Young’s Modulus

of 3.5 GPa [42, 43]. Sheets of PLA were created on on a 3D printer which used

PLA polymer as its filament material. 3D printing is an emerging technology

that allows for quick replication and rapid prototyping of designs that are cre-

ated in a CAD program like SolidWorks. A 3D printer uses a melted plastic

filament to draw layers of the desired design, ultimately creating a solid plastic

piece.

Figure 3.12: Example of a solid cylinder created on a 3D printer. The interiorhexagonal structure is called in-fill. Image courtesy of MakerBot.

In general, a object printed on a 3D printer like that available to the

author has a solid outer layer and is filled inside with a hexagonal plastic

structure, as shown in Fig. 3.12. This hexagonal structure is called the in-fill,

and it exists to maintain an object’s structural stability while reducing the

55

amount of PLA plastic used in the print. The MakerBot Replicator 2 was

used to print panels in this work. The Replicator 2 allows users to adjust the

in-fill percentage of the solid portions of a printed piece. Adjusting the in-fill

percentage adjusts the size of the hexagonal structure in the part. Fig. 3.13

shows rectangular prisms with high and low in-fills. A higher in-fill percentage

has a smaller hexagonal structure and uses a higher amount of PLA plastic in

the print. In-fill can be adjusted from 1% to 100%, but commonly solids are

printed with a 15% to 25% in-fill.

Figure 3.13: Two 3D printed pieces with high in-fill (a) and low in-fill (b)percentage.

An attempt was made to correlate in-fill percentage and panel prop-

erties. Solid square panels with the same dimensions given previously were

printed at four different in-fill percentages: 5%, 25%, 50%, and 100%. The den-

56

sities of these panels were measured to be 1.06 g/cm3, 1.08 g/cm3, 1.07 g/cm3,

and 1.04 g/cm3 respectively. An increase in in-fill percentage should increase

the percentage of the panel that is plastic, thus increasing the panel’s density.

An increase from 50% in-fill to 100% in-fill should correlate to an addition of

50% more plastic (and a rough doubling of density), but this is not shown in

the overall density calculations.

As stated above, each solid 3D printed object has a completely solid

outer layer and the hexagonal in-fill. The thickness of the solid outer layer

is usually thin in comparison to the scale of the printed object. The same

cannot be said for these panels. These panels are dominated by the solid

outer layers. This explains the disparity between in-fill and panel density.

Unlike the prisms shown in Fig. 3.13, the hexagonal layer is very thin, with

a thickness of approximately 0.5 mm compared to the total panel thickness

of 1.9 mm. Additionally, observing the panel as it was being printed showed

that the interior hexagonal layers were almost non-existent.

Though the change in in-fill percentage does not seem to affect the

bulk material properties of the thin panels, it is still worth investigating if

there will be an effect on the acoustic isolation. Panels with dimensions L =

9.8 cm, and t = 1.9 mm were printed with four holes of radius a = 4.3 mm

randomly spaced across the face. This corresponded to a void fraction of 1.3%.

Neoprene sheets of 0.8 mm thickness were attached to the PLA plastic with

neoprene based contact cement. Cubes were fabricated from panels printed at

three in-fill percentages: 5%, 25%, and 50%.

57

Figure 3.14: Isolation effect of panels with varying in-fill percentages.

Figure 3.14 shows the acoustic pressure spectra for void-filled panels of

varying in-fill percentages. Over the entire frequency range, the panels had

an isolation effect. Below 3.5 kHz, the panels behaved similarly regardless of

in-fill percentage, but above 3.5 kHz the panel behavior started to diverge. At

higher frequencies, the 5% in-fill panel still isolates, but it has a lower isolation

efficacy than the higher in-fill panels.

58

Figure 3.15: Mean insertion loss across specified frequency bands for a panelmade with PLA plastic center layer of varying in-fill percentage.

The data sets for these panels were broken down into frequency bands

and the mean insertion loss for each frequency band was calculated, as seen

in Fig. 3.15. At lower frequencies panels behave similarly, and at higher fre-

quencies their behavior diverged. The 50% in-fill panel maintained the highest

amount of isolation across all of the frequency bands.

59

Figure 3.16: Effect of in-fill percentage of a PLA panel on insertion loss forspecified frequency bands.

The effect of in-fill percentage on isolation across frequency bands is

presented in Fig. 3.16. This graph shows that, generally, an increase in in-fill

percentage leads to an increase in insertion loss. Additionally, this increase is

most drastic for the 4.5 kHz to 5 kHz frequency range. The mean insertion loss

was also calculated across the entire frequency band. The results are plotted

with the previous Delrin panel results in Fig. 3.17.

60

Figure 3.17: Mean insertion loss of the panel as a function of void fraction forDelrin plastic and PLA plastic center layers.

For both the 25% in-fill and the 50% in-fill, the PLA plastic panel

showed an improved performance over a Delrin counterpart of similar void

fraction. For the 5% in-fill, the PLA plastic panel showed a reduction in per-

formance. These results are especially interesting if one considers a Delrin

panel to be essentially 100% in-fill. One can make the hypothesis based on

the data that printed panels are ineffective at low in-fill, increase in efficacy

through in-fill up to 50%, and then decrease in efficacy again by 100%. Ul-

timately, it seems the in-fill percentage is simply a machine setting that has

more impact on the bulk material (as shown in the prisms in Fig. 3.13) than

the thin panels created here. However, based on the data, the in-fill percentage

61

of the panel material must alter isolation effect. At this point the exact phys-

ical reason for this effect is unknown. Much more detailed experiments are

required to develop firm conclusions about the isolation efficacy of 3D printed

panels.

3.4 Low Frequency (500 Hz to 1.3 kHz) Panels

Panels designed to operate in the 500 Hz to 1.3 kHz range were created

and tested as described in Section 3.1. These panels had POM layer thickness

t = 3.9 mm and total thickness T = 5.5 mm. The panels have side length

L = 15.2 cm. The inclusion has radius a = 9.525 mm. The kT value for this

system is approximately 0.01 to 0.03, compared to a kT value of 0.02 to 0.07

for the high frequency panels. All panels had two inclusions, corresponding

to a void fraction of 1.3%. The inclusions were placed randomly but care was

taken to ensure that each inclusion was one diameter away from its neighbor

and the edge of the panel. As before, three types of panels were created. The

solid POM panel, called the plain case, the POM panel with uncovered holes,

called the perforated case, and the POM panel with covered inclusions, called

the void-filled case. The three cases are shown in Fig. 3.18.

62

Figure 3.18: Cubes created from low frequency panels. From (L) to (R): Plaincase, perforated case, void-filled case.

3.4.1 Effect of Voids

Figure 3.19: Isolation effect for panels designed to operate from 500 Hz to1.3 kHz.

63

The measured transfer functions for low frequency panels are shown

in Fig. 3.19. Note the tank response is more complicated than the responses

shown in Section 3.3, due to the fact that the cylindrical tank has a more

complex modal structure than the rectangular tank used in Section 3.3. Sec-

ondly, the dynamic range of Fig. 3.19 is large with a range of 100 dB, rendering

small differences in level difficult to see. For better visualization, the data is

split into six separate frequency bands. From 150 Hz to 350 Hz the source

was exciting the tank below the resonance of the inclusion in the panel. In

Fig. 3.20, the baseline, plain, and perforated cases all had a similar received

level. The void-filled case had a received level that is higher than the base-

line. This behavior is attributed to the fact that a bubble, below resonance,

amplifies sound [8, 9] rather than attenuating it.

Figure 3.20: Isolation effect of the low frequency panel from 150 Hz to 350 Hz.

64


As the frequency was increased the resonance frequency of the bubble

was crossed and the panel began to act as an isolator, as shown in Fig. 3.21.

A significant decrease in received level is shown in this frequency band. Fig-

ure 3.22 shows the received level of the void-filled panel was consistently lower

than the baseline case. Above 650 Hz, the tank becomes increasingly diffuse;

yet, the received level for the void-filled panel remained at or below received

level for the baseline, plain and perforated cases, as shown in Fig. 3.23 through

Fig. 3.25.

65



66

Figure 3.24: Isolation effect of the low frequency panel from 800 Hz to 1 kHz.

Figure 3.25: Isolation effect of the low frequency panel from 1 kHz to 1.3 kHz.

67

3.4.2 Mean Insertion Loss

Figure 3.26: Mean insertion loss as a function of frequency for panels designedto operate from 500 Hz to 1.3 kHz.

Mean insertion loss for the low frequency panel was calculated as a

function of frequency. Figure 3.26 shows these values. An insertion loss below

zero implies an amplification relative to the baseline case, and an insertion loss

above zero indicates an isolation relative to the baseline case. As expected,

the plain and perforated cases show little deviation from baseline. Below

the resonance frequency of the inclusion, the void-filled panel demonstrated

an average amplification of 2.2 dB. Above resonance the panel had an average

isolation of 3.1 dB. This is comparable to the insertion loss for panels of similar

68

void fraction shown in Fig. 3.11. A more complete low frequency void fraction

study would be useful, but it was beyond the scope of this work.

3.5 Finite Element Analysis of 2 kHz to 5 kHz Panel

In this section, a finite element analysis of the thin-panel isolator is

developed to serve as a compliment to experimental results presented above.

The analysis of the system was done in COMSOL Multiphysics 4.3b.

3.5.1 Development of Model

The cube structure described in Section 3.1 and Section 3.3 was sim-

ulated in 3D using the COMSOL Multiphysics 4.3b Acoustic-Structure Inter-

action physics module. The geometry of the model is shown in Fig. 3.27. It

was not necessary to exactly recreate the tank experiment. In fact, the results

presented here are more preferable because they simulate the behavior of the

isolator in a free-field environment which is similar to what the panel would

experience in-situ. Instead, the model assumed the cube was located in a free-

field water environment. A water half-space was simulated by a large rectangle

bounded by perfectly matched layers on five sides. Perfectly matched layers

are finite element domains whose behavior is such that no reflections return

from the domain boundaries or edges. These types of domains can therefore

be used to simulate an infinite environment. The top face (in the orientation

shown in Fig. 3.27) of the domain was a pressure release surface modeled to

approximate the air-water interface. The cube was located directly in the cen-

69

ter of x-y plane. The source was a simulated piston located in the center of

the cube. The piston was created by modeling an empty rectangular prism

with an end-on area of 1 cm2 and applying a frequency dependent normal

acceleration condition to this plane.

Figure 3.27: 3D COMSOL geometry used to model the thin-panel system.

The material properties used in the simulation are given in Table 2.2.

The Delrin and neoprene domains were classified as linear elastic materials.

The air and water domains were classified as fluids. Isotropic loss factors were

added to both the Delrin and the neoprene domains to simulate loss in those

materials. Delrin was given a value of ηs = 0.005 and neoprene was given a

value of ηs = 0.01.

All elements in the geometry were meshed using a free tetrahedral mesh.

70

This geometry was meshed in such a way that the largest element was one-

sixth of the wavelength of the slowest sound-speed in the medium [46, 47]. On

the neoprene sheets, this mesh element size was decreased to a tenth of the

wavelength of the shear wave speed in the medium being meshed.

3.5.2 Model Analysis and Results

The excitation frequency of the piston’s normal acceleration was swept

from 2 kHz to 5 kHz in increments of 25 Hz. A COMSOL boundary probe

was used to measure the radiated power through each of the six boundaries

of the water half-space. The radiated sound power was calculated by taking

the surface integral of the component of the intensity normal to the surfaces

of the water domains,

Wboundary =

∫S

I⊥dS. (3.2)

The total radiated power is the sum of the power calculated for each of the

six surfaces of the geometry,

Wtotal =6∑i=1

Wboundary,i. (3.3)

All of these calculations were done in the COMSOL interface, and the

data was exported to MATLAB for further analysis and plotting.

71

Figure 3.28: Results finite element analysis of radiated sound power in thepanel system.

The radiated sound power predicted by the finite element model is pre-

sented in Fig. 3.28 in dB re 1 nW. Across the frequency range, the baseline,

plain, and perforated cases all maintained a similar level of radiated power.

The void-filled case showed a significant decrease in radiated power after ap-

proximately 2.25 kHz and maintained a lower radiated power up to the end

of the frequency range at 5 kHz. This behavior is consistent with what would

be expected from a panel with inclusions that have a resonance frequency

near 1.3 kHz. These results confirm what was shown in Section 3.3. A sub-

wavelength, void-filled panel will act as an isolator for sound in the 2 kHz to

72

5 kHz frequency range.

The experimental data and finite element models presented in this chap-

ter confirm that sub-wavelength panels can be constructed that surpass the

isolation capabilities of a pure panel. Their isolation capabilities rely on the

resonant effects of bubble-like cylindrical inclusions within the panel. The re-

sults also confirm that the predictions and measurements on unit cells can be

extended to multi-cell panels for sound isolation.

73

Chapter 4

Conclusions

4.1 Summary

Through a combination of experiment and finite element analysis, this

work successfully completed the three objectives presented in Chapter 1. To

reiterate, it was the goal of this work to design, fabricate, and experimentally

verify the efficacy of sub-wavelength panels for acoustic isolation. Inspired by

the previously studied isolating capabilities of bubbles and advancements in

the field of acoustic metamaterials, Chapter 1 presents the design of a three

layer panel with cylindrical air-filled inclusions.

Chapter 2 provided a derivation of the resonance of a spherical bubble

in water [7, 8] and used the radius of that sphere to approximate the radius of

an equivalent volume cylinder with thickness t. A series of unit cells with cylin-

drical inclusions of various resonance frequencies were created and tested. In

addition, several different sized cylindrical inclusions were modeled in COM-

SOL Multiphysics. From these results, improved estimates of the resonance

frequency for these cylindrical inclusions were determined.

Overall, the experimental and FEA results were in good agreement

with the Minnaert resonance model of the equivalent sphere. However, the

74

experimental results provided two important insights regarding the salient

differences between spherical and cylindrical bubbles with the same volume.

First, large cylindrical inclusions exhibit membrane resonances with amplitude

on the order of 20% the bubble resonance. This effect was shown in Fig. 2.12

and Fig. 2.13. Secondly, to closely match the Minnaert resonance, the aspect

ratio of cylinder radius, a, to cylinder thickness, t, must be such that the

inclusion maintains a flattened shape. This corresponds to an aspect ratio

a/t ≈ 2.4. An inclusion with a larger a/t aspect ratio will exhibit a resonance

that is significantly different from the Minnaert approximation, as was shown

in Fig. 2.15 and Fig. 2.16. Despite these differences, this work concludes that

the Minnaert approximation and the transformation detailed in Section 2.1.2

are appropriate design tools for the desired panel.

Chapter 3 elaborated on the design and construction of sub-wavelength

free standing panels for acoustic isolation as described in Chapter 1 and using

information obtained from the unit cell results in Chapter 2. Panels were

fabricated for two different frequency ranges: 2-5 kHz and 0.5-1.3 kHz. Panels

designed to isolate in the higher frequency range (2 kHz to 5 kHz) had a kT

value of 0.02 to 0.07. They provided an average isolation ranging from 8 dB

to 10 dB depending on the void fraction of the panel. This isolation effect

greatly exceeds what one would expect from the mass law. Increasing the void

fraction of the panel increases the isolation efficacy of the panel, as shown in

Fig. 3.11.

Additionally, the effect of center panel material was investigated for the

75

high frequency panels. A 3D printer was used to rapidly create panels from

PLA polymer at a variety of in-fill fractions. It was shown that panels with a

high in-fill fraction (25% to 50%) show a slight improvement in isolation over

the the POM panel.

Panels were also designed to isolate in the 500 Hz to 1.3 kHz range.

The isolation effect of the low frequency panel with void fraction of 1.3% was

shown in Fig. 3.26. The panel had a kT value of 0.01 to 0.03. The maximum

isolation occurs immediately following the resonance of the bubble. At this

point, the insertion loss was 3.1 dB.

Finally, the high frequency sub-wavelength panel system was modeled

using finite element analysis in COMSOL Multiphysics. The isolation effect

determined by the experimental measurements was complimented by a mea-

surement of radiated power in the system.

4.2 Future Work

This work has successfully designed and tested a sub-wavelength panel

for underwater acoustic isolation. Though the work presented here provides

strong evidence that these types of panels will be effective sub-wavelength

sound isolators, there are several aspects of the research that warrant further

investigation. First, an effort should be made to more fully characterize the

frequency dependence of the high frequency panels. The low frequency data

in Fig. 3.26 covers about two octaves, but the high frequency data in Fig. 3.10

covers only one octave. Expanding this frequency range may further illuminate

76

interesting bubble screen behavior. Additionally, the low frequency panels

should be tested at higher void fractions. Current data is for a low frequency

panel at 1.3% void fraction. Increasing the void fraction will allow for more

detailed investigation of isolation efficacy of the panel and the rich physics

associated with unit cell interaction.

It would also be interesting to investigate exactly how the panel is

isolating sound. Currently the panel is called an isolator, but performing ad-

ditional experiments on the structure would allow for further classification. It

would be pertinent to know if the panel is acting as a pure reflector, as an

absorber, or both. Additionally, it would be helpful to know what range of

kT (or another non-dimensional parameter) provides effective isolation in each

frequency range. An effort should be made to determine reflection coefficient,

transmission coefficient, and the attenuation coefficient of the material. This

information could come from measurement (like an impedance tube experi-

ment) or a more fully developed finite element analysis of the system.

Future work on this topic should attempt to more completely charac-

terize the dependence of panel isolation on panel material. This could be done

in a finite element analysis with interesting results being recreated experimen-

tally for proof of concept. It would be even easier to explore the relationship

between panel isolation and material properties using an analytical model de-

rived from the equivalent circuit model of the system. The beginning steps

of such a model is presented in Appendix B; however, it is preliminary and

requires more detailed investigation before being used as design tool for panels

77

of the type investigated by this work.

The effect of layering multiple panels and varying inclusion configu-

ration can also be explored using FEA or experiment. Creating panels with

two or more different size inclusions in hopes of increasing the range of the

isolation band is of significant interest and could easily be explored using the

methods presented in this thesis. These tests could be conducted with the

current material parameters and the scaling techniques outlined in Chapter 2.

Additionally, the panels should be tested at a variety of scaled frequencies,

including the more of the frequencies that are most associated with man-made

underwater noise [2].

Finally, it would be interesting to investigate the ability of the panels

to isolate the noise in environments where water must flow. If the panels

have a similar isolation effect when there are gaps between the edges of the

panels, they could be made to surround a flowing water source. Of particular

interest would be the isolation of aquarium filters, outboard motors, and other

underwater pumps. Fully characterized panels could be created, installed, and

tested in-situ.

In total, this thesis explores the present body of work in thin panel

acoustic metamaterials and underwater acoustic isolation using bubbles. These

ideas are then combined to develop the idea of a sub-wavelength, free-standing

underwater panel for acoustic isolation. Additionally, the spherical resonance

approximation proposed by Minnaert is shown to be valid for cylindrical inclu-

sions with a significantly flat aspect ratio. Panels with cylindrical inclusions

78

were developed and tested in the 500 Hz to 1.3 kHz range and the 2 kHz to

5 kHz range. Panels of this type show a frequency averaged insertion loss of

3 dB to 8 dB with a void fraction of 1.3% and non-dimensional kT value of 0.02

to 0.07. Additionally, it is shown that an increase in void fraction yields an

increase in panel isolation capability. It is the hope that this and future work

in the area of sub-wavelength panels will improve the underwater environment

for marine life and underwater naval applications.

79

Appendices

80

Appendix A

MATLAB Code for Data Analysis

A.1 Unit Cell - Experimental Results

The code below shows the post-processing method for determining the

resonance of the unit cell from the data collected as described in Section 2.2.2.

The code has been simplified to represent any possible file exported from the

Agilent VSA. Data is plotted using standard plot functions.

1 %Read in data and assign basic values2 Cell = importdata('DataCell.asc'); %Tank response w/ cell3 Blank = importdata('DataNoCell.asc'); %Tank respnose empty4

5 Freq Cell = Cell(:,1);6 Freq Blank = Blank(:,1);7

8 Cell Response = Cell(:,2);9 Blank Response = Blank(:,2);

10

11 %The following applies the subtraction to the data12 Real Response = Cell(:,4) - Blank(:,4);13 Imag Response = Cell(:,5) - Blank(:,5);14 Full Response = Real Response + 1i*Imag Response;15 Unit Cell Response = 20*log10(abs(Full Response));16

17 %Plot the raw data18 figure(1)19 plot(Freq Cell, Cell Response)20 hold on21 plot(Freq Blank, Blank Response)22 xlabel('Frequency [Hz]')

81

23 ylabel('Acoustic Pressure [dB]')24

25 %Plot the unit cell response26 figure(2)27 plot(Freq Cell, Unit Cell Response)28 xlabel('Frequency [Hz]')29 ylabel('Acoustic Pressure [dB]')

A.2 Unit Cell - Finite Element Analysis Results


resonance of the unit cell from the results of a finite element analysis as de-

scribed in Section 2.3.2. Data is plotted using standard plot functions.

1 %Read in data and assign basic values2 COMSOL = importdata('DataFromCOMSOL.txt');3 Freq = COMSOL(:,1);4 Velocity = COMSOL(:,2);5

6 %Normalize the velocity value7 Norm Velocity = abs(Velocity)./max(abs(Velocity));

A.3 Panel Isolation - Experimental Results


isolation effect of the panels as described in Section 3.2.3. The code has been

simplified to represent any possible file exported from the Agilent VSA. Data

is plotted using standard plot functions.

82

A.3.1 Transfer Function Graphs

Note, if there is more than one data set (as is the case for most of the

figures in this work), the maximum value will need to be determined by an

inspection of all available data sets.

1 %Read in data and assign values2 Case = importdata('Case.asc');3 Freq = Case(:,1);4 Tank Response = Case(:,2);5

6 %Determine maximum7 Max = max(Tank Response);8

9 %Normalize10 Normalized Response = 20*log10(Tank Response) - 20*log10(Max);

A.3.2 Insertion Loss Calculation

The average insertion loss is the difference between the baseline level

and the received level for various cases.

1 %Read in data and assign values2 Base = importdata('Baseline.asc');3 Case = importdata('Case.asc');4 Freq = Case(:,1);5 Base Response = Base(:,2);6 Case Response = Case(:,2);7

8 %Calculate Insertion Loss9 IL = mean(20*log10(Base Response - Case Response));

83

Appendix B

Analytical Model of Unit Cell

In an attempt to better understand the physics of the unit cell de-

scribed in Chapter 2, an analytical model was developed using an equivalent

circuit for the system. From the circuit model an impedance can be calcu-

lated. If the system is lossless, which the assumption being made here, the

resonance frequency of the inclusion will occur when the imaginary portion of

the impedance (the reactance) is equal to zero.

B.1 Equivalent Circuit Impedance

Figure B.1: Schematic of the unit cell described in Chapter 2.

Figure B.1 is a schematic drawing of the unit cell as described in Chap-

ter 2. The equivalent circuit for this system is presented in Figure B.2. Note

that in this analytical model the pressure source, P , is placed inside of the

84

cylindrical air-filled inclusion.

Figure B.2: Completed equivalent circuit of the unit cell.

Figure B.2 contains the following electro-acoustic components: The

radiation impedance of the piston-like membrane over the inclusion in the unit

cell, Zrad, the compliance of the membrane, Cmem, the mass of the membrane,

Mmem, the mass of the air near the inclusion surface, Mair, the compliance of

the air in the inclusion, Cair, and the compliance of the POM plastic walls

of the unit cell, Cwalls. The velocity components of the system are shown for

clarity in Fig. B.2. The first step to solve for the input impedance of the

system is to simplify the circuit.

85

Figure B.3: Simplified circuit model with impedances listed instead of indi-vidual components.

Figure B.3 shows the circuit model from Fig. B.2 simplified down to

the impedance components. The impedance of each individual component

is determined according to the rules of circuits. The total impedance of the

membrane can be seen as a series addition of the membrane components

Zmem = Zrad +1

jωCmem

+ jωMmem + jωMair. (B.1)

In order to show a symmetric circuit, which is easier to visualize, the compli-

ance of the wall is split into two ”‘walls”’ with each ”‘wall”’ having a com-

pliance of half the compliance of the material. The impedance of the walls is

simply

Zwalls =1

jωCwalls

2

=2

jωCwalls

. (B.2)

The impedance of the air is given by

Zair =1

jωCair

. (B.3)

86

Finally, the value for radiation impedance, Zrad, is taken from the piston func-

tions in Ref. [48], simplified to first-order when ka << 1

Zrad =ρ0c0πa2

(ω2a2

2c20+ j

8ωa

3πc20

). (B.4)

In Eq. (B.4), a is the radius of the inclusion and ρ0 and c0 are the

density and sound speed in water, respectively. The compliance and mass of

the membrane components are more difficult to determine. The thin material

layer can be treated as either a bending plate or as a membrane under tension.

As the tension of the membrane is unknown, a reasonable approximation is to

treat the thin layer as a bending elastic plate. From [53], the compliance of a

thin elastic plate is given by

Cmem =πa6

195.1(

Eh3

12(1−ν2)

) . (B.5)

In Eq. (B.5), E and ν are the Young’s modulus and Poisson’s ratio of the elastic

material, in this case neoprene. The height of the neoprene layer is given by

h as shown in Fig. B.1. According to [53] the mass of the thin neoprene layer

is well approximated as

Mmem = 1.8830ρneoh

πa2, (B.6)

and the compliance of the air volume with height h′ is dependent on the bulk

modulus of air and can be written

Cair =V

K=

πa2h′

ρairc2air. (B.7)

87

The mass of the air is given by

Mair =ρairh

′

2πa2. (B.8)

Finally, the compliance of the walls is approximated common elasticity prob-

lem of a pressurized inclusion in an infinite medium, as shown in Section 8.4.1

of Ref. [54]. The compliance is dependent on the Young’s modulus, E, and

Poisson’s ratio, ν, of the elastic material that makes up the wall of the unit

cell, which in this case is POM. The compliance is

Cwalls =πa2h′(1 + ν2)

E. (B.9)

Figure B.4: Final simplified circuit model used to determine the inputimpedance.

The circuit can be simplified once more, as shown in Fig. B.4. The

input impedance is then given by

Zin = Zair +

[2

Zmem

+2

Zwalls

]−1. (B.10)

88

B.2 Results and Discussion

The magnitude of the input impedance, Eq. (B.10), is plotted against

frequency. Additionally, the phase of the input impedance can be plotted. The

minimum values of the magnitude of the input impedance corresponds to the

resonance frequency of the system. A zero radian phase angle is also indicative

of resonance in the system. A complete MATLAB code for this circuit model is

found in Section B.4 The inclusion sizes are taken from Table 2.1. All material

properties are taken from Table 2.2.

Figure B.5: Results of circuit model analysis for the 100 Hz unit cell.

Fig. B.5 shows the circuit response of the 100 Hz unit cell. The min-

imum in the magnitude of the input impedance, and the accompanying shift

in phase angle, shows where the resonance occurs. For this unit cell, the res-

89

onance occurs at 125 Hz. For comparison, the experimental resonance of the

unit cell was found to be 131 Hz, corresponding to an error of only -4.6% be-

tween the predicted and measured values. It appears that for this radius the

circuit model is an effective model for the system.

Figure B.6: Velocity information for the 100 Hz unit cell.

Fig. B.6 shows the velocity of the membrane, umem, and the velocity of

the walls, uwalls for the 100 Hz unit cell. The velocity is determined by assuming

a unit pressure amplitude and making use of the impedance relationships so

that

umem =

[Zin−Zair

Zin

]Zmem

, (B.11)

and

uwalls =

[Zin−Zair

Zin

]Zwalls

. (B.12)

90

The peak in the velocity of the wall corresponds with the resonance of the

system at 125 Hz. The same analysis can be repeated for the remaining unit

cell sizes. The size of the cell and its resonance frequency is reported in

Table B.1.

fMinnaert (Hz) a (m) fCircuit (Hz)100 0.0853 125.3200 0.0301 211.3300 0.0164 290.6400 0.0107 385.4500 0.0076 552.5600 0.0058 853.8700 0.0046 1370800 0.0038 2111900 0.0032 31541000 0.0023 7143

Table B.1: Resonance results from the input impedance as determined by thecircuit model above. Inclusion dimensions are taken from Chapter 2

For large inclusions, the model produces reasonable values that are

close to the Minnaert resonance. This is consistent with the experimental

and FEA data presented in Chapter 2. As the inclusion gets smaller in size

(above 500 Hz), the resonances begin to significantly deviate from the Min-

naert approximation. It is believed that this is due to the aspect ratio between

membrane thickness, plastic thickness, and hole radius. This effect is investi-

gated in detail in the following section.

91

B.3 Effect of Membranes and Aspect Ratios

What effect does the presence of the membrane have on the resonance

of the system? Using the simple equivalent circuit model developed in this

appendix, the membrane can effectively be removed from the system by setting

the membrane stiffness and mass in Eq. (B.1) to zero. Eq. (B.1) becomes

Zmem = Zrad + jωMair. (B.13)

For all cases, the effective input impedance has the same characteristics

as that shown in Fig. B.5. The resonance frequency of each unit cell without

the membrane present is shown in Table B.2.

fMinnaert (Hz) a (m) fCircuit (Hz)100 0.0853 125200 0.0301 211300 0.0164 285.8400 0.0107 353.9500 0.0076 419.9600 0.0058 480.6700 0.0046 539.6800 0.0038 593.6900 0.0032 646.81000 0.0023 767.7

Table B.2: Resonance results from the input impedance as determined by thecircuit model without the membrane present.

Table B.2 shows that the resonance frequency of each unit cell above

500 Hz is shifted significantly down from the Minnaert resonance. While these

numbers are in the correct range (compared to the values give in Table B.1),

92

they do not accurately model the system. Perhaps, then, the dominating

effect is not the presence of the membrane but the aspect ratio of either the

membrane thickness to the radius of inclusion, or the POM plastic thickness

to the radius of the inclusion.

To investigate these effects, the data presented in Section B.2 are cal-

culated based on the experimental dimensions of the unit cell, which do not

adhere to the aspect ratio, given first by Eq. (2.20), but now using the con-

vention from Fig. B.1 that the height of the delrin is given by h′,

a

h′≈ 2.4. (B.14)

By imposing this aspect ratio, the radius of the inclusion and the thick-

ness of the Delrin center layer are altered from their experimental values pre-

sented in Chapter 2. The new values for radius and thickness are presented

in Table B.3, along with the resonance frequency determined by the analyt-

ical model both with and without the membrane present. Figures showing

the input impedance are not shown for all cases because they follow the same

characteristics shown in Fig. B.5.

93

fMinnaert (Hz) a (m) h’ (m) fCircuit (Hz) fCircuit,NoMembrane (Hz)100 0.0480 0.0200 94.1 94.1200 0.0240 0.0100 189.2 188.1300 0.0160 0.0067 287.6 282.2400 0.0120 0.0050 393.3 376.3500 0.0096 0.0040 511.0 470.4600 0.0080 0.0033 646.4 564.4700 0.0069 0.0029 805.1 658.4800 0.0060 0.0025 992.5 752.6900 0.0053 0.0022 1213 846.61000 0.0048 0.0020 1470 940.6

Table B.3: Resonance results from the input impedance as determined by thecircuit model of a unit cell with a modified aspect ratio for the thickness ofthe Delrin, both with and without a membrane present.

It is clear from the data in Table B.3 that above 500 Hz a component of

the membrane significantly shifts the resonance frequency. It is possible that

this shift is related to the aspect ratio between the thickness of the neoprene (h,

as shown in Fig. B.1) and the radius of the inclusion. For the cases where the

analytical model closely matches the experimental data, as shown in Fig. B.5,

the aspect ratio is on average

a

h≈ 50. (B.15)

Imposing the aspect ratios given in Eq. (B.14) and Eq. (B.15) gives the

unit cell the dimensions given in Table B.4. The resonance frequencies for this

system are also given in Table B.4.

94

fMinnaert (Hz) a (m) h’ (m) h (m) fCircuit (Hz)100 0.0480 0.0200 0.00096 94.2200 0.0240 0.0100 0.00048 188.4300 0.0160 0.0067 0.00032 282.6400 0.0120 0.0050 0.00024 376.7500 0.0096 0.0040 0.00019 470.9600 0.0080 0.0033 0.00016 565.1700 0.0069 0.0029 0.00013 659.2800 0.0060 0.0025 0.00012 753.5900 0.0053 0.0022 0.00010 847.61000 0.0048 0.0020 0.00009 941.8

Table B.4: Resonance results from the input impedance as determined by thecircuit model of a unit cell with a modified aspect ratio for the thickness ofthe Delrin and the thickness of the membrane.

These values are closer to the Minnaert resonance across the entire

range of unit cell sizes. Please note that for the experimental data (where

a close match is seen in the 100 Hz and 200 Hz cases) the aspect ratios of

Eq. (B.14) and Eq. (B.15) were not enforced.

The analytical model presented above has shown that to maintain a res-

onance frequency near the Minnaert resonance, the aspect ratios of Eq. (B.14)

and Eq. (B.15) should be enforced. Additionally, the model has matched ex-

perimentally determined resonance frequencies for the case of the 100 Hz and

200 Hz unit cell. Ultimately, this investigation of an analytical model for the

system has provided a way to analyze the effect of unit cell dimensions and

properties. This simple design tool may be helpful in future application of

these deeply sub-wavelength panels.

95

B.4 MATLAB Code

The code below shows the full code used to analyze the circuit model

described in Section B.1.

1 clear all; close all; clc;2

3 %attempt at graphing the impedance calculated from the ...circuit model to

4 %find the zeros (resonance frequencies)5

6 %Define Material properties7 a = 0.0853;8

9 %Delrin10 E del = 3.2E9;11 nu del = 0.35;12 rho del = 1.410;13 h del = 0.00635;14

15 %Neoprene16 E neo = 0.01E9;17 nu neo = 0.49;18 rho neo = 1.400;19 h neo = 0.0008;20 c neo = sqrt((E neo/(3*(1-2*nu neo)))/rho neo);21

22 %Water23 rho water = 998;24 c water = 1498;25

26 %Air27 rho air = 1.21;28 c air = 343;29 K air = (rho air*c airˆ2);30

31 %Compliances32 M mem = (1.8830*rho neo*h neo)/(pi*aˆ2);33 C mem = (pi*aˆ6)/(195.1*(E neo*(h neoˆ3)/(12*(1-(nu neoˆ2)))));34 C air = (pi*aˆ2*h del)/K air;35 M air = (rho air*h del)/(2*pi*aˆ2);

96

36 C walls = ((1+nu delˆ2)/E del)*aˆ2*h del*pi;37

38 %Radiation Impedance39 Z rad = @(w) ((rho water.*c water)./(pi.*aˆ2)).*( ...

((w.ˆ2.*a.ˆ2)./(2.*c water.ˆ2)) + ...j.*((8.*w.*a)/(3.*pi.*c water)) );

40

41 %Membrane Impedance (comment out whichever you are not ...plotting at the

42 %time)43 %Z mem = @(w) Z rad(w)+(j.*w.*M air);44 Z mem = @(w) ...

Z rad(w)+(j.*w.*M mem)+(1./(j.*w.*C mem))+(j.*w.*M air);45

46 Z input = @(w) (1./(j.*w.*C air)) + ((2./Z mem(w)) + ...(j.*w.*C walls)).ˆ-1;

47

48 %Define velocities49 u mem = @(w) ( (Z input(w) - ...

(1./(j.*w.*C air)))./(Z input(w)) ) ./ Z mem(w);50 u walls = @(w) ( (Z input(w) - ...

(1./(j.*w.*C air)))./(Z input(w)) ) ./ (2./(j.*w.*C walls));51 u tot = @(w) Z input(w).ˆ-1;52 %%53 %Define Freq range54 f = 0:0.1:250;55 x = 2.*pi.*f;56

57 %Plot as desired

97

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105

Vita

Ashley Jean Hicks is originally from Hot Springs, Arkansas. She at-

tended the University of Central Arkansas for her undergraduate studies. In

2013, she graduated Magna Cum Laude with a bachelor’s degree in Physics

and minors in Mathematics and Interdisciplinary Studies. She participated

in several research experiences throughout her undergraduate career and fol-

lowed her interest in acoustics to The University of Texas at Austin, where

she completed her Master of Science in Engineering in 2015.

Permanent address: [email protected]

This thesis was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.

106

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Copyright by Ashley Jean Hicks 2015