Nonlinear Propagation of High-Frequency Energy from Blast Waves”, in Innovations in Nonlinear Acoustics, ISNA 17, Edited by A

The Pennsylvania State University The Graduate School

NONLINEAR PROPAGATION OF HIGH-FREQUENCY ENERGY

FROM BLAST WAVES AS IT PERTAINS TO BAT HEARING

A Thesis in Acoustics by Alexandra Loubeau

°c 2006 Alexandra Loubeau

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

December 2006 The thesis of Alexandra Loubeau was reviewed and approved∗ by the following:

Victor W. Sparrow Associate Professor of Acoustics Thesis Advisor, Chair of Committee

Thomas B. Gabrielson Professor of Acoustics

John S. Lamancusa Professor of Mechanical Engineering

Anthony A. Atchley Professor of Acoustics Head of the Graduate Program in Acoustics

∗Signatures are on ﬁle in the Graduate School. Abstract

Close exposure to blast noise from military weapons training can adversely affect the hearing of both humans and wildlife. One concern is the effect of high- frequency noise from Army weapons training on the hearing of endangered bats. Blast wave propagation measurements were conducted to investigate nonlinear effects on the development of blast waveforms as they propagate from the source. Measurements were made at ranges of 25, 50, and 100 m from the blast. Par- ticular emphasis was placed on observation of rise time variation with distance. Resolving the fine shock structure of blast waves requires robust transducers with high-frequency capability beyond 100 kHz, hence the limitations of traditional microphones and the effect of microphone orientation were investigated. Measure- ments were made with a wide-bandwidth capacitor microphone for comparison 1 with conventional 3.175-mm ( 8 -in.) microphones with and without baffles. The 3.175-mm microphone oriented at 90◦ to the propagation direction did not have sufficient high-frequency response to capture the actual rise times at a range of 50 m. Microphone baffles eliminate diffraction artifacts on the rise portion of the measured waveform and therefore allow for a more accurate measurement of the blast rise time. The wide-band microphone has an extended high-frequency response and can resolve shorter rise times than conventional microphones. For a source of 0.57 kg (1.25 lb) of C-4 plastic explosive, it was observed that nonlinear effects steepened the waveform, thereby decreasing the shock rise time, from 25 to 50 m. At 100 m, the rise times had increased slightly. For comparison to the measured blast waveforms, several models of nonlinear propagation are applied to the problem of finite-amplitude blast wave propagation. Shock front models, such as the Johnson and Hammerton model, and full-waveform

iii marching algorithms, such as the Anderson model, are investigated and compared to experimental results. The models successfully predict blast wave rise times at medium distances in a homogeneous atmosphere, although rise time predictions are shorter than what was measured in an inhomogeneous atmosphere. Atmospheric turbulence, absent in the models, may be the primary cause of this diﬀerence in rise times at longer distances. Results from the measurements and models indicate that bats located within approximately 200 m of the detonation of 0.57 kg of C-4 will be exposed to audible levels of high-frequency energy, but whether those levels could be damaging to bat hearing cannot be established at this time.

iv Table of Contents

Chapter 1 Introduction 1 1.1 Introduction ...... 1 1.2 Endangered Bats ...... 2 1.3 Explosions Overview ...... 3 1.3.1 Gun Muzzle Blast Versus Explosion of a Charge ...... 4 1.4 Shock Rise Time Deﬁnition ...... 6 1.5 Propagation of Blast Noise ...... 8 1.5.1 Nonlinear Propagation ...... 8 1.5.2 Previous Blast Wave Experiments ...... 10 1.5.3 Outdoor Sound Propagation ...... 11 1.5.3.1 Geometrical Spreading ...... 12 1.5.3.2 Atmospheric Absorption and Dispersion ...... 12 1.5.3.3 Ground Impedance ...... 13 1.5.3.4 Atmospheric Refraction and Sound Speed Variability 13 1.5.3.5 Atmospheric Turbulence ...... 14 1.6 Modeling Techniques for Nonlinear Wave Propagation ...... 14 1.6.1 Hydrodynamic Models ...... 14 1.6.2 The Burgers Equation ...... 15 1.6.3 Shock Front Models ...... 15 1.6.4 Full Waveform Models ...... 21 1.7 Research Objectives ...... 23

Chapter 2 Preliminary Shock Width Model 24 2.1 Plane Wave Rise Time Model ...... 24 2.2 Spherical Wave Shock Width Model ...... 26 2.2.1 Spherical Wave Shock Width Model Applied to Blast Waves 28 2.3 Summary of Preliminary Model Results ...... 29

v Chapter 3 Experimental Setup 31 3.1 Experimental Objective ...... 31 3.2 Chronology of Field Experiments ...... 31 3.3 Microphone Equipment ...... 34 1 3.3.1 3.175-mm ( 8 -in.) Microphones ...... 34 3.3.1.1 Unbaffled Configuration ...... 34 3.3.1.2 Baffled Configuration ...... 34 3.3.1.3 Diffraction Calculations ...... 35 3.3.2 Wide-bandwidth Capacitor Microphone ...... 37 3.4 Experimental Configuration ...... 39 3.5 Meteorological Conditions ...... 41

Chapter 4 Experimental Results 43 4.1 Spectrographic Analysis ...... 45 4.2 Typical Measured Waveforms ...... 45 4.3 Rise Time Comparisons ...... 48 4.4 Sound-Exposure Levels (LE) ...... 51 4.4.1 LE Measurements ...... 52 4.5 Diﬀraction Corrections ...... 56

Chapter 5 Computational Models of Blast Wave Propagation 61 5.1 Johnson and Hammerton Model Predictions ...... 61 5.1.1 Model Equations ...... 61 5.1.2 Implementation ...... 63 5.1.3 JH Model Results ...... 65 5.1.4 Comparison to Polyakova et al. Model Predictions ...... 68 5.2 Anderson Model Predictions ...... 68 5.2.1 Model Equations ...... 68 5.2.2 Implementation ...... 69 5.2.2.1 Absorption and Dispersion ...... 71 5.2.2.2 Step Size ...... 72 5.2.2.3 Usage ...... 73 5.2.3 Anderson Model Results ...... 74 5.2.4 Comparison to Shock Front Model Predictions ...... 78

vi Chapter 6 Comparison of Experimental Results to Model Predictions 80 6.1 Rise Time Comparisons ...... 80 6.2 Atmospheric Turbulence Eﬀects on Rise Times ...... 83 6.3 Comparison to Bass et al. Predictions ...... 83 6.4 Histograms ...... 87 6.5 Comparison to Preliminary Shock Width Model ...... 87

Chapter 7 Potential Eﬀects of Blast Waves on Bat Hearing 91 7.1 Rise Times and High-frequency Energy ...... 91 7.2 Research Findings of Brittan-Powell et al...... 92

Chapter 8 Conclusions 93 8.1 Summary and Conclusions ...... 93 8.2 Suggestions for Future Work ...... 97

Appendix A Photographs from Blast Experiments 99 A.1 Microphone Baﬄe Diagram ...... 99 A.2 Photographs ...... 100 A.3 Blast Video ...... 119

Appendix B Johnson and Hammerton Model MATLAB Code 120 B.1 Introduction ...... 120 B.2 MATLAB Code ...... 121 B.2.1 JH Model Program ...... 121 B.2.2 ODE Subroutine ...... 126

Appendix C Anderson Model MATLAB Code 127 C.1 Introduction ...... 127 C.2 MATLAB Code ...... 128 C.2.1 Main Propagation Code ...... 128 C.2.2 Absorption Subroutine ...... 134 C.2.3 Example Plotting Program ...... 136 C.2.4 Rise Time Calculation Subroutine ...... 139 C.2.5 LE Subroutine ...... 141

vii Appendix D Microphone Calibrations 145

Bibliography 147

viii List of Figures

1.1 Bat audiogram showing maximum hearing sensitivity for the Big Brown Bat in the frequency range 10 kHz to 100 kHz...... 3 1.2 Blast wave signature for an unconfined explosion in free air. . . . . 4 1.3 Profile of a frozen step shock as a function of time for a fixed position and graphical definition of shock rise time...... 7 1.4 Profile of a measured blast wave shock as a function of time, including the 10% and 90% points used to calculate the rise time...... 8 1.5 Additional spreading due to nonlinearity reported theoretically by Naugol’nykh et al. and empirically by Reed...... 10 1.6 Waveform predicted by Polyakova et al. analytical model for a stationary wave in a monorelaxing fluid. Comparison of multivalued asymptotic result and correction with weak shock theory...... 17 1.7 Comparison of numerical and asymptotic solutions of Eq. 1.8 at three different length scales. Parameters used are ∆1 = 0.1, τ1 = 1, −5 ∆2 = 0.2, τ2 = 0.01, and δ = 5 × 10 . Adapted from Johnson and Hammerton...... 20

2.1 Comparison of total absorption αtotal, thermoviscous absorption αtv, nitrogen relaxation absorption αr,N, and oxygen relaxation absorp- ◦ tion αr,O as a function of frequency. Atmosphere at 20 C, 40% relative humidity, and 1 atm ambient pressure...... 26 2.2 Rise time of a plane wave vs. shock overpressure comparison for calculations with αtv and αtotal...... 27 2.3 Rise time of a spherical wave vs. range for diﬀerent source pressure values. Source frequency is 10 kHz, with an atmosphere at 20◦C, 40% relative humidity, and 1 atm ambient pressure...... 29 2.4 Characteristic frequencies as a function of range from a 10 kHz spherical source of 170 dB at 1 m. Calculated using αtv. Atmo- sphere at 20◦C, 40% relative humidity, and 1 atm ambient pressure. 30

ix 3.1 Unbaffled 3.175-mm microphone mounted on a measurement pole at 90 ◦...... 35 3.2 Baffled 3.175-mm microphone mounted on a measurement pole at 0 ◦. 36 3.3 Synthesized baffled waveform comparison. The upper waveform (unbaffled) is doubled and summed with the middle waveform (delayed), yielding the waveform in the lower plot. Synthesized waveforms from both circular and square baffled assumptions are compared to a baffled measurement...... 38 3.4 Wide-band microphone and preamplifier mounted on a measurement pole at 0 ◦...... 39 3.5 Composition C-4 plastic explosive folded in half with blasting cap inserted...... 40 3.6 Experimental setup with microphones mounted on poles at 25, 50, and 100 m...... 40 3.7 Diagram showing relative locations of blast source and microphones for the blast experiments. The 50 m baffled microphone is omitted because its measurements were unusable...... 41 3.8 Onset HOBO Weather Station at the test site. The height of the cross arm for the anemometer was 2 m above ground...... 42

4.1 Typical blast wave measured with an unbaffled microphone at 25 m. 44 4.2 Typical blast wave measured with an unbaffled microphone at 50 m. 44 4.3 Typical blast wave measured with an unbaffled microphone at 100 m. 45 4.4 Spectrogram of blast wave measured with an unbaffled microphone at 25 m...... 46 4.5 Typical waveforms measured at 25 m with unbaffled and baffled microphones. Asterisks denote the 10% and 90% amplitude points used to calculate the rise time trise...... 47 4.6 Typical waveforms measured at 50 m with unbaffled and wide-band microphones. Asterisks denote the 10% and 90% amplitude points used to calculate the rise time trise. The maximum amplitude from the wide-band microphone is set to be the same as the maximum amplitude of the unbaffled response...... 47 4.7 Typical waveforms measured at 100 m with unbaffled and baffled microphones. Asterisks denote the 10% and 90% amplitude points used to calculate the rise time trise...... 48

x 4.8 Comparison of rise times as functions of range computed for all November 2005 blasts measured with different microphone types. (a) Rise times for unbaffled (25, 50, 100 m) and wide-band (50 m) microphones. At 50 m the square markers for the unbaffled measurements are located at the intersection of the solid lines, and it appears that 6 µs is the shortest rise time that could be measured by this microphone at a 90 ◦ orientation. (b) Rise times for baffled microphones (25, 100 m). In both (a) and (b) the solid lines merely connect measurement points for individual blast events and do not represent additional data. For (b), there were no useful measurements obtained at 50 m...... 49 1 4.9 Comparison of 3 -Octave LE for a typical blast wave recorded at 25, 50, and 100 m. The 3.175-mm microphones were unbaffled and oriented at 90◦...... 53 1 4.10 Comparison of 3 -Octave LE for a typical blast wave and three ambient recordings at 25 m on the same day. The 3.175-mm microphone was unbaffled and oriented at 90◦...... 54 1 4.11 Comparison of 3 -Octave LE for a typical blast wave and three ambient recordings at 50 m on the same day. The 3.175-mm microphone was unbaffled and oriented at 90◦...... 55 1 4.12 Comparison of 3 -Octave LE for a typical blast wave and three ambient recordings at 100 m on the same day. The 3.175-mm microphone was unbaffled and oriented at 90◦...... 55 4.13 Magnitude and phase (deg.) of diffraction pressure response at the face of a semi-infinite rigid cylinder as a function of ka...... 57 4.14 Comparison of original measured blast waveforms and diffraction- corrected blast waveform at 50 m. Measurements were made with 3.175-mm microphones at 0◦ and 90◦, and the 0◦ measurement was corrected by the diffraction model...... 58 4.15 Comparison of original measured blast waveforms and diffraction- corrected blast waveform at 25 m. Measurements were made with 3.175-mm microphones at 90◦ unbaffled and at 0◦ baffled, and the baffled measurement was corrected by the diffraction model. . . . . 58 4.16 Comparison of original measured blast waveforms and diffraction- corrected blast waveform at 50 m. Measurements were made with a 3.175-mm microphone at 90◦ and with a wide-band microphone, and the wide-band measurement was corrected by the diffraction model...... 59

xi 1 4.17 Comparison of 3 -Octave LE for original measured blast waveform and diﬀraction-corrected blast waveform at 50 m. The measurement was made with a 3.175-mm microphone at 0◦...... 60

5.1 Example JH numerical model solution for a blast wave at 100 m. As- terisks denote the 10% and 90% amplitude points used to calculate ◦ the rise time trise. Input parameters are ∆p = 828 Pa, TC = 13.6 C, rh = 62.8%, and ps = 1 atm...... 66 5.2 Comparison of JH numerical and asymptotic model solutions for a blast wave at 100 m, presented at 3 different length scales. In- ◦ put parameters are ∆p = 828 Pa, TC = 13.6 C, rh = 62.8%, and ps = 1 atm. (a) Full shock with asymptotic result for the nitrogen relaxation mode. (b) Finer scale with the asymptotic result for the oxygen relaxation mode. (c) Finer scale with the asymptotic result for thermoviscosity...... 67 5.3 Comparison of JH numerical model and Polyakova et al. corrected model solutions for a blast wave at 100 m. Input parameters are ◦ ∆p = 828 Pa, TC = 13.6 C, rh = 62.8%, and ps = 1 atm...... 69 5.4 Simplified flow chart describing steps in the Anderson algorithm. . . 70 5.5 Example Anderson model solution for a blast wave at 50, 100, and 200 m. Input waveform at 25 m was measured with a 90◦ 3.175-mm microphone. Asterisks denote the 10%, 90%, and 100% amplitude points used to calculate the rise time trise. Atmospheric parameters ◦ are: TC = 13.6 C, rh = 62.8%, and ps = 1 atm...... 75 5.6 Anderson model predictions for rise times in 5 m increments (markers shown every 25 m) from 30 m out to 200 m. Input waveforms at 25 m were measured with a 90◦ 3.175-mm microphone. Atmospheric parameters match those of the 12 blasts from November 2005. . . . 76 5.7 Anderson model predictions for 30 and 200 m, with the input waveform at 25 m. Sample points are denoted with markers...... 77 5.8 Comparison of JH and Anderson model rise time (trise) predictions. The Anderson model trise predictions are shown in 5 m increments (markers shown every 25 m), and the JH model predictions are calculated at 25, 50, and 100 m. Atmospheric parameters match those of the 12 blasts from November 2005...... 79

xii 6.1 Comparison of rise times as a function of range from November 2005 experiments, JH model predictions, and Anderson model predictions. Experimental results are presented for blast waves measured with different microphone types: unbaffled microphones (25, 50, 100 m), baffled microphones (25, 100 m) and wide-band microphone (50 m). The solid lines merely connect data points for individual blast events and do not represent additional data...... 81 6.2 Comparison of shock thickness vs. ∆p from Bass et al. and JH model predictions...... 84 6.3 Comparison of shock thickness vs. ∆p from Bass et al., November 2005 experiments, JH model predictions, and Anderson model predictions...... 86 6.4 Percent occurrence of trise for different propagation distances. (a) Experimental data at 25, 50, and 100 m. (b) JH model predictions at 25, 50, and 100 m. (c) Anderson model predictions at 30, 50, and 100 m. Note that the axis limit is 60% for the 100 m data in the last row of histograms...... 88 6.5 Comparison of rise times (trise) predicted with the Naugol’nykh et al. and Anderson models. The Anderson model trise predictions are shown in 5 m increments (markers shown every 25 m)...... 89

A.1 Diagram of front and side of microphone baffle (not to scale). De- signed by Marston. 1 in = 2.54 cm...... 99 A.2 Front view of microphone baffle...... 100 A.3 Assembled microphone baffle...... 100 A.4 Microphone baffle plug and backplate...... 101 A.5 Microphone baffle with microphone...... 101 A.6 August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 102 A.7 August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 102 A.8 August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 103 A.9 August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 103 A.10 August 2004. Microphone setup. Photo courtesy of L. L. Pater. . . 104 A.11 August 2004. C-4 explosive. Photo courtesy of L. L. Pater...... 104 A.12 August 2004. Bat cage mounted with microphones. Photo courtesy of L. L. Pater...... 105

xiii A.13 August 2004. Bat cage mounted with microphones. Photo courtesy of L. L. Pater...... 105 A.14 April 2005. Microphone setup. Photo courtesy of L. L. Pater. . . . 106 A.15 April 2005. Microphone setup. Photo courtesy of L. L. Pater. . . . 106 A.16 April 2005. Bat cage with bat. Photo courtesy of L. L. Pater. . . . 107 A.17 August 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 107 A.18 August 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 108 A.19 August 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 108 A.20 August 2005. C-4 explosive. Photo courtesy of L. L. Pater...... 109 A.21 August 2005. C-4 explosive. Photo courtesy of L. L. Pater...... 109 A.22 August 2005. C-4 explosive. Photo courtesy of L. L. Pater...... 110 A.23 August 2005. C-4 explosive. Photo courtesy of L. L. Pater...... 110 A.24 November 2005. B & K power supply. Photo courtesy of L. L. Pater.111 A.25 November 2005. B & K power supply. Photo courtesy of L. L. Pater.111 A.26 November 2005. B & K power supply. Photo courtesy of L. L. Pater.112 A.27 November 2005. Closeup of microphone baffle. Photo courtesy of L. L. Pater...... 112 A.28 November 2005. Closeup of microphone baffle. Photo courtesy of L. L. Pater...... 113 A.29 November 2005. Closeup of microphone baffle. Photo courtesy of L. L. Pater...... 113 A.30 November 2005. Wide-band microphone and B & K power supply. Photo courtesy of L. L. Pater...... 114 A.31 November 2005. Wide-band microphone. Photo courtesy of L. L. Pater...... 114 A.32 November 2005. Wide-band microphone. Photo courtesy of L. L. Pater...... 115 A.33 November 2005. Wide-band and baffled microphones. Photo courtesy of L. L. Pater...... 115 A.34 November 2005. Wide-band and baffled microphones. Photo courtesy of L. L. Pater...... 116 A.35 November 2005. Wide-band and baffled microphones. Photo courtesy of L. L. Pater...... 116 A.36 November 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater...... 117 A.37 November 2005. Mounted baffled microphone. Photo courtesy of L. L. Pater...... 117

xiv A.38 November 2005. Mounted baﬄed microphone. Photo courtesy of L. L. Pater...... 118 A.39 November 2005. C-4 explosive. Photo courtesy of L. L. Pater. . . . 118 A.40 August 2004 blast video. Video courtesy of L. L. Pater...... 119

xv List of Tables

3.1 Chronology and summary of ﬁeld experiments...... 32

4.1 Comparison of measured rise times at 25, 50, and 100 m for all November 2005 blasts. ¤ is the unbaffled 3.175-mm microphone, ° is the baffled 3.175-mm microphone, and F is the wide-band microphone...... 50 4.2 Comparison of the mean (trise) and standard deviation (σt) of measured rise times at 25, 50, and 100 m for November 2005 blasts. ¤ is the unbaffled 3.175-mm microphone, ° is the baffled 3.175-mm microphone, and F is the wide-band microphone...... 51

6.1 Comparison of the mean (trise) and standard deviation (σt) of measured and predicted rise times at 25, 50, and 100 m for November 2005 blasts. ¤ is the unbaﬄed 3.175-mm microphone, ° is the baﬄed 3.175-mm microphone, F is the wide-band microphone, M is the Johnson and Hammerton prediction, and O is the Anderson model prediction...... 82

xvi List of Symbols

a Radius

c0 Small-signal sound speed

P 0 c∞ Frozen sound speed, = c0 + ci

0 ci Net increase in phase speed due to ith relaxation process as frequency varies from zero to inﬁnity

cp, cv Specific-heat coefficients at constant pressure, volume D Ratio of relaxation effects to nonlinear effects

f Frequency fr,N, fr,O Relaxation frequency of nitrogen, oxygen FFT Fast Fourier Transform

h Molar concentration of water vapor in the atmosphere

Im Imaginary part

k Wavenumber, = ω/c0

LE Sound exposure level m Geometrical spreading parameter

md Dispersion parameter p Acoustic pressure

p0 Reference pressure, = 20 µPa

xvii pinc Incident pressure

pref Reﬂected pressure

pdiﬀ Diﬀracted pressure

ps Atmospheric pressure P Fourier transform of acoustic pressure p

Pr Prandtl number

PSD Power spectral density

r Range from source

r0 Source radius

rh Relative humidity (%) Re Real part

SPL Sound pressure level

t0 Reference time, = 1 s

tarr Nominal time of shock arrival

t+ Positive phase duration of a blast wave

td Total wave duration of a blast wave

tr Relaxation time

trise Rise time of a shock

trise Mean of rise time

◦ TC Temperature in C u Acoustic particle velocity

Z0 Non-dimensional source radius

Z1 Non-dimensional shock formation distance

Z2 Non-dimensional distance for onset of sawtooth region

xviii Z3 Non-dimensional distance for beginning of old age region

αc Complex absorption coeﬃcient

αr,N, αr,O Vibrational relaxation absorption coeﬃcient for nitrogen, oxygen

αtv Thermoviscous absorption coeﬃcient

1 β Coeﬃcient of nonlinearity, = 2 (γ + 1) for gases £ ¤ 1 4 δ Diﬀusivity of sound, = µ + µB + µ (γ − 1) /Pr ρ0 3 ∆ Non-dimensional shock width ∆p Shock overpressure

∆N , ∆O Non-dimensional sound speed increment due to nitrogen relaxation, oxygen relaxation ∆r Range step size

² Acoustic Mach number, = u0/c0 η Step size factor

γ Ratio of speciﬁc heats, = cp/cv Γ Gol’dberg number µ Shear viscosity

µB Bulk viscosity ω Radian frequency, = 2πf Φ Dispersion

ρ0 Ambient density

σt Standard deviation of rise time

τ Retarded time, = t − r/c0

τN , τO Non-dimensional relaxation time for nitrogen relaxation, oxygen relaxation

τ˜N ,τ ˜O Relaxation time for nitrogen relaxation, oxygen relaxation ξ Non-dimensional position, = x − V t

xix Acknowledgments

There are many people whose encouragement and direct assistance have been invaluable to my completion of this thesis. First I’d like to thank Penn State’s Graduate Program in Acoustics and my advisor Vic Sparrow, who has tirelessly encouraged me throughout my research. Many thanks to my doctoral committee members, Tom Gabrielson, John Lamancusa, and Anthony Atchley, for their guid- ance and support. Thanks especially to Dr. Gabrielson for research advice even before becoming a part of my committee, and then for joining the committee so late in the process. This research was supported by the U.S. Army Engineering Research and De- velopment Center Construction Engineering Research Laboratory (ERDC-CERL). I especially appreciate the support and camaraderie of the project leader Larry Pa- ter. I am also grateful to Jeff Mifflin for his jack-of-all-trades expertise in blast measurements and to the Edgewood staff for assistance in coordinating and conducting the tests. In the end stages of thesis writing, I was supported by the Applied Research Laboratory’s Exploratory & Foundation program. I specifically would like to thank Karen Brooks and Richard Stern for making this possible. I have also received research help from outside of Penn State. I am indebted to Wayne Wright for lending me a wide-band microphone and for accommodating my last-minute visit to Texas to learn about the microphone. Thanks to Paul Hammerton for answering my “cold” emails and for clarifying the JH model by graciously providing a Maple code solution. There are many friends, from both within and outside the acoustics world, to

xx whom I owe thanks. It would be impossible to list them all, but their contributions are no less worthy than those specified here. I gratefully acknowledge Tim Marston for constructing the microphone baffles, for providing the diffraction code, and for friendly discussions of the elusive “perfect” microphone. Thanks to Ann Ward for not only being a great friend, but for introducing me to useful tools for addressing journal reviewer comments and for tracking of references. I owe much to my family, for their never-ending support over the years. Lastly, I would like to thank Brian Tuttle. His tireless help with writing, research, and LATEX have been invaluable. Most importantly, his continued encouragement and ability to lift my spirits have given me the strength to complete my graduate work.

xxi Dedication

Pour ma grand-m`ereMarcelle Roumain Loubeau (“Mamie”)

xxii Chapter

1 Introduction

1.1 Introduction

Current environmental regulation [1] requires that the Department of Defense assess the potential impact of noise from military training on endangered wildlife [2, 3]. One particular concern is the effect of noise from Army weapons training on the hearing of endangered bats. High-frequency energy is generated from nonlinear propagation of finite-amplitude shock waves created by explosions and firing of large weapons. Energy at these high frequencies may be harmful to bats because their auditory systems are sensitive to high-frequency information which they use for flight navigation, communication, and hunting [4]. The purpose of this study is to determine the spatial extent of high-frequency energy around a blast event that might be harmful to bat auditory systems. This region near the blast event is where high frequencies generated by nonlinear propagation effects persist despite the effects of dissipation. Determining the spatial extent of high-frequency noise requires an understanding of the short rise times associated with the propagating blast waves. The finite-amplitude pressure and the transient nature of blast waves pose challenges for both measurements and model predictions. In particular, resolving the shock structure requires special instrumentation and modeling techniques. This thesis describes research on defining blast wave characteristics as a function of range. It is one part of a larger project collaboration between the U.S. Army, The Pennsylvania State University, and the University of Maryland, and it is the first research project to combine quantitative definition of the blast stimulus 2

(U.S. Army and The Pennsylvania State University) and measured bat response (University of Maryland).

1.2 Endangered Bats

Army installations are often located in remote areas where wildlife exists. In particular, certain endangered bat species have been found on Army training grounds. The endangered bats that have been identified for study by the Army include the Indiana Bat (Myotis sodalis), Gray Bat (Myotis grisescens), and Lesser Long-nosed Bat (Leptonycteris curasoae yerbabuenae)[2]. Environmental concerns over the safety of these endangered bats limit the Army’s training capabilities by restricting land use. For example, Army training in areas around bat caves is restricted to foot maneuvers [2]. Very little is known about the hearing of these bats because of restrictions on testing of endangered species. Therefore, surrogate species with similar auditory system physiology have been identified and are used in bat auditory systems research. A surrogate species commonly used for endangered bats is the Big Brown Bat (Eptesicus fuscus). Most studies of this surrogate species, however, have not involved quantitative measurements of the sound stimulus or experimentally field- tested animal responses to sound [2]. Previous work has focused on laboratory experiments and passive observations of animal behavior in the field. The publication Hearing by Bats, edited by Popper and Fay, presents a comprehensive compilation of bat hearing research. In the third chapter by Moss and Schnitzler [4], behavioral studies of hearing in bats are discussed. The Big Brown Bat uses short frequency-modulated (FM) vocalizations for echolocation and communication. These FM signals sweep downward from about 55 kHz to 20–25 kHz, and the vocalizations also include higher harmonics. The bats use echolocation to detect the range of targets, localize the targets in space, and determine the dimensions, movement, and surface texture of targets. Although the Big Brown Bat auditory system is most sensitive to frequencies at which the bat vocalizes, it is sensitive to noise over a broad range of frequencies from about 5 kHz to 100 kHz. This is demonstrated in Fig. 1.1, where behavioral bat audiograms from Dalland [5] are compared to more recent Auditory Brainstem Response (ABR) thresholds 3 measured by Brittan-Powell et al. at the University of Maryland [6]. Hearing at the lower frequencies is believed to aid in social communication and in passive listening of prey [4].

Figure 1.1. Bat audiogram showing maximum hearing sensitivity for the Big Brown Bat in the frequency range 10 kHz to 100 kHz. From Brittan-Powell et al. [6].

1.3 Explosions Overview

This section presents a short overview of explosions and blast waves to provide a background for the main part of the thesis research. Many sources are available for further information, including works by Raspet [7], Baker [8], Kinney and Graham [9], Cooper [10], Sachdev [11], Klingenberg and Heimerl [12], and Zel’dovich and Raizer [13]. An explosion is caused by a rapid release of energy. The resulting accumulation of this energy radiates from the source in the form of a blast wave. In air, which is a compressible gas, this ﬁnite-amplitude pressure wave steepens as it propagates and forms a shock. The blast wave signature for an unconﬁned explosion in free 4 air can be approximated by the following equation [14], µ ¶ µ ¶ " µ ¶ # t t t 2 p(t) = ∆p 1 − 1 − 1 − , (1.1) t+ td td where p(t) is the acoustic pressure at time t, ∆p is the peak shock overpressure, t+ is the positive phase duration, td is the total wave duration, and the range of 1 validity is 0 ≤ t ≤ td. As shown in Fig. 1.2, an idealized blast wave consists of an instantaneous rise in acoustic pressure above the ambient pressure, followed by a decay that dips below ambient pressure, and an eventual return to ambient pressure.

1 ←− ∆p

0.8

0.6

0.4

0.2 t+ td − ← ←− Normalized Pressure 0

−0.2

−0.4 0 0.2 0.4 0.6 0.8 1 Normalized Time Figure 1.2. Blast wave signature for an unconﬁned explosion in free air [14].

1.3.1 Gun Muzzle Blast Versus Explosion of a Charge

The purpose of this research is to deﬁne accurately the blast wave characteristics, speciﬁcally the high-frequency content of the wave contained mainly in the shock rise. Military training activities often involve artillery weapons, which may

1Note that Reed used τ for the total wave duration, but this notation is modified here to avoid confusion with the retarded time τ. 5 form blast waves with characteristics that are different from that of the explosion of an unconfined charge. Because of financial and logistical constraints, however, all of the experimental and numerical data reported in this thesis assume that the explosion is unconfined. This section describes some of the differences between gun and charge blasts and the significance thereof. Blast waves are formed by the firing of a gun when gun muzzle gases suddenly displace the air around a gun. In addition, these gun muzzle gases chemically react with the air, resulting in gun muzzle flash and sometimes a secondary blast wave. The shock overpressure of the primary blast wave from a gun decays with range and varies with angle. In most cases, the highest overpressures are observed in the direction of the shot, or a 0◦ angle [15, 16]. The use of muzzle brakes results in a more circular directivity pattern [16]. Recoilless rifles, however, radiate the most sound towards both the front and rear [16]. The secondary blast wave is most often found in large caliber firings [15]. It develops in an intermediate region between deflagration and detonation, when gun muzzle gases and entrained oxygen cause combustion reactions. Some propellants and mechanical attachments to the muzzles of guns can reduce the growth of the secondary blast. The blast wave resulting from the unconfined explosion of a charge, such as Composition C-4, does not vary as a function of angle [16, 9]. No secondary blast waves are produced. In reality, variations in blast overpressure with angle are due to atmospheric conditions, most notably wind direction. In addition, a blast wave’s interaction with the ground surface results in reflection, which can strengthen the shock front through oblique reflection or formation of a Mach stem [9]. These variations in blast characteristics are due to atmospheric conditions and terrain composition [17], and not due to the charge explosion itself. In fact, the same effects would be present in measurements of gun muzzle blasts.

The sound-exposure level (LE) is a parameter used to describe the energy content in a transient wave, such as a blast wave. It is deﬁned as ÃR ! T 2 0 p(t) dt LE = 10 log10 2 , (1.2) p0t0 where the integral is performed over the period T of the pressure signal p(t), the 6

reference pressure p0 = 20 µPa, and the reference time t0 = 1 s.

Schomer et al. [16] compared the C-weighted LE from several different Army weapons at a distance of 250 m. The reference blast was produced with C-4 plastic explosive. For example, the LE measured from an M60 105-mm tank loaded with a propelling charge weight of 5.45 kg was 111.4 dB. This same LE can be achieved with only 0.31 kg of C-4. The long tube of the tank gun constrains the charge, and this results in a lower noise level than an equal mass of unconstrained C-4 [16]. The data in Ref. [15] for the existence of the secondary blast wave is discussed only for near-field distances of a few meters. It is therefore difficult to determine whether this blast characteristic extends to the range of interest in this research. Even though a gun muzzle blast wave is directional, the problem can be simplified by considering only the angle of largest shock overpressures. Assuming spherical propagation, where the propagation is the same at all angles, will result in a worst-case scenario with larger shock overpressures and shorter rise times than would occur off axis from a real gun blast. Post-processing to scale the blast data for different weaponry is also possible.

1.4 Shock Rise Time Deﬁnition

A blast wave’s shock jump is not a perfect discontinuity. Because of dissipation, it takes time for the pressure to increase to the shock overpressure ∆p. The rise time trise associated with a shock is a parameter used to assess the presence of high-frequency energy in the blast wave, and it can be defined in several ways. Assuming a frozen step shock in a thermoviscous medium, the profile of the shock as a function of time for a fixed position [18] is given as ½ ·µ ¶ ¸¾ 1 2 p = ∆p 1 + tanh (t − tarr) , (1.3) 2 trise where the nominal time of shock arrival, tarr, occurs when the pressure has reached 1 2 ∆p. This shock profile was derived by Taylor [19] for weak shocks in a thermoviscous gas. In a step shock the pressure is constant ahead of and behind the shock. In addition, the shock moves at a constant velocity without a change in form because the shock profile is frozen. 7

For this hyperbolic tangent shock proﬁle, shown in Fig. 1.3, the maximum slope occurs at the half-peak point. A tangent to the waveform at its half-peak point is found, and the rise time is then the time interval over which the tangent increases from p = 0 to p = ∆p. This rise time is given by

4ρ δ t = 0 , (1.4) rise β∆p where ρ0 is the ambient density, δ is the diﬀusivity, β is the coeﬃcient of nonlinearity, and ∆p is the shock overpressure.

∆ p

1 ∆ 2 p Pressure

trise 0 0 tarr Time

Figure 1.3. Profile of a frozen step shock as a function of time for a fixed position and graphical definition of shock rise time. Adapted from Blackstock et al. [18].

A similar equation for the rise time of a frozen step shock in a relaxing medium can also be derived [20]. Depending on the relative strength of nonlinearity to molecular relaxation, the profile is not always perfectly skew-symmetric. Therefore the maximum slope does not always occur at the half-peak point. For a real blast wave, the shock profile changes as the wave propagates, and thus a different method for computing rise times is needed. Figure 1.4 presents the shock of a measured blast wave, and it is seen that a shock from an explosion is not skew-symmetric. Following the work of Taylor [19] and established convention in the sonic boom community, the rise time trise associated with a blast wave’s initial 8 shock jump is defined in this thesis as the time it takes for the sound pressure to rise from 10% to 90% of the maximum amplitude. This method is limited by two operations. First, finding the maximum amplitude is not always straightforward because many experimental waveforms have double-peaks or are “noisy” due to turbulence (see Sec. 1.5.3.5). In addition, the sampling frequency may not be high enough for accurate determinations of the 10% and 90% time locations.

∆ p 0.9∆ p Pressure

0.1∆p trise

Time

Figure 1.4. Proﬁle of a measured blast wave shock as a function of time, including the 10% and 90% points used to calculate the rise time.

1.5 Propagation of Blast Noise

1.5.1 Nonlinear Propagation

Sounds such as speech cause small pressure fluctuations in the air at amplitudes that are nearly infinitesimal. Explosions, however, cause large pressure fluctuations, and the blast waves that they generate are referred to as finite-amplitude waves. Blast pressures of finite amplitude result in nonlinear wave propagation. Nonlinear steepening occurs in finite-amplitude waves generated by blasts because the local wave speed increases with local wave amplitude, and thus peaks of the waveform travel faster than troughs [18]. In other words, parts of the waveform 9 with higher values of particle velocity propagate faster than parts of the waveform with lower values of particle velocity. The propagation speed of a particular point on the waveform is given by ¯ ¯ dx¯ ¯ = c0 + βu, (1.5) dt u where u is the particle velocity at the particular point, β is the coefficient of nonlinearity, and c0 is the small-signal sound speed associated with linear wave propagation. As a finite-amplitude sound wave propagates, the cumulative effect is a distortion of the waveform and the formation of shocks. The interaction of nonlinearity and dissipation controls the rise portion of the shocks [21, 22, 23]. In particular, the combined effects of dissipation, which is more pronounced at higher frequencies, and dispersion due to molecular relaxation in air tend to increase shock rise times. Another nonlinear effect of propagation from a finite-amplitude source involves a transfer of energy from the wave’s fundamental frequency to its harmonics. This alone does not modify the net acoustic energy, but, once a shock forms, some of the energy is lost into the shock as heat. It follows that the decrease in amplitude with distance is greater than for linear spherical spreading. The rate of energy loss in the waveform is especially important over longer propagation distances. Including the nonlinear effect of energy loss at shocks, Naugol’nykh et al. [24] provided the amplitude dependence on distance for a finite-amplitude spherical source as ³ √ ´ p∝1/ r ln r . This relation shows that the overall pressure amplitude derived for a nonlinear spherical wave decays faster than 1/r. The equation by Naugol’nykh et al. matches the decay at large distances for an outgoing spherical N wave reported by Blackstock et al. [18]. An N wave has a head shock and a tail shock, and the pressure decreases linearly in between the two; these properties result in an N- shaped waveform [25]. Earlier reports of the finite-amplitude spherical wave decay were given by Landau [26] and DuMond et al. [25]. Reed developed a formula for the amplitude dependence on distance for explosion waves based on his measurements of the noise propagation from various types of explosives [14, 27]. This empirical formula is given by p ∝ 1/r1.2. As seen in Fig. 1.5, this experimentally observed decay with range is similar to the 10 theory reported by Naugol’nykh et al., with both showing a faster decay than for linear spherical spreading. The Naugol’nykh et al. relation is valid only at large distances compared to the source wavelength. Estimating that a typical 20-ms blast waveform has a fundamental wavelength of approximately 7 m, the range shown for the calculations in Fig. 1.5 begins at 10 m.

1.1

/r 0.9

0.8 Linear 1 /r √ 0.7 Naugol’nykh 1/(r ln r ) Reed 1/r1.2 0.6

0.5 Pressure normalized to 1 0.4

0.3 1 2 10 10 Range (m)

Figure 1.5. Additional spreading due to nonlinearity reported theoretically by Nau- gol’nykh et al. [24] and empirically by Reed [14, 27].

1.5.2 Previous Blast Wave Experiments

Several blast wave experiments have been conducted to characterize the blast waveform as a function of distance, and a few significant studies are mentioned here. Reed [14, 27] reported on several blast wave studies conducted at Sandia Laboratories and compared results to empirical models of blast wave attenuation (see Sec. 1.5.1). Reed also advocated the use of transducers with adequate frequency response that could respond to a frequency of 1/trise. Work by Ford et al. [28] emphasized the importance of measuring blasts at 11 medium distances to enhance understanding of blast waveform propagation in the transition area between the close-range region of very high pressure and the linear acoustics zone much farther away. Ford et al. acquired extensive data on blast wave propagation over concrete, grass, and water for small unconfined charges of plastic explosive. A large set of medium- and long-range blast wave propagation measurements in Norway were documented in a series of articles [29, 30, 31, 32, 33], which included the acoustic data, detailed meteorological data for summer and winter conditions, and discussions of ground impedance effects. For most measurements, typical commercial microphones and DAT recorders with a sampling frequency of 12.8 kHz were used. Although rise time measurements are not discussed, it should be noted that this equipment would not be sufficient to measure short rise times accurately. Another experimental study of blast noise that was performed at the U.S. Army’s Fort Drum facility in New York was recently documented in a series of articles [34, 35, 36, 37, 38]. The focus of the tests was the effect of sound-absorbing surfaces both underneath the blast detonation point and along the propagation path. No high-frequency equipment was used in this set of tests either. In other research, wide-bandwidth capacitor microphones, capable of resolving rise times of less than 1 µs, have been used to measure intense acoustic impulses from sparks [39, 20, 40] and ballistic shock waves [41], but no accounts of their use in measuring medium-range blast waves from explosives have been reported previously. There is therefore little or no blast wave data available that is accurate to high frequencies and able to resolve the short rise times accurately.

1.5.3 Outdoor Sound Propagation

There are several physical processes involved in outdoor sound propagation, and their eﬀects on blast noise propagation will be discussed in this section. Some sources of comprehensive information include works by Embleton [42], Sutherland and Daigle [43], Salomons [17], and the American National Standard Estimat- ing Air Blast Characteristics for Single Point Explosions in Air, with a Guide to Evaluation of Atmospheric Propagation and Eﬀects [44]. 12

1.5.3.1 Geometrical Spreading

A spherical wave emitted from a compact source propagates equally in all directions. In a lossless medium, the ﬁnite-amplitude pressure decreases inversely ³ √ ´ with range as 1/ r ln r [24]. This spherical spreading constitutes a decrease in amplitude with distance not found in plane wave propagation. This in turn results in a reduction of nonlinear distortion compared to planar propagation. In the far ﬁeld, propagation of a spherical wave approaches that of a plane wave. For this research, however, the propagation distances considered are close enough to the source for the waves to be spherical.

1.5.3.2 Atmospheric Absorption and Dispersion

Atmospheric absorption and dispersion affect the shape of a waveform as it propagates through the atmosphere. Atmospheric absorption is defined as the summation of individual loss mechanisms, including thermoviscous and molecular vibration relaxation losses. Thermoviscous absorption is associated with thermal conduction and viscous losses, as well as molecular absorption due to rotational relaxation. Thermoviscous absorption is proportional to f 2 and depends on the temperature and ambient pressure of the medium. Another absorption and dispersion mechanism includes molecular absorption due to vibrational relaxation of diatomic nitrogen and oxygen in the air [45, 21, 22, 46, 47, 48]. The nitrogen and oxygen relaxation frequencies depend strongly on the temperature and on the concentration of water vapor in the air; therefore the total absorption is dependent on humidity. Sutherland and Bass [49] have found that molecular relaxation loss by carbon dioxide and ozone is important at high altitudes. However, vibrational relaxation by carbon dioxide is negligible in the lower atmosphere and is not considered here. These loss factors combine to dissipate energy and reduce the wave amplitude. Thermoviscous absorption has been included in many studies of nonlinear sound propagation. However, molecular vibration relaxation is another important factor which is sometimes overlooked. Frequency-dependent dispersion and absorption due to molecular relaxation affect the shock thickness [21, 22, 23]. Dispersion causes different components of the wave to travel at different speeds. This disper- 13 sion smooths out the shock front, and absorption introduces losses; both result in longer shock rise times. Because of the high pressure amplitudes of blast waves, oxygen relaxation in air is the dominant relaxation mechanism and is important for modeling the propagation of explosion noise [20, 50, 51, 52]. Thermoviscous dispersion is considered negligible and is usually ignored [53].

1.5.3.3 Ground Impedance

It is simplest to model propagation from a source in free space. In practice, the presence of a ground surface will affect blast wave propagation [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 54, 55]. The sound waves reflect off the ground, and this can result in interference between direct and reflected waves at the receiver. Determi- nation of these reflections and interference patterns is further complicated by the ground impedance. For example, studies have shown that blast propagation over grass differs from that over water [28]. The water presents a higher impedance to the wave and therefore reflects more energy; in contrast, a grass-covered ground absorbs more energy. This reflection or absorption of energy will affect the radiation of sound and will further modify the pressure dependence on 1/r. Explosive charges are typically suspended 1-2 m above ground for detonation [54], but the composition of the terrain still plays a role. Considering nonlinear propagation, incident and reflected shocks may form a single shock front, or Mach stem, which has the properties of a stronger source wave [44, 28].

1.5.3.4 Atmospheric Refraction and Sound Speed Variability

Atmospheric conditions, such as temperature, wind, and humidity, also play an important role in wave propagation. Sound velocity depends on the square root of the temperature, and temperature varies as a function of altitude. For example, during the day air near the ground is warmer than air at higher altitudes in the troposphere, the lowest section of the atmosphere; this is known as a temperature lapse. The opposite situation occurs at night, and this is known as a temperature inversion. The resulting sound velocity gradients cause spherical waves to distort because wave front normals bend toward slower sound speeds. Similarly, wind profiles cause sound rays to refract. A temperature inversion 14 or downwind propagation causes downward refraction, and a temperature lapse or upwind propagation causes upward refraction. An effective sound speed can be used to account for wind speeds in the surface layer. Furthermore, upper atmospheric winds can increase the effective sound speed in the stratosphere, causing sound to refract back to the ground, typically at a range of 200 km or more from the source [56]. In addition, humidity increases the speed of sound by the factor (1 + 0.16h), where h is the molar concentration of water vapor in the atmosphere, expressed as a fraction (0 ≤ h ≤ 1) [17, 56].

1.5.3.5 Atmospheric Turbulence

Another atmospheric factor that affects sound propagation is atmospheric turbulence. Turbulence is caused by random fluctuations in temperature and wind velocity. For example, temperature fluctuations can be caused by thermal mixing from the sun heating the ground. As a sound wave propagates through turbulent eddies, sound energy is scattered, which results in random fluctuations in acoustic amplitude and phase. For nonlinear wave propagation, turbulence generally decreases the shock overpressure and increases the shock rise time [57, 58].

1.6 Modeling Techniques for Nonlinear Wave Propagation

1.6.1 Hydrodynamic Models

Hydrodynamic models exist for numerical computations of blast wave propagation [44, 11]. These models are intended for use with intense shocks where nonlinearity is high and dissipation is low. Absorption due to molecular relaxation is not included in these models, and thus shock rise times cannot be predicted accurately at medium ranges where nonlinearity weakens. Nonlinear acoustics models for weak nonlinearity (|u| ¿ c0) are therefore needed to couple the hydrodynamic models to linear acoustics models, in which nonlinearity is not signiﬁcant. 15

1.6.2 The Burgers Equation

The evolution of a finite-amplitude wave in one dimension can be described by the extended generalized Burgers equation, µ ¶ ∂p βp ∂p δ ∂2p X ∂2p m = + + R − p, (1.6) ∂r ρ c3 ∂τ 2c3 ∂τ 2 ν ∂τ 2 r 0 0 0 ν where p is acoustic pressure, r is range, τ is the retarded time τ = t−r/c0, β is the coefficient of nonlinearity, ρ0 is the ambient density, c0 is the small-signal sound speed, δ is the diffusivity, Rν is a relaxation operator for each relaxation process ν, and m is a geometrical spreading parameter. This equation accounts for the physical processes of second-order nonlinearity, thermoviscous absorption, absorption and dispersion due to molecular relaxation, 1 and geometrical spreading. For spherical spreading m = 1, m = 2 for cylindrical waves, and m = 0 for plane waves. This simplified partial differential equation assumes one-dimensional propagation in a homogeneous dissipative fluid. Although there is some inconsistency in terminology in the literature [59], in this thesis the generalized form of the Burgers equation indicates inclusion of the effects of geometrical spreading, and the extended form indicates inclusion of the effects of relaxation processes. A detailed history, description, and derivation of the Burgers equation are included in Refs. [60, 18].

1.6.3 Shock Front Models

Several models of shock front propagation exist and are explored in this thesis. These shock front models assume a frozen step-shock profile that propagates without change in form. This means that nonlinearity and the opposing effects of absorption and dispersion are in balance, and the shock moves at c0. Additionally, plane wave propagation is assumed, and this allows for a simplified model. These assumptions are not generally applicable to blast waves, mainly because the wave shape behind the shock does influence future shocks and rise times. Spherical spreading and losses at the shock also decrease the shock amplitude and therefore affect the strength of nonlinear steepening. In other words, a frozen shock profile implies that the rise time has reached a stable value, but this is not valid for the 16 medium-range blast waves of interest. However, the models presented here are used because they are computationally fast. Comparisons to full-waveform models are included in Sec. 5.2.4. Polyakova et al. present an analytical model based on the Burgers equation for a plane stationary wave in a monorelaxing fluid [61, 53]. As a preliminary model, blast wave shock propagation is investigated with molecular vibration of oxygen molecules, which is the dominant relaxation mechanism in air for the pressure amplitudes being considered [51, 50, 52]. Note that thermoviscous absorption and nitrogen vibration relaxation absorption are not included in this model. The ratio of relaxation effects to nonlinear effects can be described by the parameter 2 2 2 2 2 0 D = mdρ0c0/2βp0, where the dispersion parameter md = (c∞ − c0)/c0 = 2c /c0, c∞ is the frozen sound speed at f = ∞, c0 is the equilibrium sound speed at f = 0, and c 0 is the net increase in phase speed due to the relaxation process as frequency varies from zero to infinity [53]. The model equation for the plane step shock is

D−1 (1 + p/p0) τ = tr ln D+1 , (1.7) (1 − p/p0)

3 where τ is the retarded time τ = t − x/c0, tr is the relaxation time, p0 is half the shock overpressure, and D is the relaxation/nonlinear parameter discussed above. To compare to blast wave propagation, parameters from blast wave measurements at 50 m from the source are used in the model. The parameter D = 0.0426, which means that there are strong nonlinear effects. The model predicts a multivalued waveform with such high nonlinearity, and the solution must be corrected by weak shock theory. The multivalued solution is corrected by drawing a straight line down to zero for the remainder of the shock front that falls below the pressure value ∆p = (1 − 2D)p0. The multivalued asymptotic and corrected analytical results are presented in Fig. 1.6. There is an instantaneous pressure jump at the shock arrival, and relaxation causes the subsequent rounded wave profile. Another shock front model was developed by Kang and Pierce [52, 62, 51] for predicting the effect of molecular relaxation processes on sonic boom waveforms. Their model incorporates effects of both nitrogen and oxygen vibration relaxation

2Note that although the dispersion parameter is usually referred to as m, it is modiﬁed here with the subscript d to avoid confusion with the geometrical spreading parameter m. 3For plane waves, r is replaced by x for the position variable. 17

1500

Asymptotic 1000 Corrected

Pressure (Pa) 500

−40 −20 0 20 40 60 80 100 Time (µs) Figure 1.6. Waveform predicted by Polyakova et al. analytical model for a stationary wave in a monorelaxing fluid. Comparison of multivalued asymptotic result and correction with weak shock theory. on the shock structure. An extended form of the Burgers equation is solved numerically using asymptotic expressions for the shock pressure both ahead of and behind the shock. Results show that N2 relaxation delays reaching the maximum shock overpressure, and O2 relaxation causes the shock profile to have a gentle slope. The O2 relaxation controls the lower portion of the shock, and the N2 relaxation controls the upper part. Johnson and Hammerton [63] developed a shock front model similar to that of Kang and Pierce. Johnson and Hammerton present an asymptotic analysis and numerical results for the effect of molecular relaxation processes on the structure of a shock. Specifically, the authors focus on sonic boom propagation through the atmosphere. The analysis is for one-dimensional plane wave propagation, and it includes the effects of nonlinearity, thermoviscosity, and relaxation effects. The waves are assumed to have reached a steady-state structure, so that the wave is a step shock that is propagating without a change in form. Starting with the extended Burgers equation, which includes thermoviscous 18 and relaxation effects, the following non-dimensional equation is derived: µ ¶ µ ¶ µ ¶ d d p dp 1 − τ 1 − τ (p − 1) − δ = N dξ O dξ 2 dξ dp d2p (τ ∆ + τ ∆ ) − τ τ (∆ + ∆ ) , (1.8) N N O O dξ N O N O dξ2 where p is the non-dimensional acoustic pressure,4 ξ = x−V t is a non-dimensional position, τN and τO are non-dimensional relaxation times for nitrogen and oxygen 5 relaxation, δ is the non-dimensional thermoviscous diffusivity, and ∆N and ∆O are the non-dimensional sound speed increments due to nitrogen and oxygen relaxation. The boundary conditions are p → 1 as ξ → −∞ and p → 0 as ξ → ∞. Because an exact solution can only be obtained for one relaxation process, asymptotic analysis of the solution with two relaxation mechanisms is presented. The relaxation processes of interest are due to internal vibration of diatomic nitrogen and oxygen in the air. The model can be generalized to N relaxation processes, with the resulting ordinary differential equation (ODE) being of order N+1. In this case, there are two relaxation processes, and therefore the model is a third-order ODE.

The authors consider the case δ ¿ τ2 ¿ τ1, where δ is thermoviscous diffusivity, and τi are the dimensionless relaxation times for the relaxation processes. This case is valid in air, where the relaxation times for nitrogen and oxygen differ by two orders of magnitude. In the following discussion, nitrogen relaxation is referred to as the first relaxation process, and oxygen relaxation is referred to as the second relaxation process.

Johnson and Hammerton describe three cases which depend on the value of ∆1, the nondimensional increment in sound speed due to the ﬁrst relaxation process.

Case I involves ∆1 > 1/2, which yields a single-valued waveform whose shape depends on the ﬁrst relaxation process; the second relaxation process and viscosity have little eﬀect on the waveform shape. This result is termed fully-dispersed.

Case II involves ∆1 < 1/2 and ∆1 + ∆2 > 1/2. The waveform is considered

4Note that here and throughout the Johnson and Hammerton model discussion, dimensional parameters are denoted with a tilde, such as δ˜. Non-dimensional parameters are identified by the lack of a tilde. This convention is not followed throughout the thesis for simplicity. 5 ˜ 1 Johnson and Hammerton define δ as 2 the quantity usually defined as diffusivity. 19 partly dispersed, and the second relaxation process creates an inner fully-dispersed shock, where the shock slope decreases. Viscosity has little effect on the waveform shape.

In Case III, ∆1 +∆2 < 1/2, and three regions appear. The waveform contains a partly-dispersed shock due to the first relaxation process, an inner partly-dispersed shock due to the second relaxation process, and a narrower inner fully-dispersed shock due to viscosity. Johnson and Hammerton compare the asymptotic analysis for Case III, with the three regions, to numerical solutions of Eq. 1.8. The asymptotic cases match the numerical results in the corresponding regions. Because the length scales of the different shock regions are different, it is difficult to resolve the changes in shock structure in one figure. Therefore the shock profile is presented in subplots with three different length scales, as shown in Fig. 1.7. Parameters used are ∆1 = 0.1, −5 τ1 = 1, ∆2 = 0.2, τ2 = 0.01, and δ = 5 × 10 . In Fig. 1.7(a), the full shock is seen with the asymptotic result for the first relaxation mode. Results match for 1 > p > P0 = 1 − 2∆1. Here P0 = 0.8, and this marks the position at which the partly-dispersed shock due to the first relaxation process meets the inner partly-dispersed shock due to the second relaxation process. In Fig. 1.7(b), the scale is zoomed-in to the same numerical result as in (a), but now with the asymptotic result for the second relaxation mode. Here the results match for

P0 > p > P1 = 1−2(∆1+∆2) = 0.4. The lower limit P1 marks the position at which the partly-dispersed shock due to the second relaxation process matches the inner fully-dispersed shock due to viscosity. In Fig. 1.7(c), the same numerical result is shown at an even ﬁner scale compared to the asymptotic result for viscosity.

Results match for P1 > p > 0. Johnson and Hammerton also apply their model to propagation of sonic booms in the atmosphere. Atmospheric parameters are assumed to be air at 20◦C, standard pressure, and a relative humidity of 50%. The relaxation parameters are −4 −6 ˜ 2 −10 τÑ = 4.73 × 10 s,τ Õ = 4.42 × 10 s, and δ/c = 1.58 × 10 s. These values agree with the assumption δ ¿ τ2 ¿ τ1, where nitrogen is the first relaxation process, and oxygen is the second relaxation process. For low shock overpressures less than 14.76 Pa, Case I is valid. For shock overpressures 14.76 Pa < ∆p < 93.76 Pa, Case II is valid. For large shock overpressures 20

(a) p 1

0.8

0.6

0.4

0.2

on-dimensional pressure 0 N −2 −1.5 −1 −0.5 0 0.5 1 Non-dimensional position ξ (b) p 1

0.8

0.6

0.4

0.2

on-dimensional pressure 0 N −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 Non-dimensional position ξ (c) p 1 Numerical 0.8 Asymptotic

0.6

0.4

0.2

on-dimensional pressure 0 N −4 −3 −2 −1 0 1 2 Non-dimensional position ξ −3 x 10

Figure 1.7. Comparison of numerical and asymptotic solutions of Eq. 1.8 at three different length scales. Parameters used are ∆1 = 0.1, τ1 = 1, ∆2 = 0.2, τ2 = 0.01, and δ = 5 × 10−5. Adapted from Johnson and Hammerton [63]. 21 greater than 93.76 Pa, Case III is valid. The authors compare numerical predictions to data from Pierce and Kang [62], where ∆p = 39.5 Pa. This corresponds to Case II, where the waveform is partly dispersed due to the first relaxation process of nitrogen, and the second relaxation process of oxygen creates an inner fully-dispersed shock. However, the authors seem to have erroneously listed oxygen as the first relaxation process that governs the outer shock and nitrogen as the second relaxation process that governs the inner shock. The numerical results match those of Pierce and Kang. Johnson and Hammerton conclude that the asymptotic model can be used to describe shock structure when numerical calculations are prohibitive. However, molecular relaxation times are strongly dependent on humidity, which varies with altitude. This effect should be considered before comparing sonic boom measurements to numerical solutions. Because the blast propagation considered in this thesis is close to the ground and only the direct path is being considered, humidity variations with altitude are of no concern for the present research. As mentioned above, the portion of a blast wave behind its shock is important. The increase in wave speed due to nonlinearity, βu, is not negligible, and the wave shape is continuing to distort. Hence propagation of a blast wave can not be predicted with a step shock. However, the shock front models can be used as an approximation of the shock structure if the shock overpressure and atmospheric conditions are known.

1.6.4 Full Waveform Models

In order to predict propagation of a full waveform, a different type of numerical model is needed. Researchers have developed computational methods to account for the major physical processes involved in the propagation of finite-amplitude sound. Algorithms generally solve a form of the Burgers equation numerically using a marching scheme and the method of fractional steps [64]. In this method, it is assumed that the physical processes act independently over small distances. The Burgers equation is separated into different equations for each effect by operator splitting, each equation is solved independently over small incremental range steps, and the effects are superposed at each propagation step. This method allows 22

flexibility to select which terms to include to study certain effects. Pestorius [65] developed an algorithm to investigate propagation of finite-amplitude noise in pipes. His algorithm, based on weak shock theory, includes the effects of nonlinearity and tube wall boundary layer attenuation and dispersion. The hybrid time-frequency domain algorithm applies nonlinearity in the time domain, applies a fast Fourier transform (FFT), and then applies attenuation and dispersion in the frequency domain. Then an inverse FFT is taken to return to the time domain to propagate to the next step. Anderson [39] extended this original work by Pestorius to investigate spherical N-wave propagation. The Anderson algorithm does not use weak shock theory, but instead includes thermoviscous absorption in the frequency domain to avoid multivalued waveforms. Nonlinearity is applied in the time domain, and spherical spreading and absorption are applied in the frequency domain. The Anderson algorithm, in turn, has been modified by several researchers to include molecular relaxation effects. Bass and Raspet [21] investigated the rise time of explosion waves and included absorption due to relaxation of nitrogen and oxygen. Orenstein [20] further added dispersion due to molecular relaxation of oxygen for an investigation of spark-generated N waves. Absorption and dispersion vibrational relaxation effects for both nitrogen and oxygen were added to the Anderson algorithm by Bass et al. [22] in their investigation of shock wave propagation in the atmosphere. Results from this research were readdressed by Bass et al. [23] to correct for numerical errors due to undersampling of the waveforms. Recently, a modified Anderson algorithm has been implemented to predict the propagation of finite-amplitude jet noise [66]. The Cleveland algorithm [59, 67] is based on an algorithm by Lee and Hamilton [68] that models the propagation of pulsed finite-amplitude sound beams in a thermoviscous fluid. The Cleveland method solves the Burgers equation purely in the time domain by using a finite-difference approach and solving a tridiagonal matrix system at each propagation step. This time-domain algorithm can be more computationally efficient than a hybrid algorithm for propagation of pulses or step shocks. Cleveland et al. developed this algorithm to model sonic boom propagation and included effects of nonlinearity, geometrical spreading, thermoviscous absorption, molecular relaxation absorption and dispersion, and a stratified atmosphere. 23

1.7 Research Objectives

The research goal is to define the profile of a blast wave as a function of distance, with particular attention to the shock structure. It is the shocks that contain the high pressure amplitudes and high-frequency content that may be harmful to bat hearing. Accurate predictions of blast wave characteristics will enable the U.S. Army to assess the potential risk of bat hearing damage within a range of proximity to firing of weapons or detonations of explosives. The research presented in this thesis includes results of blast wave experiments and computer model predictions. The main goal of the experiments was to capture the shock structure accurately using high-frequency instrumentation. The blast noise measurements include comparisons between a wide-bandwidth capacitor microphone and conventional microphones with and without baffles. Recording equipment with a sampling rate of 1 MHz was used to get good resolution at the shock. Because these experiments are expensive to perform and require a large amount of personnel time, several computational models are explored to predict the blast wave evolution with distance. Existing shock front models and a full waveform model based on the Anderson algorithm are investigated. Blast wave propagation is predicted in the presence of atmospheric absorption and dispersion; ground effects, sound speed gradients, and turbulence are not included. Comparisons of experimental data with outputs from the computational models are presented. It is shown that the models predict a lower bound for the rise times of blast waves and therefore can predict a worse-case scenario for the level of high-frequency energy around a blast event. Combined with bat audiogram and auditory threshold shift data from the University of Maryland [6], the Anderson computational model could serve as a tool for the Army to plan training in bat-sensitive areas. Chapter

2 Preliminary Shock Width Model

A simple preliminary model of blast wave shock rise times was desired to guide the design of blast wave experiments. The model had to be easy to implement and computationally inexpensive. An idea of the distances from an explosion where high-frequency energy persists was needed to guide locations of microphones. In addition, a ballpark range of rise time values was needed to guide the selection of microphone models and a recording unit. The two models presented here account for nonlinear eﬀects and thermoviscous absorption, the two most important factors that aﬀect shock rise times. The second model also includes spherical spreading, which serves as an additional mechanism for decreasing shock amplitude.

2.1 Plane Wave Rise Time Model

As discussed earlier in Sec. 1.4, the rise time of a plane frozen step shock in a thermoviscous medium is given by a simple relationship [18]. The rise time equation is reprinted here as 4ρ δ t = 0 , (2.1) rise β∆p where ρ0 is the ambient density, β is the coeﬃcient of nonlinearity, ∆p is the shock overpressure, and δ is the diﬀusivity given by · ¸ 1 4 µ (γ − 1) δ = µ + µB + , (2.2) ρ0 3 Pr 25

where µ is shear viscosity, µB is bulk viscosity, γ is the ratio of specific heats, and Pr is the Prandtl number. Equation 2.1 reveals that the rise time is inversely proportional to the shock overpressure. Therefore, larger shock overpressures yield smaller rise times. In addition, the rise time of a shock is proportional to the ratio of δ over β. This ratio demonstrates the tradeoff between dissipation and nonlinearity. As dissipation is increased, the rise time increases; however, as nonlinearity increases, the rise time decreases. Thermoviscous absorption, which is a function of diffusivity, is defined as

ω2 αtv = δ 3 , (2.3) 2c0 where ω is the radian frequency and c0 is the small-signal speed of sound. As seen in Eq. 2.3, thermoviscous absorption increases as frequency squared. The rise time can now be rewritten as 8ρ c3 α t = 0 0 tv . (2.4) rise β ω2∆p The total absorption in the atmosphere is a combination of thermoviscous and vibrational relaxation losses [45],

αtotal = αtv + αr,N + αr,O, (2.5)

where αtv is thermoviscous absorption and αr,N and αr,O are vibrational relaxation absorption for nitrogen and oxygen, respectively. A comparison of αtv and αtotal is presented in Fig. 2.1. Included are the separate eﬀects of nitrogen and oxygen vibration relaxation. At lower frequencies, the total absorption αtotal, including thermoviscous eﬀects and molecular relaxation, is larger than the thermoviscous absorption αtv. At higher frequencies, around 1 MHz, αtv approaches αtotal.

If Eq. 2.4 is used to calculate the rise time, αtotal can be substituted for αtv to approximate the rise time of a plane wave in a relaxing ﬂuid. Figure 2.2 compares the shock rise time of a 10 kHz sinusoid plane wave as a function of shock overpressure for calculations with αtv and αtotal. Notice that the rise times calculated with αtotal are larger. However, at large overpressure values, such as 170 dB, the diﬀerence between the two calculations is substantially reduced. 26

2 10 αtotal αtv 0 αr,N 10 αr,O

−2 10

−4 10

−6 10 Absorption (Np/m)

−8 10

−10 10 1 2 3 4 5 6 10 10 10 10 10 10 Frequency (Hz)

Figure 2.1. Comparison of total absorption αtotal, thermoviscous absorption αtv, nitrogen relaxation absorption αr,N, and oxygen relaxation absorption αr,O as a function of frequency. Atmosphere at 20◦C, 40% relative humidity, and 1 atm ambient pressure.

2.2 Spherical Wave Shock Width Model

The preliminary model described in this section approximates the blast source as an initially sinusoidal spherical source. As shown earlier in Sec. 1.3, a blast event produces a wave that is more complicated than a monofrequency sinusoid [28, 14]. The pressure waveform contains an initial shock which decays to a level below ambient pressure to form a negative acoustic pressure phase; the increase back to ambient pressure is nearly asymptotic. However, this research is concerned with short rise times, which are only associated with the shock jump; the remainder of the waveform is inconsequential. Therefore the shock jump can be modeled approximately by a sine wave that undergoes nonlinear steepening, and a sinusoidal spherical source is an appropriate ﬁrst estimate. Naugol’nykh, Soluyan, and Khokhlov derived an expression for the non-dimensional shock width of a spherical wave as a function of distance [69]. This model accounts for spherical spreading, nonlinearity, and thermoviscous absorption. Al- 27

25 αtv αtotal 20

s) 15 µ ( rise

t 10

0 130 140 150 160 170 ∆p (dB re 20 µPa) Figure 2.2. Rise time of a plane wave vs. shock overpressure comparison for calculations with αtv and αtotal. though it does not include the effects of molecular relaxation, ground impedance, or atmospheric conditions, this model is useful for determining the effect of nonlinearity and thermoviscous absorption on finite-amplitude spherical propagation. Naugol’nykh et al. use non-dimensional parameters to define the source radius

Z0, the shock formation distance Z1, the onset of the sawtooth region Z2, and the beginning of the old age region Z3. The non-dimensional shock width ∆, given by the expression r µ ¶ 1 + Z0 |ln( )| r ∆ = r0 , (2.6) πΓ/2 r0 is valid in the sawtooth region between Z2 and Z3. Z0 represents the non-dimensional source radius distance β²kr0, where ² is the acoustic Mach number, k is the wave number, and r0 is the source radius. The shock formation distance Z1

1/Z0 is Z0e , which is equivalent to Eq. 286 in Ref. [18]. The Gol’dberg number Γ is equal to β²k/α and represents the relative strength of nonlinearity compared to dissipation. Naugol’nykh and his colleagues refer to a quantity that is half the Gol’dberg number as the Reynolds number Re. However, this is not the common deﬁnition of the Reynolds number. Thus it has been replaced with Γ/2 in Eq. 2.6 to avoid confusion. At Z3, the shock wave collapses into a sinusoidal wave, and 28 deﬁnition of a shock width is no longer appropriate.

2.2.1 Spherical Wave Shock Width Model Applied to Blast Waves

The spherical wave shock width model is used to predict shock rise times from blast waves. The spherical spreading reduces wave amplitude, thereby reducing nonlinearity faster than in the plane wave model. Here, the shock width ∆ is defined by Naugol’nykh et al. as the interval between the peak and trough of the shock. This is different from the rise time definition introduced in Sec. 1.4. For a sinusoid with a period of 2π, the shock width is π. Thus the non-dimensional shock width ∆ divided by 2πf yields the shock rise time trise. The rise time for a 10 kHz finite-amplitude spherical wave as a function of range and source pressure level is shown in Fig. 2.3. A 10 kHz source is used because this is at the lower range of bat hearing. For low source amplitudes, the rise time increases rapidly with distance. In contrast, the rise time does not increase as quickly for larger source amplitudes such as 170 dB. The rise time increases in the sawtooth region until reaching the old-age region, where the wave reverts to a sinusoidal form. At this point, ∆ = π, or trise = 50 µs, and definition of a rise time has no meaning. The characteristic frequency of a shock wave is found from the reciprocal of the rise time. Figure 2.4 indicates the spatial extent of the shock characteristic frequencies for a 10 kHz source of 170 dB re 20 µPa at 1 m. Because only thermoviscous absorption is included in this calculation, the results probably overpredict the extent of high-frequency content. As the wave propagates away from the source, the shock thickens and the rise time increases; thus the characteristic frequency decreases. High-frequency energy is present at substantial distances from the source because of the nonlinear effects discussed in Sec. 1.5.1. For example, frequencies of 50 kHz and above are present in the range 200 m from the source in all directions. Although it is not shown in the figure, the 10 kHz contour is at approximately 800 m. It is within this area where bats may be affected by sound from the blast event. 29

50 130 dB 140 dB 40 150 dB 160 dB 170 dB

s) 30 µ ( rise

t 20

0 0 50 100 150 200 Range (m) Figure 2.3. Rise time of a spherical wave vs. range for diﬀerent source pressure values. Source frequency is 10 kHz, with an atmosphere at 20◦C, 40% relative humidity, and 1 atm ambient pressure.

2.3 Summary of Preliminary Model Results

The analysis presented in this chapter illustrates how the shock structure of a ﬁnite-amplitude waveform changes as it propagates. Due to the interaction of nonlinear eﬀects, spherical spreading, and atmospheric absorption, the shock fronts thicken as the wave propagates. The larger shock widths result in longer rise times and less high-frequency energy. Rise time predictions from these preliminary models were published by Loubeau and Sparrow [70] in 2004, but the results were in error. Errors in calculating the dimensional rise times have been corrected, and the new results are presented in this chapter. As a result of the corrections, it was found that using αtotal in the Naugol’nykh et al. spherical model yielded rise time values that were not within the range of validity of the model. Therefore, these αtotal calculations were removed from the analysis. Shock rise times on the order of a few µs are predicted for close ranges less than 100 m. These predicted rise times are for the entire shock, trough to peak. For comparison to other models, the 10–90% rise time can be approximated by multiplying 30

200 50 200 kHz 50 50 180

50 160 100 100 100

150 50 140 200 120 150 100 50 200

0 100 100

150 150 100 50 200 Distance (m) 80 200 50 150 60 −100 100 100

50 40 50 20 50 −200 50 −200 −100 0 100 200 Distance (m)

Figure 2.4. Characteristic frequencies as a function of range from a 10 kHz spheri- ◦ cal source of 170 dB at 1 m. Calculated using αtv. Atmosphere at 20 C, 40% relative humidity, and 1 atm ambient pressure. by 0.8. Preliminary calculations show that high-frequency energy above 10 kHz extends to a range of approximately 800 m. This range defines the area around a blast event where bats may be affected. Note that this result was calculated with thermoviscous absorption only. The plane wave rise time model suggests that rise times are longer when molecular relaxation absorption is included. It is therefore assumed that the high-frequency energy will dissipate well before reaching 800 m. Dispersion is another factor that has not been included in the calculations, and this can have a significant effect on shock rise times [23]. In addition, factors such as ground impedance and atmospheric conditions are not included. Furthermore, the shock width is approximated by using a steepening sinusoid in these initial models, but the waveforms created by actual explosions are more complicated [28, 14]. Chapter

3 Experimental Setup

3.1 Experimental Objective

The main goal of the experiments was to capture the shock structure of the blast wave accurately using high-frequency instrumentation. Preliminary model results guided the initial selection of microphones, recording equipment, and microphone locations. Several experiments were conducted over one-and-a-half years, and improvements were made to the test setup each time. It was determined that conventional sensors were inadequate, and special equipment had to be built or acquired. The importance of the experimental results is that they are used as benchmarks for the numerical predictions of blast wave propagation. Additionally, the waveforms recorded at the closest ranges are used as inputs to the Anderson code. The validity of the Anderson predictions inherently depends on the quality of the input waveform.

3.2 Chronology of Field Experiments

There were six different field experiments conducted at the Edgewood site of Aberdeen Proving Grounds in Maryland. A brief summary of each experiment is provided in Table 3.1. All experiments were directed by Dr. Larry L. Pater of the U.S. Army Engineer Research and Development Center Construction Engineer- ing Research Laboratory (ERDC-CERL), and the majority of the equipment was acquired from the CERL laboratory. The first experiment, in June 2004, took place at I Field. An equipment failure 32 of the digital oscilloscope prevented the completion of the full test plan, and only three waveforms were recovered. These waveforms did give an idea of what sound pressure levels to expect, and the levels were in general agreement with results from empirical models by Ford et al. [28] and Raspet and Bobak [71]. Familiarity with the microphone and recording equipment and the test site was obtained. However, it was determined that the site was not ideal for the blast experiments because of the limited space and non-flat terrain.

Date Brief Summary of Blast Experiment June 9, 2004 Equipment failure, three waveforms recovered

1 ◦ August 25-26, 2004 New M Field site, 4 -in. mics oriented at 0 and 90 , bat cages, system noise ﬂoor measured, 2 radials, 10 blasts

1 1 November 8, 2004 Bats exposed for ﬁrst time, 8 - and 4 -in. mics, 25-400 m in main radial, 50 m in 2nd radial, windy, 3 blasts

1 April 5, 2005 Bats exposed, 8 -in. mic baﬄes at 25 and 50 m, 3 blasts August 30-31, 2005 Test complicated with large-scale simultaneous test, large 1 baﬄe, more 8 -in. mics, several sensor failures, 16 blasts and ambient recordings

1 November 7-9, 2005 Bats exposed, wide-band mic, seven 8 -in. mics, baffled vs. unbaffled at 90◦, cable lengths minimized, best consistent data, 14 blasts and ambient recordings Table 3.1. Chronology and summary of field experiments.

The second experiment in August 2004 was the first successful experiment. It was held at M Field, as were all future tests. The microphones were G.R.A.S. 1 ◦ ◦ 40BF 6.35-mm ( 4 -in.) free-field microphones oriented at both 0 and 90 , with the microphone axis parallel and perpendicular to the blast wave direction, respectively. Microphones were also mounted inside two different kinds of bat cages to determine if the cage structure had an effect on the received blast waveform; no appreciable effect was observed. Microphones were mounted along two radials that were oriented 90◦ from each other. The primary radial had measurement sites at 50, 100, 200, and 400 m from the blast, and the second radial had measurement sites at 50 and 100 m. There were recordings of ten blasts. In addition, the system noise floor was measured with ambient recordings with various components 33 removed successively. The results of this test were presented at the 148th Meeting of the Acoustical Society of America [72, 73]. The third experiment occurred in November 2004 and was the first experiment to include bat exposure to blast noise. The tests were delayed for several days due to windy weather conditions. Finally, on November 8, the tests were completed and three blasts were recorded, although the wind was still around 5 m/s gusting to 1 7 m/s. Four G.R.A.S. 40DP 3.175-mm ( 8 -in.) pressure microphones were used for the first time. Because of their increased dynamic range and frequency response (see Sec. 3.3.1), these microphones were able to be used at a closer site 25 m from the blast. A combination of 3.175-mm and 6.35-mm microphones were placed at 25, 50, 100, 200, and 400 m from the blast. The fourth experiment in April 2005 was short, with only three blasts being recorded at ranges of 25 and 50 m. The purpose of this test was to investigate the effect of microphone baffles used with the 3.175-mm microphones (see Sec. 3.3.1.2) and to expose bats to a blast. The results of this test were presented at the 17th International Symposium on Nonlinear Acoustics [74]. In August 2005, the fifth experiment was part of a large-scale experiment collaboration with Wyle Laboratories, Arlington, VA. A large 1 m baffle was introduced in this test, but the results were inconclusive, probably due to unstable microphone placement in the baffle. Despite many equipment failures, a comprehensive set of waveforms was obtained for 16 blasts and ambient noise. The sixth and final experiment was held in November 2005 and was successful in many ways. More 3.175-mm microphones were available for use in this test, and a wide-bandwidth capacitor microphone (see Sec. 3.3.2) was acquired for comparison to the conventional microphones with and without baffles. The waveforms were consistent across all 14 blasts, and weather conditions were favorable. In collaboration with the University of Maryland, bats were exposed to blast noise once again. The data from this final test will be used throughout the rest of the thesis. A portion of these results was reported previously by Loubeau et al. [75]. 34

3.3 Microphone Equipment

The equipment used in the November 2005 experiment is discussed in this sec- 1 tion. Commercial 3.175-mm ( 8 -in.) pressure microphones and a wide-bandwidth capacitor microphone were used to measure the blast wave. In addition, microphone baﬄes (see Sec. 3.3.1.2) were used with 3.175-mm microphones.

1 3.3.1 3.175-mm (8 -in.) Microphones The G.R.A.S. 40DP 3.175-mm pressure microphones have a low sensitivity which allows for their use in finite-amplitude measurements. The dynamic range extends to 178 dB re 20 µPa with 3% distortion [76]. In addition, the nominal frequency response extends to 140 kHz. These microphones were chosen because their size is small compared to a wavelength for frequencies up to 100 kHz. They 1 therefore have a superior high-frequency response compared to 6.35-mm ( 4 -in.) 1 and 12.7-mm ( 2 -in.) microphones. Typical preamplifiers (G.R.A.S. 26AB [77] and Brüel& Kjær 2633 [78]) and power supplies (Brüel& Kjær 2804 [79]) compatible with these condenser microphones were used in the blast experiments.

3.3.1.1 Unbaﬄed Conﬁguration

Unbaffled microphones were oriented with the microphone axis perpendicular to the blast wave direction, as shown in Fig. 3.1; this will be referred to as the 90 ◦ orientation. This orientation gives a more accurate pressure measurement than a 0 ◦ orientation (microphone axis parallel to wave direction) because it minimizes diffraction effects [80]. An example of this diffraction effect is included in Sec. 4.5 in the following chapter. All protection grids were removed to avoid any undesired acoustic resonances [80].

3.3.1.2 Baﬄed Conﬁguration

Microphone baffles were also used to reduce diffraction artifacts of the microphone housing on the rise portion of the blast waves. Built by Timothy M. Marston at Penn State [81], the baffles used in the experiments were 20.32-cm (8-in.) squares, and the microphones fit flush with the front of the baffle. The 35

Figure 3.1. Unbaffled 3.175-mm microphone mounted on a measurement pole at 90 ◦. microphones and baffles were oriented at 0 ◦ from the blast, as shown in Fig. 3.2. The baffles were mounted with taut plastic cable ties between two rods. Details of baffle construction and additional photographs are included in Appendix A. Assuming a plane wave, a rigid circular baffle, and measurement at a point, there is pressure doubling at the microphone due to the superposition of the incident and reflected waves. The diffracted wave is delayed according to the radius, out of phase, and half the amplitude [82]. For the square baffles used in the experiments, the travel time from the midpoint of the baffle edge to the center is associated with the first arrival of the diffracted wave. The diffracted wave would first arrive with a delay of approximately 296 µs. Therefore, even though the blast waveform was distorted due to the finite size and square shape of the baffle, the rise portion was unaffected. This means that microphone baffles should be used with microphones at 0 ◦ incidence to measure rise times accurately. Blast wave measurements made with baffled microphones are compared to measurements with unbaffled microphones in the next chapter.

3.3.1.3 Diﬀraction Calculations

As mentioned above in Sec. 3.3.1.2, the diffracted wave from the edge of the baffle affects the received waveform recorded by the microphone. It is possible to predict this effect of the baffle on the received waveform using diffraction theory. The diffraction calculations give some insight into the shape of the baffled waveforms reported in the next chapter. Starting with the assumption of a circular baffle, the edge-diffracted wave re- 36

Figure 3.2. Baffled 3.175-mm microphone mounted on a measurement pole at 0 ◦. ceived at the center of the baffle is equal to the incident wave inverted and delayed. The circumferential edge of the baffle is considered to be made up of many small circular arcs. The diffraction contribution from each one of these arcs is the total diffracted wave divided by the number of arcs. In other words, the total diffracted wave is the superposition of all the little diffracted waves. However, each diffraction wave has a travel time from the edge to the center. Accounting for spherical spreading, the edge diffracted wave emitted from each edge arc is computed; all edge diffracted waves have the same strength because they all have the same travel distance (a) to the center. The total pressure received at the center is given by µ ¶ a p = pinc + pref + pdiff = p(t) + p(t) − p t − , (3.1) c0 where pinc is the incident wave, pref is the reflected wave, pdiff is the diffracted wave, and c0 is the speed of sound. Now consider the case of a square plate with sides of length 2a. The edges are divided into little edge lengths equal to those used for the circular baffle. Assuming the strength per unit edge length is the same, spherical spreading is applied to each wave. This time, the path lengths differ, with the shortest path being a and the 37

√ longest path from the corner being a 2. The consequences are that the amplitudes of the diffracted waves received at the center are not all the same, and they√ do not 2 2 all arrive at the same time. The time delays for the square baffle case are a +bn , c0 where bn is the position of the nth edge piece. Taking into account spherical spreading and these varied time delays, the contributions from each edge piece at the center of the baffle are summed into a synthesized diffracted wave. Finally, a synthesized baffled wave for the square baffle is created by summing the synthesized diffracted wave with the incident wave doubled in amplitude; the doubling accounts for the superposition of incident and reflected waves. The result, shown in Fig. 3.3, is similar to the measured wave with the baffle. The results from the simpler circular baffle assumption are also compared to the square baffle computation. The shape of the baffled wave synthesized with the square baffle computation more closely matches the actual baffled measurement. This is mostly due to the time delays computed for the different travel paths of the edge-diffracted waves. The varied time delays result in smoother transitions, which are also observed in the baffled measured waveform. However, the synthesized wave for the square baffle case still is very similar to the simpler circular baffle case. The measurements made with a baffle possibly could be corrected by using diffraction theory, and this is discussed in Sec. 4.5. A larger baffle could also be used to further delay the diffracted wave; this was attempted in August 2005, but the results were inconclusive. Another approach to the problem is reported by Menounou et al. [83], who investigated the diffraction effect from a ragged- edge disk on measurement of an underwater spark. They were able to reduce the coherent pressure of the diffracted wave. In one case, the edge wave, with a low amplitude and long duration, disappeared into the background noise.

3.3.2 Wide-bandwidth Capacitor Microphone

The wide-band microphone and preampliﬁer used in this study, designed by Wayne M. Wright [40, 84, 85], has a high resonance frequency and therefore has a better high-frequency response than a conventional 3.175-mm microphone. This type of broadband electrostatic transducer, or “solid-dielectric”, generally has a 38

1 Unbaffled

0.5

Circular Synthesized Diffracted Wave 0 Square Synthesized Diffracted Wave

−0.5 Pressure (kPa) −1

2 Circular Synthesized Square Synthesized 1 Baffled Measurement

0 1 2 3 4 Time (ms) Figure 3.3. Synthesized baffled waveform comparison. The upper waveform (unbaffled) is doubled and summed with the middle waveform (delayed), yielding the waveform in the lower plot. Synthesized waveforms from both circular and square baffled assumptions are compared to a baffled measurement.

flat frequency response to above 500 kHz and can measure a rise time of less than 0.5 µs under ideal laboratory conditions. With the particular microphone used in the experiments, shown in Fig. 3.4, rise times of less than 1 µs were recorded for an acoustic impulse from a spark source in the laboratory. The particular microphone used in the experiments had an active area diameter of 1.27 cm (0.5 in.) embedded in a insulator with a diameter of 3.81 cm (1.5 in.). This insulator acted as a built-in baffle when the microphone was oriented at 0 ◦ from the blast. Therefore a wave diffracted from the edge would arrive with a delay of 55.5 µs. Because this was outside the range of expected shock rise times, 39

Figure 3.4. Wide-band microphone and preamplifier mounted on a measurement pole at 0 ◦. the rise portion of the waveform was not affected, and an additional exterior baffle was not needed.

3.4 Experimental Conﬁguration

The explosive used in the experiments was 0.57 kg (1.25 lb) of unconfined Com- position C-4 plastic explosive. Although the C-4 had been molded into spheres for the early tests, it was determined that this was not necessary. In November 2005, the C-4 was simply folded in half, as shown in Fig. 3.5. A Fidelity Blasting Sys- tem Model 50 connected to a blasting cap inserted in the C-4 was used to detonate the charges at a height of 3 m. Handling of the explosive and charge detonation were undertaken by representatives of the Edgewood Area of Aberdeen Proving Grounds. A picture of the experimental setup is shown in Fig. 3.6, and a diagram de- tailing the locations of the C-4 and microphones is shown in Fig. 3.7. Additional photographs from several of the experiments and a blast video are included in Appendix A. The 3.175-mm microphones were mounted on poles at distances of 25, 50, and 100 m from the blast. The baffled 3.175-mm microphones were colocated at the same distances of 25, 50, and 100 m; however, the 50 m baffled microphone failed. The single wide-band microphone was placed at 50 m. The 40

Figure 3.5. Composition C-4 plastic explosive folded in half with blasting cap inserted. microphones were mounted nominally 5 m above ground to minimize contamina- tion from ground reflections. At this height, the time delay of the ground-reflected wave associated with the path length difference between the direct and reflected waves was sufficiently long to not affect the shock.

Figure 3.6. Experimental setup with microphones mounted on poles at 25, 50, and 100 m.

Blast waveforms were recorded on a Yokogawa DL750 ScopeCorder [86] digital oscilloscope at a sampling rate of 1 MHz. This high sampling frequency was 41

3 Blast Source Height (m) Unbaffled Microphone 2 Baffled Microphone Wide−band Microphone 1

0 0 25 50 100 Range (m) Figure 3.7. Diagram showing relative locations of blast source and microphones for the blast experiments. The 50 m baﬄed microphone is omitted because its measurements were unusable. chosen to give good resolution at the shock. Recording equipment was situated so that twisted-pair cable [87] lengths were minimized where possible. For example, the shortest cable lengths, approximately 22 m, were connected to the 50-m site. Minimizing cable lengths was a precaution taken to avoid radio-frequency noise and any undesired signal distortion due to possible reﬂections in the cable [88].

3.5 Meteorological Conditions

The November 2005 experiment was conducted over three days, during which time the weather conditions varied. Two Onset HOBO Weather Stations [89] (see Fig. 3.8) were used to measure temperature, relative humidity, rainfall, barometric pressure, solar radiation, wind speed, and wind direction. The results presented in this thesis are for the first two days of the experiment. Out of a total of twelve blasts, five blasts were measured on the first day in the afternoon, when the temperature was about 16 ◦C, the relative humidity was 38%, and the wind speed was 2.8 m/s gusting to 3.3 m/s. The remaining seven blasts were measured on the 42 second day in the morning, when the temperature was about 14.2 ◦C, the relative humidity was 62%, and the wind speed was 1.6 m/s gusting to 1.9 m/s. Solar radiation was much higher on the second day, which could have led to the presence of more turbulence, possibly resulting in longer shock rise times [57, 58]. However no correlation was evident between the measured rise times and the measured local meteorological conditions.

Figure 3.8. Onset HOBO Weather Station at the test site. The height of the cross arm for the anemometer was 2 m above ground. Chapter

4 Experimental Results

This chapter presents rise times results for all waveforms in the first two days of the November 2005 test. The measured waveforms for the 12 blasts were consistent for the 25 and 50 m sites. However, the waveforms measured at 100 m varied considerably. The shape of some peaks and the existence of double peaks in some waveforms suggest that turbulence likely affected these waveforms recorded at 100 m [57, 58]. A brief description of the calibration procedure and an estimate of the uncertainty in measured pressure levels is included in Appendix D. A typical waveform from the unbaffled microphone at 25 m is presented in Fig. 4.1. This waveform is similar to the ideal blast waveform presented in Fig. 1.2, but it is complicated with reflections and turbulence. For example, the first ground reflection is expected to occur at 3.6 ms, which closely corresponds to the location of a secondary jump in pressure observed in Fig. 4.1. The shock overpressure is 3670 Pa, or 162 dB re 20 µPa. Note that this dB value is calculated by dividing the √ peak shock overpressure by 2 because the reference pressure of 20 µPa is a RMS pressure. Typical waveforms measured by unbaffled microphones at 50 and 100 m are presented in Fig. 4.2 and Fig. 4.3, respectively. 44

1 Pressure (kPa) 0

−1

−10 0 10 20 30 40 Time (ms) Figure 4.1. Typical blast wave measured with an unbaﬄed microphone at 25 m.

1.5

0.5

0 Pressure (kPa)

−0.5

−1

−10 0 10 20 30 40 Time (ms) Figure 4.2. Typical blast wave measured with an unbaﬄed microphone at 50 m. 45

0.5

Pressure (kPa) 0

−0.5 −10 0 10 20 30 40 Time (ms) Figure 4.3. Typical blast wave measured with an unbaﬄed microphone at 100 m.

4.1 Spectrographic Analysis

Spectrographic analysis of the waveforms conﬁrms that the high-frequency energy is concentrated near the shocks. Spectrograms are found by sliding a narrow time window across the waveform and taking an FFT at each point. The spectrogram in Fig. 4.4, showing sound pressure level (SPL) in dB re 20 µPa as a function of frequency and time, is representative of a blast wave measured with an unbaﬄed microphone at 25 m. This particular spectrogram was calculated using a Chebyshev window with 50% overlap and a 15 Hz frequency resolution at a 1 MHz sampling frequency. Notice that the peaks in high-frequency content occur at the shocks, with substantial energy at the initial shock. It is these shocks, therefore, that contain the high frequencies that may be harmful to bat hearing.

4.2 Typical Measured Waveforms

Typical blast waveforms measured at 25, 50, and 100 m are presented in Figs. 4.5, 4.6, and 4.7. The asterisks denote the 10% and 90% amplitude points used to 46

100 110dB

100 80

90 60 80 40 70 Frequency (kHz)

20 60

0 50 0 5 10 15 20 4 2 0 p (kPa) −2 0 5 10 15 20 Time (ms) Figure 4.4. Spectrogram of blast wave measured with an unbaﬄed microphone at 25 m.

calculate the rise time trise. The waveforms were measured at 1 µs intervals, and an additional 0.1 µs precision in the rise portion was deduced from interpolation. For the particular blast shown in Fig. 4.5, the first 50 µs of the baffled and unbaffled waveforms are compared. The baffled rise time of 19.6 µs is nearly the same as the unbaffled rise time of 21.1 µs. Because of the pressure doubling at the baffled microphone, explained above in Sec. 3.3.1.2, the baffled response was halved. However, the peak amplitude does not match that of the unbaffled microphone because the pressure doubling assumption is approximate. The experiments involved a square baffle of finite area. As a result, pressure doubling occurs only for frequencies with wavelengths that are small compared to the baffle size. Because nonlinear steepening leads to decreased rise times, a decrease in rise times at 50 m was expected. The rise times of the waveforms at 50 m were indeed shorter. The experiment was designed to compare all three microphone types at 50 m. Unfortunately, as mentioned earlier, the baffled microphone at 50 m failed, and the only useful comparison at this location is between the unbaffled microphone and the wide-band microphone. Figure 4.6 shows that the wide-band response is 47

4000

3000

2000 Pressure (Pa)

1000

Unbaffled trise = 21.1µs Baffled trise = 19.6µs 0 −10 0 10 20 30 40 50 Time (µs) Figure 4.5. Typical waveforms measured at 25 m with unbaffled and baffled microphones. Asterisks denote the 10% and 90% amplitude points used to calculate the rise time trise.

1600

1200

800 Pressure (Pa)

400

Unbaﬄed trise = 6.1µs 0 Wide-band trise = 4.5µs

−10 0 10 20 30 40 50 Time (µs) Figure 4.6. Typical waveforms measured at 50 m with unbaffled and wide-band microphones. Asterisks denote the 10% and 90% amplitude points used to calculate the rise time trise. The maximum amplitude from the wide-band microphone is set to be the same as the maximum amplitude of the unbaffled response. 48 clearly steeper than the unbaffled response. The rise time from the wide-band microphone was 4.5 µs, shorter than the unbaffled rise time of 6.1 µs. In Fig. 4.7, the 100 m unbaffled and baffled waveforms are similar, and the rise times were 12.4 µs and 11.6 µs. These rise times are longer than at 50 m. It is interesting to note that the difference in peak values observed at 25 m due to the pressure doubling approximation is not present in the 100 m measurement. The cause of this difference at 100 m is unknown.

800

600

400 Pressure (Pa) 200

Unbaﬄed = 12.4 s 0 trise µ Baﬄed trise = 11.6µs

−10 0 10 20 30 40 50 Time (µs) Figure 4.7. Typical waveforms measured at 100 m with unbaﬄed and baﬄed microphones. Asterisks denote the 10% and 90% amplitude points used to calculate the rise time trise.

4.3 Rise Time Comparisons

So far, results for one representative waveform have been presented. Figure 4.8 presents the rise times as a function of range for all measurements. The same results are listed in Table 4.1, and the mean (trise) and standard deviation (σt) of the measured rise times are presented in Table 4.2. The rise times of the waveforms measured with unbaﬄed 3.175-mm microphones at 25 m varied from 17.5 µs to 49

21.3 µs. The baﬄed responses at 25 m had nominally the same rise times, ranging from 17.9 µs to 21.9 µs. The average shock overpressure was 3330 Pa.

(a) (b) 25 25

20 20 s)

µ 15 15

10 10 Rise time (

5 5 Unbaffled Wide−band Baffled 0 0 25 50 100 25 100 Range (m) Range (m) Figure 4.8. Comparison of rise times as functions of range computed for all November 2005 blasts measured with different microphone types. (a) Rise times for unbaffled (25, 50, 100 m) and wide-band (50 m) microphones. At 50 m the square markers for the unbaffled measurements are located at the intersection of the solid lines, and it appears that 6 µs is the shortest rise time that could be measured by this microphone at a 90 ◦ orientation. (b) Rise times for baffled microphones (25, 100 m). In both (a) and (b) the solid lines merely connect measurement points for individual blast events and do not represent additional data. For (b), there were no useful measurements obtained at 50 m.

The rise times of the waveforms measured with unbaﬄed microphones at 50 m were shorter, varying from 5.9 µs to 6.2 µs. At 50 m, the average shock overpressure was 1480 Pa. The small variations in measured rise times at 50 m seem to indicate that a 6 µs rise time was the limit of the 3.175-mm microphone system in the current conﬁguration. Therefore one can not be sure if the true rise time of the wave was recorded. It is because of this microphone limitation that a wide-band microphone was also used. At 50 m, the wide-band microphone measured rise times of 3.2 µs to 7.5 µs. As shown in Fig. 4.8 and Table 4.1, rise times from the wide-band microphone were less than 6 µs for all blasts except one. These rise times are within the capabilities of the microphone, so it is likely that these were the true rise times of 50

Blast Rise Time (µs) 25 m 50 m 100 m i F i

1 20.8 21.1 5.9 4.4 16.3 18.0 2 20.7 20.8 6.1 4.8 7.3 16.0 3 17.7 17.9 5.9 4.5 9.2 14.5 4 20.1 20.8 6.0 4.6 7.3 21.0 5 20.3 18.1 6.1 4.5 7.1 15.2 6 21.1 19.6 6.1 4.5 12.4 11.6 7 19.9 20.9 6.0 7.5 11.2 8.6 8 17.5 20.3 6.1 4.0 11.1 16.1 9 21.3 21.9 6.1 5.7 12.5 14.3 10 20.8 21.9 6.2 4.5 12.0 9.0 11 21.1 20.1 6.2 3.2 14.9 15.0 12 20.7 19.7 6.0 4.2 12.4 8.6

Legend ¤ Unbaﬄed ° Baﬄed F Wide-band

Table 4.1. Comparison of measured rise times at 25, 50, and 100 m for all Novem- ber 2005 blasts. ¤ is the unbaffled 3.175-mm microphone, ° is the baffled 3.175-mm microphone, and F is the wide-band microphone. the blast waves. Although the unbaffled microphone rise times at 50 m appear to be limited to 6 µs, the rise times from the wide-band microphone exhibit a larger spread, indicating that these rise times were not limited by the response of the microphone. At 100 m, the rise times of the recorded waveforms were longer, with the unbaffled rise time measurements ranging from 7.1 µs to 16.3 µs and the baffled rise time measurements ranging from 8.6 µs to 21 µs. The average shock overpressure 51

Statistic 25 m 50 m 100 m i F i

trise (µs) 20.2 20.3 6.1 4.7 11.1 14.0

σt (µs) 1.3 1.3 0.1 1.0 3.0 3.9

Legend ¤ Unbaﬄed ° Baﬄed F Wide-band

Table 4.2. Comparison of the mean (trise) and standard deviation (σt) of measured rise times at 25, 50, and 100 m for November 2005 blasts. ¤ is the unbaffled 3.175-mm microphone, ° is the baffled 3.175-mm microphone, and F is the wide-band microphone. was 680 Pa. The differences between the unbaffled and baffled rise times at 100 m are probably due to some combination of the microphone housing diffraction effect and localized atmospheric turbulence. Overall, the experimental results show that between 50 and 100 m nonlinearity began to lose strength, and losses began to dominate. However, at 100 m the rise time was still shorter than at 25 m.

4.4 Sound-Exposure Levels (LE)

The sound-exposure level (LE) is a parameter that can be used to describe the energy content in a transient wave, such as a blast wave. Because the LE is proportional to signal energy, the length of the time record containing the blast wave should not affect the result. For example, the LE of a 50-ms signal containing a 25-ms blast wave would be the same as that of a 1-s signal containing the same short-duration blast wave. Therefore the energy of events with different time durations can be compared. The LE is defined as [56, 54] ÃR ! T 2 0 p(t) dt LE = 10 log10 2 , (4.1) p0t0 where the integral is performed over the period T of the pressure signal p(t), the reference pressure p0 = 20 µPa, and the reference time t0 = 1 s. 52

In MATLAB, the LE can be computed from the power spectral density (PSD) [90, 91], 2 ∆t2 |FF T (1 : N/2 + 1)|2 PSD = , (4.2) T p0 where the Fast Fourier Transform (FF T ) is given by [91]

XN −j2π (n−1)(m−1) FF T (pn) = pne N , (4.3) n=1 pn is the sampled time-domain waveform, n is the index for the time samples, and m is the index for the frequency-domain points. In Eq. 4.2, ∆t is the time increment of pn, ∆f is the sample increment in the frequency domain, and thus T 1 the number of samples is N = ∆t = ∆t ∆f . The single-sided PSD is computed by including the first N/2 + 1 unique values from the FF T that correspond to frequencies 0 to (N/2)∆f and then multiplying by 2. In fact, this explanation is an oversimplification, because the 0 Hz value is unique in a full FF T and thus is not doubled for the single-sided FF T . Also, if the number of points in the FF T is even, the N/2 + 1 value at (N/2)∆f Hz is also unique and is not doubled for the single-sided FF T either. 1 The next step is to calculate the 3 -octave-band frequencies according to the ANSI standards in Refs. [92, 93]. The PSD is linearly interpolated to calculate PSD values at the band edges, and then the PSD is integrated across each 1 -octave band ³ ´ 3 2 to find the mean-square pressure p in each band. Finally, the LE is computed as [91] Ã ! p2 T LE = 10 log10 2 , (4.4) p0t0

2 with units of dB re (20 µPa) s. Although the LE appears to have a dependence on T in Eq. 4.4, it is in fact independent of T because the PSD contains a factor of 1/T .

4.4.1 LE Measurements

1 The 3 -octave LE computed for the November 2005 data are presented here. 1 In Fig. 4.9 the 3 -octave LE is compared for a typical blast wave recorded at 25, 53

50, and 100 m. These results are for unbaﬄed 3.175-mm microphones oriented at 90◦. It is shown that the highest energy in the blast wave is contained in Band 17, 1 2 44.7 − 56.2 Hz. Peak 3 -octave LE values are 131, 128, and 120 dB re (20 µPa) s for distances of 25, 50, and 100 m, respectively. Overall, the energy in the blast wave decreases with increasing distance. At the frequencies above 50 kHz, however, the

LE at 50 m is higher than the LE at 25 m. At the low-frequency end, it appears that the measurement at 50 m has a rolloﬀ in response below approximately 50 Hz. This trend is present in all blast wave measurements made with this particular microphone, yet no reasonable explanation exists.

140 25 m 50 m s) 130 2 100 m Pa)

µ 120 (20

re 110 (dB

E 100 L

90 -Octave

1 3 80

70 1 2 3 4 5 10 10 10 10 10 Frequency (Hz)

1 Figure 4.9. Comparison of 3 -Octave LE for a typical blast wave recorded at 25, 50, and 100 m. The 3.175-mm microphones were unbaﬄed and oriented at 90◦.

It is also instructive to have measurements of the system noise floor. In August 2004, the system noise floor was measured with various components removed successively, starting with everything connected and finishing with only a terminator attached to the digital recorder. At frequencies below 200 Hz, the microphone and preamplifier noise dominated, while the digital recorder noise dominated at high frequencies. It was found that the system noise floor was well below the blast levels, up to approximately 80 kHz. These measurements were made with 6.35-mm 54

1 ( 4 -in.) microphones at 50 m. In November 2005, a less vigorous system noise floor measurement was made with all components connected and mounted in place. This test did not allow for shielding of the microphone from ambient sound. At least the system noise could be no higher than what was measured. The ambient noise LE recorded by an unbaffled 3.175-mm microphone at 90◦ is compared to a typical blast wave measurement made with the same microphone at 25 m on the same day in Fig. 4.10. Corresponding system noise floor comparisons for 50 and 100 m are included in

140 25 m Blast 130

s) 25 m Ambient 2 120 Pa) µ 110 (20

re 100

(dB 90 E L 80

70 -Octave 1 3 60

50 1 2 3 4 5 10 10 10 10 10 Frequency (Hz)

1 Figure 4.10. Comparison of 3 -Octave LE for a typical blast wave and three ambient recordings at 25 m on the same day. The 3.175-mm microphone was unbaﬄed and oriented at 90◦.

Figs. 4.11 and 4.12, respectively. The blast wave energy is above the system noise ﬂoor at all frequencies up to 100 kHz, although the signal-to-noise ratio rapidly decreases above 60-80 kHz. 55

130 50 m Blast

s) 120 50 m Ambient 2

Pa) 110 µ

(20 100 re 90 (dB E

L 80

70 -Octave 1 3 60

50 1 2 3 4 5 10 10 10 10 10 Frequency (Hz)

1 Figure 4.11. Comparison of 3 -Octave LE for a typical blast wave and three ambient recordings at 50 m on the same day. The 3.175-mm microphone was unbaﬄed and oriented at 90◦.

130 100 m Blast

s) 120 100 m Ambient 2

Pa) 110 µ

(20 100 re 90 (dB E

L 80

70 -Octave 1 3 60

50 1 2 3 4 5 10 10 10 10 10 Frequency (Hz)

1 Figure 4.12. Comparison of 3 -Octave LE for a typical blast wave and three ambient recordings at 100 m on the same day. The 3.175-mm microphone was unbaﬄed and oriented at 90◦. 56

4.5 Diﬀraction Corrections

In Sec. 3.3.1.3, the effect of diffraction from the edges of microphone baffles was discussed. Predicting a diffracted wave from the unbaffled measurement and summing the two resulted in a waveform similar to the baffled measurement (see Fig. 3.3). It is also possible to correct for diffraction to obtain a waveform that is free of diffraction artifacts. Gifford [94] numerically implemented a high-frequency diffraction model by Jones [95] for a semi-infinite rod of circular cross section. For a given microphone radius, the diffraction at the face of the microphone is calculated, and the deflection of the diaphragm is modeled by a Bessel function to account for maximum sensitivity at the diaphragm center. This algorithm was enhanced by Marston [81] and used to reverse the effects of diffraction on a received waveform. Although free-field microphones exist in certain sizes to correct for diffraction artifacts in magnitude response, these microphones do not correct the phase response. In addition, the free field correction is only valid for 0◦ incidence. Marston demonstrated the algorithm’s effectiveness on pulses recorded from balloon popping at one end of a tube. Sensors used were 3.175-mm and 12.7-mm microphones, as well as the same wide-band microphone used in the current blast wave research. The diffraction response used in the model is presented in Fig. 4.13. The magnitude and phase of the diffraction response at the face of a microphone of radius a are plotted as a function of ka. For small ka, the microphone is smaller than a wavelength, and the diffraction magnitude is 1. For large ka, the microphone is large compared to the wavelength, and the diffraction magnitude approaches 2, indicating pressure doubling. The transition region is where the model is especially useful. It should be noted that this model, like the free-field microphones, assumes a 0◦-incident wave, and additionally assumes a plane wave. To apply the diffraction correction, a measured pulse waveform is transformed into the frequency domain, the diffraction response is calculated, the pulse spectrum is divided by the diffraction response, and finally the spectrum is transformed back into the time domain. Marston’s diffraction code was used to correct for diffraction effects in the measured blast waveforms. Examples of diffraction model corrections are included in Figs. 4.14, 4.15, and 4.16. In Fig. 4.14, a blast wave 57

2.5 2 1.5

Magnitude 1 0.5 −1 0 1 10 10 10 30

10 Phase (deg.) 0 −1 0 1 10 10 10 ka Figure 4.13. Magnitude and phase (deg.) of diffraction pressure response at the face of a semi-infinite rigid cylinder as a function of ka. Adapted from Marston [81]. measurement at 50 m with a 3.175-mm microphone at 90◦ is compared to a 0◦ measurement that has been corrected by the diffraction model. The original 0◦ measurement exhibits an overshoot in amplitude at the shock, and this is corrected by the diffraction model to yield a waveform similar to that measured with the 90◦ orientation. The slope of the shock is also affected by the correction, and the resulting rise time is longer. Therefore the original 0◦ measurement has an erroneously short rise time and shows more high-frequency energy than is actually present in the blast wave. This is important when considering the impact of blast noise on bat hearing. The diffraction model does not work well for the baffled or wide-band microphone because the model assumes that the diaphragm covers the face of the instrument. In the baffled case, the microphone encompasses only the center 3.175 mm of the 20.32-cm diameter face. For the wide-band microphone, the active area encompasses only the center 1.27 cm of the 3.81-cm diameter face. Marston therefore modified the diffraction correction code to account for a fractional active area. Figure 4.15 compares a diffraction-corrected baffled measurement to the 90◦ measurement at 25 m. In this case, the model still does not work as well because 58

2000

1500

1000 Pressure (Pa)

500

Measured 0◦ Corrected 0◦ 0 Measured 90◦

0 20 40 60 80 100 Time (µs) Figure 4.14. Comparison of original measured blast waveforms and diﬀraction- corrected blast waveform at 50 m. Measurements were made with 3.175-mm microphones at 0◦ and 90◦, and the 0◦ measurement was corrected by the diﬀraction model.

6000 Measured Baffled 5000 Corrected Baffled Measured Unbaffled

4000

3000

2000 Pressure (Pa)

1000

−1000 0 0.2 0.4 0.6 0.8 1 Time (ms) Figure 4.15. Comparison of original measured blast waveforms and diffraction- corrected blast waveform at 25 m. Measurements were made with 3.175-mm microphones at 90◦ unbaffled and at 0◦ baffled, and the baffled measurement was corrected by the diffraction model. 59 the baffle is square, and the model assumes a circular cross section. Nevertheless, a shape closer to the unbaffled measurement is recovered. Finally, the diffraction-corrected wide-band measurement is compared to the 90◦ measurement at 50 m in Fig. 4.16. Although the microphone face is circular, the circumference is chamfered, and the clamping of diaphragm edges is different from the 3.175-mm microphone. Hence the model’s assumptions are not entirely met.

1800 1600 1400 1200 1000 800 600 Pressure (Pa) 400 200 Measured Wide-band 0 Corrected Wide-band Measured Unbaffled −200 −50 0 50 100 150 200 250 Time (µs) Figure 4.16. Comparison of original measured blast waveforms and diffraction- corrected blast waveform at 50 m. Measurements were made with a 3.175-mm microphone at 90◦ and with a wide-band microphone, and the wide-band measurement was corrected by the diffraction model.

The diffraction model effectively reduces high-frequency energy in the waveform, as demonstrated by the diffraction response in Fig. 4.13. The effect on the spectrum of the 0◦ diffraction-corrected waveform is presented in Fig. 4.17. It is shown that the diffraction correction decreases the high-frequency energy above approximately 10 kHz. The diffraction correction model is a useful tool for eliminating diffraction artifacts on measured waveforms. It should be noted, however, that several other 60

130 Measured 0◦ Corrected 0◦ s)

2 120 Pa) µ 110 (20 re 100 (dB E L 90 -Octave

1 3 80

70 1 2 3 4 5 10 10 10 10 10 Frequency (Hz)

1 Figure 4.17. Comparison of 3 -Octave LE for original measured blast waveform and diffraction-corrected blast waveform at 50 m. The measurement was made with a 3.175- mm microphone at 0◦. factors, such as diaphragm resonance and preamplifier rolloff, also affect the measurement. Therefore, applying the diffraction correction to measurements made with different microphones will not necessarily yield the same waveform. Further- more, the diffraction model must be applied in post-processing and thus is not available in real time. Chapter

5 Computational Models of Blast Wave Propagation

5.1 Johnson and Hammerton Model Predictions

The Johnson and Hammerton (JH) algorithm was implemented in MATLAB, and the code (hammertonfn_numerical.m) is provided in Appendix B. Personal communication with P. W. Hammerton clariﬁed several aspects of the model, and Hammerton graciously provided a version of the code in Maple. It should be noted that the model equation manipulations presented here and used in the code do not all follow Hammerton’s development. The following sections describe the algorithm and include example results relevant to blast waves.

5.1.1 Model Equations

The Johnson and Hammerton (JH) model is based on the extended Burgers equation, which includes thermoviscous and relaxation eﬀects. The following non- dimensional version of the Burgers equation is derived: µ ¶ µ ¶ µ ¶ d d p dp 1 − τ 1 − τ (p − 1) − δ = N dξ O dξ 2 dξ dp d2p (τ ∆ + τ ∆ ) − τ τ (∆ + ∆ ) , (5.1) N N O O dξ N O N O dξ2 62 where p is the non-dimensional acoustic pressure,1 ξ = x−V t is a non-dimensional position, τN and τO are non-dimensional relaxation times for nitrogen and oxygen 2 relaxation, δ is the non-dimensional thermoviscous diﬀusivity, and ∆N and ∆O are the non-dimensional sound speed increments due to nitrogen and oxygen relaxation. The boundary conditions are p(ξ) → 1 as ξ → −∞ and p(ξ) → 0 as ξ → +∞. Equation 5.1 can be expanded to give · µ ¶¸ d3p 1 d2p −δτ τ + δ (τ + τ ) + τ τ ∆ + ∆ − N O dξ3 N O N O N O 2 dξ2 · ¸ 1 dp p + (τ + τ ) − (∆ τ + ∆ τ + δ) − 2 N O N N O O dξ 2 τ τ d2(p2) 1 d(p2) p2 + N O − (τ + τ ) + = 0, (5.2) 2 dξ2 2 N O dξ 2 where terms of like order are grouped. Nonlinear terms consisting of p2 and derivatives of p2 are found in the third line of the equation. In order to get rid of these derivatives of p2, the following relation is used: · ¸ d2(p2) d d(p2) = dξ2 dξ dξ · ¸ d dp = 2p dξ dξ µ ¶ d2p dp 2 = 2p + 2 . (5.3) dξ2 dξ

Equation 5.2 can then be rewritten as · µ ¶¸ d3p 1 d2p −δτ τ + δ (τ + τ ) + τ τ ∆ + ∆ − N O dξ3 N O N O N O 2 dξ2 · ¸ 1 dp p + (τ + τ ) − (∆ τ + ∆ τ + δ) − 2 N O N N O O dξ 2 " µ ¶ # · ¸ τ τ d2p dp 2 1 dp p2 + N O 2p + 2 − (τ + τ ) 2p + = 0. (5.4) 2 dξ2 dξ 2 N O dξ 2

1Note that here and throughout the Johnson and Hammerton model discussion, dimensional parameters are denoted with a tilde, such as δ˜. Non-dimensional parameters are identified by the lack of a tilde. This convention is not followed throughout the thesis for simplicity. 2 ˜ 1 Johnson and Hammerton define δ as 2 the quantity usually defined as diffusivity. 63

For programming in MATLAB, Eq. 5.4 must be broken down into a system of ﬁrst-order equations. Because it is a third-order ODE, the system consists of 3 equations. Setting

A = δτN τO ¡ ¢ 1 B = δ (τN + τO) + τN τO ∆N + ∆O − 2 1 C = (τN + τO) − (∆N τN + ∆OτO + δ) 2 (5.5) 1 D = 2

E = τN τO

F = τN + τO,

0 dp writing p = dξ , and replacing in Eq. 5.4 yields

−Ap000 + Bp00 + Cp0 − Dp + Epp00 + E (p0)2 − F pp0 + Dp2 = 0, (5.6) or, equivalently, · ¸ B + Ep C − F p E D p000 = p00 + + p0 p0 − p(1 − p). (5.7) A A A A

Equation 5.7 is then re-expressed as a system of 3 ﬁrst-order ODEs:

0 0 p = p1 = p2 p00 = p0 = p 2 3 · ¸ (5.8) B + Ep C − F p E D p000 = p0 = 1 p + 1 + p p − p (1 − p ). 3 A 3 A A 2 2 A 1 1

5.1.2 Implementation

The Johnson and Hammerton model is implemented numerically using a fourth- order Runge-Kutta marching algorithm [96, 97]. For implementation in MATLAB, the third-order ODE is rewritten as a system of 3 ﬁrst-order ODEs, given in Eqs. 5.8. The fourth-order Runge-Kutta algorithm numerically solves the initial-value 64 problem 0 p = f(ξ, p); p(ξ0) = p0 (5.9) by a step-by-step method. The solution is computed for the equation

∆ξ p(ξ + ∆ξ) = p(ξ) + (k + 2k + 2k + k ), (5.10) 6 1 2 3 4 where

k1 = f(ξ, p) ∆ξ ∆ξ k2 = f(ξ + , p + k1) 2 2 (5.11) ∆ξ ∆ξ k = f(ξ + , p + k ) 3 2 2 2 k4 = f(ξ + ∆ξ, p + ∆ξ k3).

For the JH model, p0 is given by Eqs. 5.8, and the above procedure is repeated in steps of ∆ξ for the range of ξ. The ﬁxed step size ∆ξ is chosen to be small enough to resolve the shock. A possible improvement of computational eﬃciency could be achieved with implementation of an adaptive step size. Boundary conditions are determined by considering that p(ξ) → 0 as ξ → +∞. This allows for linearization of Eq. 5.4, resulting in · µ ¶¸ d3p 1 d2p −δτ τ + δ (τ + τ ) + τ τ ∆ + ∆ − N O dξ3 N O N O N O 2 dξ2 · ¸ 1 dp p + (τ + τ ) − (∆ τ + ∆ τ + δ) − = 0. (5.12) 2 N O N N O O dξ 2

Solutions of the form p ∝ e−λξ are desired, where

τN λ τOλ 1 δλ + ∆N + ∆O = ; λ > 0. (5.13) 1 + τN λ 1 + τOλ 2

The root of the equation, λ1, is found numerically using the MATLAB function fzero. This function requires an initial guess, which is found by evaluating Eq. 5.13 for a range of λ and calculating the minimum of the absolute value of the function.

The approximate boundary conditions for p and its derivatives at ξ = ξ0, where 65

ξ0 À 1, are then

0 00 2 p(ξ0) = ² ; p (ξ0) = −λ1² ; p (ξ0) = λ1², (5.14) where ² = e−λ1ξ0 ¿ 1. The algorithm uses these “initial” values at the top of the domain and marches in the negative direction in steps of ∆ξ, which is a negative value. Results are dimensionalized by setting µ ¶ x˜ t t˜= − ξ ref (5.15) c0tref V and by multiplying p by the shock overpressure ∆p. The reference time tref is chosen as tref = V τ˜N to eliminate V , andτ ˜N is the dimensional relaxation time for

N2. For simplicity,x ˜ is chosen asx ˜ = 0, resulting in t˜= −ξτÑ . The solution to Eq. 5.7 was also determined using the MATLAB function ode45, and results matched those of the Runge-Kutta solution explained above. The function ode45 is a Runge-Kutta algorithm itself, and it requires the same system of first-order ODEs (Eqs. 5.8) and boundary conditions (Eqs. 5.14) discussed above. In order to obtain a stable solution while integrating in the negative direction, the error tolerances RelTol and AbsTol must be set with odeset to sufficiently small values.

5.1.3 JH Model Results

The JH numerical model is used to approximate the shock structure of blast waveforms using shock overpressure values and meteorological conditions from the November 2005 blast experiments. The JH solution is computed for each blast at the distances 25, 50, and 100 m, and rise times are calculated. An example solution is presented in Fig. 5.1 for a blast waveform at 100 m. The solution consists of 3 regions with diﬀerent length scales, corresponding to

N2 relaxation, O2 relaxation, and thermoviscosity. It is found that, for the high amplitudes of the blast waves being considered, the shock shape is dominated by thermoviscosity. However, the top knee of the shock is controlled by O2 relaxation, and the N2 relaxation determines the eventual rise to ∆p. The rise time calculated 66 for this waveform is 0.4 µs, which is much less than the 12.4 µs rise time measured with the 90◦ 3.175-mm microphone at 100 m.

800

600

trise =0.4 µs

400 Pressure (Pa)

200

0 0 10 20 30 40 50 Time (µs) Figure 5.1. Example JH numerical model solution for a blast wave at 100 m. Asterisks denote the 10% and 90% amplitude points used to calculate the rise time trise. Input ◦ parameters are ∆p = 828 Pa, TC = 13.6 C, rh = 62.8%, and ps = 1 atm.

Figure 5.2 compares the same example waveform from the JH numerical model to the result from the JH asymptotic model discussed in Sec. 1.6.3. Here the 3 regions are clearly represented by the dashed asymptotic solutions for N2 relaxation, O2 relaxation, and thermoviscosity, in (a), (b), and (c), respectively. The asymptotic solutions can be used to determine the overall shock structure, but they can not be used to calculate the shock rise time. 67

(a) 800

600

400

Pressure (Pa) 200

0 −1500 −1000 −500 0 Time (µs) (b) 800

600

400

Pressure (Pa) 200

0 −10 0 10 20 30 40 50 Time (µs) (c)

800

600

400

Pressure (Pa) 200 Numerical Asymptotic 0

0 0.2 0.4 0.6 0.8 1 Time (µs)

Figure 5.2. Comparison of JH numerical and asymptotic model solutions for a blast wave at 100 m, presented at 3 diﬀerent length scales. Input parameters are ∆p = 828 Pa, ◦ TC = 13.6 C, rh = 62.8%, and ps = 1 atm. (a) Full shock with asymptotic result for the nitrogen relaxation mode. (b) Finer scale with the asymptotic result for the oxygen relaxation mode. (c) Finer scale with the asymptotic result for thermoviscosity. 68

5.1.4 Comparison to Polyakova et al. Model Predictions

The JH numerical solution is compared to the Polyakova et al. model solution for a monorelaxing fluid. Because the Polyakova et al. model only includes the effect of one relaxation mechanism, the solution is expected to differ from the JH solution, which includes N2 relaxation, O2 relaxation, and thermoviscosity. An example of the comparison between models is presented in Fig. 5.3. The JH numerical solution is the same as in Fig. 5.1. The smooth, albeit fast, rise of the JH solution is contrasted by the discontinuity in the Polyakova et al. solution. This discontinuity is a result of correcting the original multivalued waveform by weak shock theory, which does not account for thermoviscous effects (see Sec. 1.6.3). A rise time can not be calculated for the Polyakova et al. solution because of this discontinuity; in other words, the predicted rise time is 0 µs. The observed effect of

O2 relaxation, however, is similar to the rounding at the knee of the shock exhibited in the JH solution. Finally, the Polyakova et al. solution reaches ∆p much faster than the JH solution because it does not include the eﬀects of N2 relaxation. It can be concluded that the JH model oﬀers more detail about the shock structure than the Polyakova et al. model, and thus the JH model is more appropriate for the study of shocks in blast waves.

5.2 Anderson Model Predictions

The Anderson algorithm also was implemented in MATLAB, and it is a mod- iﬁcation of Gee’s Anderson code [66]. The code (anderson_propagation.m) is provided in Appendix C, along with subroutines to calculate absorption, LE, and trise. The following sections describe the algorithm and include example results relevant to blast waves.

5.2.1 Model Equations

The Anderson model is based on the extended generalized Burgers equation, introduced in Sec. 1.6.2: µ ¶ ∂p βp ∂p δ ∂2p X ∂2p m = + + R − p, (5.16) ∂r ρ c3 ∂τ 2c3 ∂τ 2 ν ∂τ 2 r 0 0 0 ν 69

800

700

600

500

400

Pressure (Pa) 300

200 JH Numerical 100 Polyakova et al. 0

−2 0 2 4 6 8 Time (µs) Figure 5.3. Comparison of JH numerical model and Polyakova et al. corrected model ◦ solutions for a blast wave at 100 m. Input parameters are ∆p = 828 Pa, TC = 13.6 C, rh = 62.8%, and ps = 1 atm. where p is acoustic pressure, r is range, τ is retarded time, β is the coeﬃcient of nonlinearity, ρ0 is the ambient density, c0 is the small-signal sound speed, δ is the diﬀusivity, Rν is a relaxation operator for each relaxation process ν, and m is a geometrical spreading parameter. This equation accounts for the physical processes of second-order nonlinearity, thermoviscous absorption, absorption and dispersion due to molecular relaxation, and geometrical spreading. For the spherical spreading observed in blast waves, m = 1.

5.2.2 Implementation

The Anderson model is used in the present blast wave research to model the propagation of blast waves at medium distances. As shown in Fig. 5.4, the algorithm takes an initial waveform in time, Fourier transforms the waveform into the frequency domain, applies absorption, dispersion, and spherical spreading over one range step ∆r, inverse Fourier transforms the spectral information back into the time domain, steepens the modiﬁed time waveform via the Earnshaw solution 70

[18] over one range step ∆r, and then re-samples the waveform for uniform time samples. The algorithm proceeds similarly for each range step until the desired propagation distance is reached.

Input data file

Define initial variables

Define ranges

Apply absorption, Apply Take Calculate time FFT dispersion, and IFFT nonlinear Resample Save range step distortion spreading distortion

N Final range?

Plot

Figure 5.4. Simpliﬁed ﬂow chart describing steps in the Anderson algorithm.

The Burgers equation is implemented by splitting Eq. 5.16 into 2 independent equations [39, 98]. The ﬁrst equation accounts for nonlinear eﬀects and is given as

∂p βp ∂p = 3 . (5.17) ∂r ρ0c0 ∂τ The second equation accounts for spherical spreading, atmospheric absorption, and dispersion, and is given as

∂P (f) 1 = −α (f)P (f) − P (f), (5.18) ∂r c r where P (f) is the Fourier transform of p(τ) and αc(f) is the complex absorption coeﬃcient. Each of these equations has an analytical solution [39, 98] which is applied at each range step ∆r. The Earnshaw solution for nonlinear distortion is used to shift 71 the arrival time of each point on the waveform and is implemented numerically as

βp(τr)∆r τr+∆r = τr − 3 , (5.19) ρ0c0 where τr is the retarded time of a sample before distortion, and τr+∆r is the retarded time for the same sample after distortion. Each point on the waveform is modiﬁed so that positive pressure points are shifted earlier and negative pressure points are shifted later in time; this accounts for nonlinear steepening. The solution for spherical spreading, absorption, and dispersion in the frequency domain is implemented as r P (r + ∆r, f) = e−αc(f)∆rP (r, f). (5.20) r + ∆r

5.2.2.1 Absorption and Dispersion

As discussed in Sec. 2.1, the total absorption coeﬃcient in air is the superposition of three factors: thermoviscous absorption, nitrogen vibrational relaxation absorption, and oxygen vibrational relaxation absorption. The total complex absorption is given by

αc = αtv + αr,N + αr,O − jΦ, (5.21) where Φ is the dispersion due to nitrogen and oxygen vibrational relaxation. Fol- lowing the development by Blackstock [99] and Hamilton et al. [53], the complex absorption due to a single relaxation process is

2 mdω τr αˆr = , (5.22) 2c0(1 + jωτr) where the hat denotes a complex quantity, and τr represents the relaxation time for the relaxation process. Therefore,

2 mdω τr Re[ˆαr] = αr = 2 2 (5.23) 2c0(1 + ω τr ) 72 and

3 2 mdω τr Im[ˆαr] = Φ = 2 2 2c0(1 + ω τr ) Φ = αrωτr. (5.24)

For multiple (i) relaxation processes, X Φ = αr,i ωτr,i i X α Φ = f r,i , (5.25) f i r,i where fr,i = 1/(2πτr,i) is the relaxation frequency. For air, · ¸ α α Φ = f r,N + r,O . (5.26) fr,N fr,O

Finally, the total complex absorption coefficient in air as a function of frequency is · ¸ αr,N αr,O αc(f) = (αtv + αr,N + αr,O) − jf + . (5.27) fr,N fr,O The empirical formulas needed to solve for the absorption coefficients and relaxation frequencies in Eq. 5.27 are given by Bass et al. [46, 47]. The subroutine code absorption.m, a modification of original code by Brian C. Tuttle, is used to calculate absorption and dispersion and is included in Sec. C.2.2 in Appendix C.

5.2.2.2 Step Size

The range step size must be small enough to prevent the formation of a multivalued waveform. As outlined by Anderson [39], a multivalued waveform will not develop if

τi−1(r + ∆r) ≤ τi(r + ∆r), (5.28) where τi(r + ∆r) is the retarded time for the ith sample at range position r + ∆r. Substituting the Earnshaw solution from Eq. 5.19,

β∆r β∆r τi−1(r) − 3 pi−1(r) ≤ τi(r) − 3 pi(r), (5.29) ρ0c0 ρ0c0 73 and the step size condition becomes [39]

ρ c3 1 ∆r ≤ 0 0 . (5.30) β [∆p(r)/∆τ(r)]

All the points on the waveform must propagate the same ∆r, so the step size is limited by the largest ∆p(r)/∆τ(r) derivative [39], leaving

3 ρ0c0 1 ∆rmax = . (5.31) β [∆p(r)/∆τ(r)]max

Equation 5.31 is essentially the local plane-wave shock formation distance for lossless propagation. The actual step size used in the code is determined by multiplying Eq. 5.31 by a factor η, where η ≤ 1 [98]. Finally, the adaptive step size used which prevents the formation of multivalued waveforms is given by

ρ c3 1 ∆r = η 0 0 . (5.32) β [∆p(r)/∆τ(r)]max

5.2.2.3 Usage

The Anderson program, written in MATLAB, takes as input a blast waveform at an initial range and propagates the waveform downrange. For example, the current implementation assumes that the waveform is a measurement at 25 m, and it produces predicted waveforms at 50, 100, and 200 m. This setup is useful for comparing experimentally measured waveforms with computer program predictions. Step-by-step instructions for running the code are included in Appendix C. Another important point should be made about the MATLAB computer algorithm. The last operation performed for every range step ∆r is to resample the waveform after nonlinear steepening has occurred. This resampling can be a slow process, and thus a MATLAB MEX-function compiled from a Fortran 95 function written by Gee [66] is used. The MATLAB statement which calls the MEX-function rsamp.dll is given as xnew=rsamp(tdistort,x,t);

One should note that since this is an external function to MATLAB, the compiled MEX-function may fail to work in some future version of MATLAB. If that occurs, 74 one can use an alternate, albeit slower, resampling function within MATLAB: xnew=interp1(tdistort,x,t,‘linear’,‘extrap’,0);

5.2.3 Anderson Model Results

The Anderson model is used to predict the propagation of blast waveforms using an input waveform from the unbaffled 25 m measurements and meteorological conditions from the November 2005 blast experiments. The Anderson solution is computed for each blast out to the distance of 200 m, in increments of 5 m. Rise times are calculated for each resulting waveform. Example predictions for blast waveforms at 50, 100, and 200 m are presented in Fig. 5.5, along with the initial input waveform at 25 m. The predicted rise times are 1.6, 2.3, and 6.7 µs at 50, 100, and 200 m, respectively. These predicted rise times are much shorter than the measured rise times. For the same blast presented here, the measured rise times at 50 m were 4.5, 4.3, and 6.1 µs for measurements with the wide-band microphone, 0◦ 3.175-mm microphone, and 90◦ 3.175-mm microphone, respectively. At 100 m, the measured rise times were 12.4 and 11.6 µs for measurements with the 90◦ 3.175-mm microphone and the baffled 3.175-mm microphone, respectively. The short rise time predicted at 50 m confirms that the propagation is nonlinear, and the wave has steepened. From 50 to 200 m, the predicted rise times increase slowly, indicating that nonlinearity is weakening and that dissipation is starting to become a stronger factor. Figure 5.6 presents rise time predictions as a function of range using the An- derson model. The atmospheric conditions for the twelve blasts in November 2005 were used in the model. Out to a distance of 100 m, there are some variations in rise times due to atmospheric conditions and the shape of the 25 m input waveform, but in general the predictions are similar. Beyond 100 m, however, the atmospheric conditions appear to have an influence on the predicted rise times. The blue markers represent rise times predicted for the atmospheric conditions on the afternoon of November 7, when the relative humidity was 38%. In contrast, the red markers represent rise times predicted for the atmospheric conditions on the morning of November 8, when the relative humidity was 62%. It is shown that the rise times predicted for the second, more humid, day are shorter than for the drier 75

25 m 50 m 1500 3000 1000 2000 500 1000 ∆ =3670 Pa ∆p=1380 Pa Pressure (Pa) p Predicted =1.6 s Measured t =21.1 µs trise µ 0 rise 0 0 10 20 30 40 50 0 10 20 30 40 50 100 m 200 m 600 200 400

200 100 ∆ =560 Pa ∆ =240 Pa Pressure (Pa) p p Predicted t =2.3 µs Predicted =6.7 s 0 rise 0 trise µ 0 10 20 30 40 50 0 10 20 30 40 50 Time (µs) Time (µs) Figure 5.5. Example Anderson model solution for a blast wave at 50, 100, and 200 m. Input waveform at 25 m was measured with a 90◦ 3.175-mm microphone. Asterisks denote the 10%, 90%, and 100% amplitude points used to calculate the rise time trise. ◦ Atmospheric parameters are: TC = 13.6 C, rh = 62.8%, and ps = 1 atm.

first day. This trend has been reported by Kang [51] in his sonic boom research, in which an increase in humidity at a given temperature resulted in a decrease in shock rise times. This is due to the sensitivity of relaxation times to humidity. At close ranges less than 100 m from the blast source, the humidity does not seem to have a large effect on rise times because nonlinearity is very strong. However, as nonlinearity weakens and relaxation effects become important, humidity begins to affect the rise times. The effect of step size on blast rise time predictions was investigated. The step size must be small enough to assume that the physical processes are independent. The step size also must be smaller than the true local shock formation distance to avoid a multivalued waveform. For the steep shocks in blast waves, very small steps must be taken. However, taking very small steps could result in an appreciable accumulation of errors due to FFT conversions and resampling [59]. In addition, 76

12 Nov. 7 Nov. 8 10

8 s) µ

Rise time ( 4

0 25 50 100 200 Range (m) Figure 5.6. Anderson model predictions for rise times in 5 m increments (markers shown every 25 m) from 30 m out to 200 m. Input waveforms at 25 m were measured with a 90◦ 3.175-mm microphone. Atmospheric parameters match those of the 12 blasts from November 2005. computation time may become prohibitive with extremely small step sizes. Cleve- land et al. [98] recommended using η = 0.2 for sonic boom calculations. Although a blast wave is a transient similar to a sonic boom, it was expected that the step size requirements would not be the same. Thus blast wave predictions were tested with η = 0.2, 0.1, 0.01, and 0.001. Results showed that the rise times converge with η = 0.01, where ∆r ≈ 0.14 mm near the initial shock and ∆r ≈ 6 mm at 200 m. Therefore, η = 0.01 was chosen for the Anderson calculations. The computation time required for a step size with η = 0.01 was approximately 3 hr on a modern desktop computer with a 3.2 GHz processor. The number of range steps taken was slightly less than 190000 on the average. For η = 0.001, the computation time increased dramatically to 64 hr. For η = 0.1 and 0.2, the computation time decreased noticeably to 12 and 9 min, respectively. Another issue investigated by Cleveland [59] and Bass and Raspet [100] is the number of sample points on the shock. It is recommended to have at least 77

10 points on the shock for good numerical resolution and to decrease resampling errors. In fact, the entire waveform must be sampled at this high rate because a uniformly sampled waveform is required by the algorithm. For the initial waveform, the number of points on the full shock is approximately 50, while the number of points on the 10-90% rise portion is approximately 20. After propagating to 50 m, however, the number of full shock points is 20 and the number of 10-90% shock points is only 1. After propagating to 200 m, the number of full shock points is 40 and the number of 10-90% shock points increases to 6. These results are nominally the same regardless of step size, and examples showing sample points are presented in Fig. 5.7. The number of points on the shock is determined by

0.8

0.6

0.4 Normalized Pressure 0.2 25 m 30 m 0 200 m

−10 0 10 20 30 40 50 Time (µs) Figure 5.7. Anderson model predictions for 30 and 200 m, with the input waveform at 25 m. Sample points are denoted with markers. the sampling frequency from the experimental input waveform. This can not be changed in postprocessing, except by interpolation, which causes additional errors. Thus the original waveforms are used, despite the small number of points on the shock when propagated. This shows that the main limitation of the Anderson model is that it depends on the quality of the input waveform. 78

5.2.4 Comparison to Shock Front Model Predictions

The Anderson rise time predictions are compared to the JH numerical solutions for a relaxing ﬂuid. Atmospheric parameters corresponding to the 12 blasts from November 2005 are used in both codes. While the Anderson model receives the 25 m waveform as an input to propagate, the JH model uses the shock overpressure of the experimental waveforms at the ranges 25, 50, and 100 m as an input. A comparison of predicted rise times from both models is presented in Fig. 5.8. Both models predict extremely short rise times close to the blast; the JH model predicts a rise time of 0.05 µs at 25 m, and the Anderson model predicts a rise time of 1.5 µs at 30 m. Nonlinearity is predicted to be exceptionally strong in both cases. At longer distances, the rise time predictions increase for both models. The Anderson predictions are bracketed by the JH predictions at 100 m. Additional data out to longer distances are not available for the JH model because waveforms were not measured beyond 100 m. Both the JH and Anderson predictions show shorter rise times for the more humid conditions on November 8. Additionally, the measured shock overpressures at 100 m were higher on November 8, and this could contribute to shorter rise times in the JH model. Comparisons of these computer model predictions to the experimental data are discussed in the following chapter. 79

JH Nov 7 10 JH Nov 8 Anderson Nov 7 8 Anderson Nov 8 s)

µ 6 ( rise t 4

0 25 50 100 200 Range (m)

Figure 5.8. Comparison of JH and Anderson model rise time (trise) predictions. The Anderson model trise predictions are shown in 5 m increments (markers shown every 25 m), and the JH model predictions are calculated at 25, 50, and 100 m. Atmospheric parameters match those of the 12 blasts from November 2005. Chapter

6 Comparison of Experimental Results to Model Predictions

This chapter compares the experimental data from Chapter 4 to the computational model predictions from Chapter 5. Limitations in both the experimental setup and in the validity of the models are discussed. Finally, the validity and applicability of the simple shock width model presented in Chapter 2 is investigated.

6.1 Rise Time Comparisons

The rise times from the November 2005 experiments are compared to predictions from both computational models in Fig. 6.1. These results were presented at the 151st Meeting of the Acoustical Society of America [101]. In addition, the mean (trise) and standard deviation (σt) of the measured and predicted rise times are presented in Table 6.1. Included are the measured rise times with unbaffled 3.175-mm 90◦ microphones at 25, 50, and 100 m, baffled 3.175-mm microphones at 25 and 100 m, and the wide-band microphone at 50 m. These experimental results were presented previously in Fig. 4.8. The JH and Anderson model predictions, presented previously in Fig. 5.8, are also included for comparison. As mentioned earlier, atmospheric parameters corresponding to the 12 blasts from November 2005 are used in both models. The Anderson model receives the unbaffled 25 m measured waveform as an input to propagate, and the JH model uses the shock overpressure of the experimental waveforms at the ranges 25, 50, and 100 m as an input. Close to the blast, both models predict extremely short rise times that are 81

15 s) µ ( rise

t 10

Unbaffled Baffled 5 Wide−band JH Anderson

0 25 50 100 Range (m)

Figure 6.1. Comparison of rise times as a function of range from November 2005 experiments, JH model predictions, and Anderson model predictions. Experimental results are presented for blast waves measured with different microphone types: unbaffled microphones (25, 50, 100 m), baffled microphones (25, 100 m) and wide-band microphone (50 m). The solid lines merely connect data points for individual blast events and do not represent additional data. much shorter than those measured. Nonlinearity is predicted to be exceptionally strong in both cases. For the Anderson model, the wave immediately steepens and the rise time decreases to less than 2 µs in the first meter of numerical propagation from 25 to 26 m. At longer distances, the rise time predictions increase for both models. For the most part, however, the rise time predictions are still shorter than those measured at 50 and 100 m. The Anderson predictions are bracketed by the JH predictions at 100 m. As discussed in Chapter 5, both the JH and Anderson models predict shorter rise times for the more humid conditions on November 8. This trend is not observed in the experimental data. Limitations in the measurement system and limitations in the model predictions both contribute to the discrepancies between rise times observed in Fig. 6.1. 82

Statistic 25 m 50 m 100 m i M F M O i M O

trise (µs) 20.2 20.3 0.1 6.1 4.7 0.1 1.6 11.1 14.0 2.8 2.6

σt (µs) 1.3 1.3 0.0 0.1 1.0 0.0 0.1 3.0 3.9 3.0 0.4 Legend ¤ Unbaﬄed ° Baﬄed F Wide-band M JH O Anderson

Table 6.1. Comparison of the mean (trise) and standard deviation (σt) of measured and predicted rise times at 25, 50, and 100 m for November 2005 blasts. ¤ is the unbaﬄed 3.175-mm microphone, ° is the baﬄed 3.175-mm microphone, F is the wide- band microphone, M is the Johnson and Hammerton prediction, and O is the Anderson model prediction.

The bandwidth of the measurement system is limited mainly by the frequency response of the microphones. This limits the ability to measure short blast rise times accurately. It was found that the wide-band microphone, with its extended high-frequency response, recorded the shortest rise times; these rise times were within the capabilities of the microphone. Thus the wide-band microphone rise time measurements are considered to be more accurate and free of transducer effects. The only wide-band microphone available for the experiments was mounted at 50 m, and its measurements are within 1-3 µs of the Anderson predictions at the same range. The JH model assumes a plane frozen step-shock profile that propagates without change in form. On the other hand, blast waves at medium distances are continuing to distort. Hence this frozen step-shock simplification is expected to predict shock structures and rise times that are different from those measured. 83

6.2 Atmospheric Turbulence Eﬀects on Rise Times

Both the JH and Anderson models assume a simplified one-dimensional homogeneous atmosphere. Therefore neither model in its current implementation includes sound speed variability due to turbulence, wind, or atmospheric stratification. Explanations of turbulence effects on sonic boom rise times were reported by Pierce and Maglieri [102] in 1972, and Pierce’s wave front folding model was verified more recently by Lipkens [103]. Experimental studies of spark-generated N waves by Lipkens and Blackstock [104, 105] to model sonic boom propagation through turbulence show that turbulence almost always increases rise times. The measured wind speeds were very low throughout the November 2005 experiments, but the turbulence may have been significant, especially on the second day when the solar radiation was higher. It is hard to quantify the effect that turbulence might have had on the measured waveforms because the meteorological measurements were not sufficiently detailed. However, the shapes of waveforms at longer ranges, where nonlinearity is not as strong, qualitatively match turbulized waveforms modeled by Blanc-Benon et al. [57] and confirmed experimentally by Ollivier and Blanc-Benon [58]. These studies of turbulence have shown that rise times generally increase with turbulence, due to the random fluctuations in acoustic amplitude and phase and the formation of caustics. Thus it is believed that this turbulence effect is a major cause of differences in rise times at longer ranges between the measured and predicted waveforms presented here.

6.3 Comparison to Bass et al. Predictions

The experimental rise times and model predictions are compared to shock thickness predictions by Bass et al. [23] for a 68-kg TNT explosion. Atmospheric parameters are T = 295 K and rh = 30%. Figure 6.2 compares the Bass et al. predictions using an Anderson-type algorithm to JH model predictions with the same atmospheric parameters. The thin solid lines labeled a, b, and c are theoretical predictions for a steady-state waveform in an atmosphere with absorption Af 2, where A is determined with thermoviscous effects, thermoviscous and oxygen re- 84 laxation effects, and thermoviscous combined with oxygen and nitrogen relaxation effects, respectively.

2 10 JH Bass et al. dispersive Bass et al. nondispersive

0 10

−2 10 Shock thickness (ft.)

−4 10

−1 0 1 2 10 10 10 10 ∆p (psf) Figure 6.2. Comparison of shock thickness vs. ∆p from Bass et al. [23] and JH model predictions. Atmospheric parameters are T = 295 K and rh = 30%. The thin solid lines labeled a, b, and c are theoretical predictions for a steady-state waveform in an atmosphere with absorption Af 2, where A is determined with thermoviscous effects, thermoviscous and oxygen relaxation effects, and thermoviscous combined with oxygen and nitrogen relaxation effects, respectively. The bold line is a model prediction by Bass et al. for a blast wave with dispersion included, and the dashed line is the same model with dispersion numerically removed. Shock thickness in ft. can be converted to rise time in s by trise = lrise ∗ 0.3048/c0. Shock overpressure in psf can be converted to Pa by multiplying by 47.88 Pa/psf. Used by permission.

Lines a, b, and c are calculated as follows:

• isolate the straight sections of loglog plots of αtv, αtv + αr,O, and αtv + αr,O +

αr,N vs. frequency

• calculate A, the leading coeﬃcient of a quadratic polynomial that ﬁts the data, for each case

• ﬁnd a frequency f where a relaxation transition is not occurring, for each case 85

• multiply A by f 2 to obtain α for each case 8ρ c3 α • calculate the steady-state rise time t = 0 0 for each case for a range rise β ω2∆p of ∆p

• calculate the shock thickness lrise = trisec0 for each case

The bold line is a model prediction by Bass et al. for a blast wave with dispersion included, and the dashed line is the same model with dispersion numerically removed. Differences between the two are mainly noticeable in the transition region from b to c. The Bass et al. model shows that, as the blast wave propagates and its shock overpressure decreases, the shock thickness evolves from depending on thermoviscous and oxygen relaxation absorption to depending additionally on nitrogen relaxation absorption. The JH model is used to investigate how the shock thickness dependencies change at higher overpressures. As shown in Fig. 6.2, the JH model confirms that thermoviscous absorption dominates at high overpressures. As the blast wave propagates and its shock overpressure decreases, the shock thickness transitions from a to b, signifying the shift in dependence to thermoviscous and oxygen relaxation absorption. As with the Bass et al. model, the JH model also predicts a transition from b to c as the shock overpressure continues to decrease. Although the results are similar, the two models transition at different overpressures. Additionally, the JH model transitions fully to c, whereas the Bass et al. model levels off below c. One reason for this difference could be that the JH model is for a plane wave, but the Bass et al. model includes spherical spreading. Units common in the sonic boom literature are used. Shock thickness in ft. can be converted to rise time in s by trise = lrise ∗ 0.3048/c0. Shock overpressure in pound-force per square foot (psf) can be converted to Pa by multiplying by 47.88 Pa/psf. Fig. 6.3 compares the same Bass et al. results to the November 2005 experimental data, JH model predictions, and Anderson model predictions. In this case, the data do not have the same atmospheric parameters as the Bass et al. model. Hence a direct quantitative comparison is not warranted, but it is still a useful qualitative comparison. Note that the lines a, b, and c would be shifted downward for the higher humidities recorded during the blast experiments. The experimental data included are shock thicknesses measured with unbaffled 3.175-mm microphones 86

2 10

0 10

−2 10 Unbaﬄed

Shock thickness (ft.) Baﬄed Wide-band JH −4 10 Anderson Bass et al. dispersive Bass et al. nondispersive

−1 0 1 2 10 10 10 10 ∆p (psf) Figure 6.3. Comparison of shock thickness vs. ∆p from Bass et al. [23], November 2005 experiments, JH model predictions, and Anderson model predictions. Atmospheric parameters for the Bass et al. results are T = 295 K and rh = 30%. The thin solid lines labeled a, b, and c are theoretical predictions for a steady-state waveform in an atmosphere with absorption Af 2, where A is determined with thermoviscous effects, thermoviscous and oxygen relaxation effects, and thermoviscous combined with oxygen and nitrogen relaxation effects, respectively. The bold line is a model prediction by Bass et al. for a blast wave with dispersion included, and the dashed line is the same model with dispersion numerically removed. Shock thickness in ft. can be converted to rise time in s by trise = lrise ∗ 0.3048/c0. Shock overpressure in psf can be converted to Pa by multiplying by 47.88 Pa/psf. Used by permission. at 90◦, baffled 3.175-mm microphones, and the wide-band microphone. The JH model is used to compute lrise with the experimental blast wave parameters, and the Anderson model uses an experimental waveform as input. The experimental lrise at the highest shock overpressures are from both the unbaffled and baffled 25 m measurements, and they lie close to line c. The next group of experimental data to the left are from the unbaffled and wide-band 50 m measurements, and these lie closer to line b, signifying an increase in nonlinearity and a loss of dependency on nitrogen relaxation absorption. The scattered group of data between 10 and 20 psf represents the unbaffled and baffled 100 m measurements, where the lowest 87 shock overpressures were recorded. These data still show a dependency on oxygen relaxation absorption. The Anderson predictions mostly follow line b, indicating the importance of oxygen relaxation absorption in shock structure. Interestingly, the Anderson predictions at 4–7 psf are similar to the Bass et al. predictions in the same range, despite the differences in input parameters. As in the previous figure, the JH model indicates a transition from a to b as ∆p decreases. The transition, however, occurs at lower values of ∆p and is sharper than when the model is run with the Bass et al. parameters. This demonstrates that atmospheric parameters, especially relative humidity, affect the shock thickness.

6.4 Histograms

Histograms of rise times were calculated for the experimental data, JH model predictions, and Anderson model predictions. A comparison of the percent occurrence of rise times is presented in Fig. 6.4. The unbaffled and baffled rise times were both included in the experimental histograms for 25 and 100 m; similarly, the unbaffled and wide-band rise times were both included in the experimental histograms for 50 m. Although the number of data points included in the histogram statistics is small, conclusions from the histograms are the same as for Fig. 6.1. Measurements show that rise times decrease from 25 to 50 m, and there is a scatter of longer rise times at 100 m. Both models predict shorter rise times than what was measured, and the models predict longer rise times with increasing propagation distances.

6.5 Comparison to Preliminary Shock Width Model

The validity and applicability of the simple shock width model presented in Chapter 2 is investigated. It would be worthwhile to note if this simple model is adequate for predicting blast wave shock rise times. The Naugol’nykh et al. spherical wave shock width model is used to predict rise times as a function of 88

(a) (b) (c) 100 100 100 25 m 25 m 30 m

50 50 50

0 0 0 100 100 100 50 m 50 m 50 m

50 50 50 % Occurrence

0 0 0 60 60 60 100 m 100 m 100 m 40 40 40

20 20 20

0 0 0 0 10 20 0 10 20 0 10 20 trise (µs) trise (µs) trise (µs)

Figure 6.4. Percent occurrence of trise for diﬀerent propagation distances. (a) Exper- imental data at 25, 50, and 100 m. (b) JH model predictions at 25, 50, and 100 m. (c) Anderson model predictions at 30, 50, and 100 m. Note that the axis limit is 60% for the 100 m data in the last row of histograms.

range for a 10 kHz sinusoidal source with a source radius r0 = 25 m and a pressure amplitude in the range of experimental values at 25 m. Figure 6.5 compares these Naugol’nykh et al. model results to the Anderson model predictions. The predictions are remarkably similar, despite diﬀerences in the model assumptions. For example, the Naugol’nykh et al. model includes only thermoviscous absorption, while the Anderson model additionally includes molecular vibrational relaxation 89 absorption. It is interesting to note that the Naugol’nykh et al. rise times follow the same trend as the November 8 Anderson rise times starting at a range of approximately 80 m.

Naugol’nykh et al. 10 Anderson

8 s)

µ 6 ( rise t 4

0 25 50 100 200 Range (m)

Figure 6.5. Comparison of rise times (trise) predicted with the Naugol’nykh et al. and Anderson models. The Anderson model trise predictions are shown in 5 m increments (markers shown every 25 m).

Although the results in Fig. 6.5 look promising, a word of caution is warranted. The Naugol’nykh et al. model assumes an initially sinusoidal waveform, and con- sequently a specific frequency must be chosen. The frequency 10 kHz was chosen because it corresponds to a period of 100 µs; the rise time of a wave with this period is on the order of the blast wave rise times. Therefore the 10 kHz sinusoid is used as an approximation for the leading edge of a steepening blast wave. If the frequency is modified, however, the results change drastically. Lower frequencies result in shorter rise times, and higher frequencies result in longer rise times. It can be concluded that the Naugol’nykh et al. shock width model can be used as a quick indication of expected rise times for finite-amplitude spherical waves, provided the input parameters are known. The main benefit is that the model requires little computational power. Nevertheless, it is recommended to 90 implement models such as the Anderson model for more accurate predictions of the effects of nonlinearity and dissipation on the evolution of blast waves. Chapter

7 Potential Eﬀects of Blast Waves on Bat Hearing

This chapter presents an interpretation of thesis results in the context of the possible effects of blasts on bat hearing. Military live fire drills near endangered bat populations could potentially affect the well-being and survival of these bat species. Because of the nonlinearity inherent in the propagation of finite-amplitude blast noise, simple linear calculations are not sufficient to describe the blast wave evolution with distance.

7.1 Rise Times and High-frequency Energy

Characterization of blast waves was mainly focused on rise time because it is an indicator of high-frequency energy contained in the blast wave. The frequency range of hearing for the Big Brown Bat, a candidate surrogate for endangered species of bats, is roughly 10–100 kHz, which corresponds to blast rise times of 10–100 µs. Special considerations were made to ensure measurement systems and models were capable of resolving the fine structure of shocks. It has been demonstrated that the majority of high-frequency energy in a blast wave is contained in the shock. Spectrograms of measured blast waves reveal that this high-frequency energy at the shock extends into the bat hearing range. Rise times in the range 3–22 µs were measured at medium ranges from an explosion of 0.57 kg of C-4. Model results for the same conditions predicted even shorter rise times in the range of approximately 0.1–8 µs. It is likely that the true rise times 92 of the blast waves lie between the measured and predicted results. Atmospheric turbulence is known to affect the rise time and structure of shocks, thereby affecting the frequency content of blast waves. Turbulence is likely to affect rise times at a range of 100 m from the explosion. Lengthening of the rise times at this distance, especially in the afternoon, potentially would make the blast noise less severe for bat hearing systems. However, this effect diminishes when the sun sets and turbulence near the ground decreases, and it is at this time of the day when bats are likely to leave their roosts. There is no easy formula or simple plot that can describe how bats will be affected by blast noise. This research is an attempt to quantify one part of the problem, namely the structure of the blast wave as it propagates. Although dif- fering in exact rise times, the experiments and model predictions both show that nonlinearity causes a steep shock to form, indicating the presence of a considerable amount of high-frequency energy in the wave. Even at 200 m from the explosion, this high-frequency energy persists, despite spreading and absorption losses.

7.2 Research Findings of Brittan-Powell et al.

Complementary work by Brittan-Powell et al. [106] at the University of Mary- land has focused on deﬁning the hearing sensitivity of Big Brown Bats and assessing their hearing damage after exposure to blast noise. The main conclusion from this work is that there is little to no hearing damage in Big Brown Bats exposed to a single blast at 25 m or at longer ranges. Brittan-Powell et al. note that behavioral changes were not monitored, and the bats were exposed to only one blast impulse. Thus it is diﬃcult to assess whether the bats could survive this exposure, or repeated exposures, in the wild. Chapter

8 Conclusions

8.1 Summary and Conclusions

The presence of endangered bats on military training grounds has prompted this investigation of high-frequency energy in blast waves. Defining the blast wave evolution with distance is the first step to answering the question of whether the sound from large weapon firing may adversely affect these bats. In this section, a summary of thesis results is presented, followed by a list of key conclusions. Spectrograms of blast waves reveal that the majority of high-frequency energy is contained in the shocks, and that this high-frequency energy extends into the bat hearing range. Hence this research focused on defining the shock structure and how it changes with increasing propagation distances. Characterization of blast waves was mainly focused on rise time because it is an indicator of high-frequency energy contained in the blast wave. Special considerations were made to ensure measurement systems and models were capable of resolving the fine structure of shocks. The shock structure of a blast wave is an important factor affecting the high- frequency content of the wave. Several factors contribute to the shape of the shock, including nonlinearity, thermoviscous absorption, and absorption and dispersion due to molecular vibrational relaxation of N2 and O2. Nonlinearity due to the finite-amplitude of the blast wave causes the waveform to steepen and form a shock. Absorption counters the effects of nonlinearity and tends to round out the shock. Spherical spreading reduces the pressure amplitude, resulting in a decrease of nonlinearity at increasing distances. At high frequencies, thermoviscous absorp- 94 tion is the dominant absorption mechanism. However, as the wave propagates and nonlinearity weakens, the high-frequency energy is reduced, oxygen relaxation begins to affect the shock slope, and nitrogen relaxation delays the rise to maximum pressure. Dispersion smooths out the shock front even further through frequency- dependent phase changes. Measurements of blast wave propagation from an explosion of 0.57 kg of C-4 show that out to a distance of 50 m, where the peak pressure was around 1480 Pa (corresponding to a SPL of 154 dB re 20 µPa), nonlinear effects had steepened the initial rise of the blast waveform. This resulted in a decrease in shock rise time and therefore an increase in high-frequency energy. Measurements with a wide- bandwidth microphone gave shorter rise times than measurements with conventional 3.175-mm microphones. For example, rise times measured at 50 m were 5.9– 6.2 µs for unbaffled 3.175-mm microphones and 3.2–5.7 µs for the wide-band microphone. For laboratory measurements of balloon pops with a Brüel& Kjær 4138 3.175-mm microphone at 90 ◦, Marston [81] found that the shortest rise time the microphone could measure was 4.5 µs even though the true rise time was shorter. Marston’s microphone was similar to the G.R.A.S. microphone used in this research, thus his results support the conclusion that the unbaffled microphone can not respond fast enough to give the true rise time at 50 m. In contrast, Marston estimated that the wide-band microphone’s rise time limit is 0.7 µs. Marston arrived at this estimate by considering that the exponential time constant is 0.32 µs for the wide-band microphone’s high-frequency rolloff of 500 kHz. For a step-function pressure input, the 10–90% rise time is 2.2 times the exponential time constant, or 0.7 µs. Because of its extended high-frequency response, the wide-band microphone gives more confidence that one is observing true pressure variation and not the effect of the transducer. Therefore the results of this research suggest that wide-band microphones should be used for measuring rise times of finite-amplitude blast waves. Microphone baffles were also used in the blast measurements to eliminate microphone diffraction effects on the rise portion of the wave. Although the full waveform is not recovered, the rise time measurements with baffled microphones at 0◦ incidence are more accurate than with the unbaffled microphones at 90◦ incidence. Rise times for the baffled microphone were 17.9–21.9 µs at 25 m and 95

8.6–21 µs at 100 m. In general, rise times at 100 m were still shorter than at 25 m due to nonlinearity. The large spread in rise times at 100 m is believed to be caused by atmospheric turbulence. Model results for the same temperature and humidity conditions predict even shorter rise times in the range of approximately 0.1–8 µs. Close to the blast, both the JH and Anderson models predict extremely short rise times due to high nonlinearity. At longer distances, the rise time predictions increase for both models. This is in contrast to the initial decrease in rise time observed in measurements. For the most part, however, the rise time predictions are still shorter than those measured at 50 and 100 m. An increase in humidity results in shorter rise times for both the JH and Anderson models, even though this trend is not observed in the experimental data. The bandwidth of the measurement system is limited mainly by the frequency response of the microphones. Limitations in the 3.175-mm microphone bandwidth resulted in measured rise times at 50 m that were longer than the true rise times. The accuracy of the JH model rise time predictions is limited by assumptions about the shape and propagation of the wave and by exclusion of turbulence effects. The Anderson model predictions also are limited by exclusion of turbulence effects. The combination of these factors contributes to the discrepancies between measured and predicted rise times. The JH model assumes a plane frozen step-shock profile that propagates without change in form. On the other hand, blast waves at medium distances are continuing to distort. Hence this frozen step-shock simplification is expected to predict different shock structures and rise times that are shorter than those measured in this study. Moreover, both models assume a simplified one-dimensional homogeneous atmosphere. Therefore neither model in its current implementation includes sound speed variability due to turbulence, wind, or atmospheric stratification. Ground effects are also ignored. Studies of turbulence reported in the sonic boom literature have shown that rise times generally increase with turbulence, due to the random fluctuations in acoustic amplitude and phase and the formation of caustics. Thus it is believed that this turbulence effect is a major cause of differences in rise times at longer ranges between the measured and predicted waveforms presented here. 96

Furthermore, higher-order nonlinearities are not considered by the models. The model equations are based on second-order approximations [18], which possibly are not sufficient to completely describe the near-field propagation of such high- amplitude blast waves. The simplifications assumed in the models resulting from the second-order approximations lead to an approximation of the actual physical phenomena. The main conclusions resulting from this thesis research are:

1. Bats will be exposed to audible levels of high-frequency energy contained in the shocks of blast waves out to distances of several hundred meters from a blast event. At 100 m, the measured sound exposure level was 75-95 dB re (20 µPa)2s in the bat hearing range of 10–100 kHz. At 50 m the sound exposure level above 50 kHz was higher than at 25 m, indicating an increase in high-frequency energy due to nonlinearity despite spherical spreading and absorption eﬀects.

2. The measurement system bandwidth is an important factor to consider when measuring blast waves. Microphones with suﬃcient high-frequency response are needed to resolve the shock structure. In particular, a microphone with high-frequency capability above 300 kHz is necessary to resolve a blast rise time of 3 µs. One such microphone is the wide-band microphone introduced in Sec. 3.3.2, which measured blast rise times as short as 3 µs.

3. Microphones oriented at 0◦ incidence from the blast should be mounted in microphone baﬄes to eliminate diﬀraction artifacts on the rise portion of the measured waveform, therefore allowing for a more accurate measurement of the blast rise time.

4. Modeling of blast waves at medium distances (≈ 25–100 m) requires inclusion of molecular vibrational relaxation absorption. Including only thermoviscous absorption results in underprediction of shock rise times.

5. For the meteorological conditions tested in the models, a higher relative humidity of 62% resulted in shorter rise time predictions than for an atmosphere with a lower 38% relative humidity. This eﬀect occurs because molecular relaxation times are humidity-dependent. 97

6. Neither model, in its current implementation, can predict accurately the true rise times of blast waves at medium distances (≈ 25–100 m). The JH model, however, is useful for examining the effects of relative humidity, shock overpressure, and different absorption mechanisms on the structure of the shock. Similarly, the Anderson model is useful for predicting the effects of different propagation parameters and additionally accounts for the shape of the entire waveform.

7. Both the JH and Anderson models predict shorter rise times than what were measured. Atmospheric turbulence present in the measurement, but not included in the models, is likely responsible for increasing the measured rise times, at least at longer distances (≈ 100 m).

8.2 Suggestions for Future Work

The measurement and modeling of blast waves is an area of research that requires further understanding. Several studies merit additional attention, including improvements to the measurement system as well as the numerical models. Additional measurements with wide-band microphones at each measurement site are suggested, and additional sites at longer ranges could be employed again. An additional direct comparison of unbaffled, baffled, and wide-band microphones should be performed, with all sensors at the same distance. Uniform cable lengths should be utilized to eliminate the possibility of bias in the measurements. An even higher sampling rate could be applied to supply a finely-sampled shock as an input to computational models. Moreover, detailed meteorological measurements should be taken to characterize atmospheric turbulence accurately. The Anderson model could be refined to include atmospheric turbulence, based on the detailed meteorological measurements. For example, following a method described by Blanc-Benon et al. [57], realizations of turbulent fields are generated, rays through the turbulent fields are calculated using linear geometrical acoustics (ray tracing), the Anderson model is applied to each eigenray, and finally the eigenrays are combined to predict the received waveform. Note that this method is not free of approximations either, but predictions have been shown to agree with 98 scale-model experimental studies of nonlinear propagation through turbulence. In addition, other atmospheric effects and ground effects could be investigated. Modification of the Anderson model to incorporate the Nonuniform Discrete Fourier Transform [107] would allow for nonuniform time samples. This would eliminate the need for resampling after the application of nonlinear distortion, thereby eliminating resampling errors. Equally important, the number of points on the shock would not decrease as the wave steepens. Another area of interest would involve analyzing the transient blast waveforms with wavelet transforms [90]. Although similar to a spectrogram, wavelet analysis does not sacrifice frequency resolution for time resolution. Accordingly, wavelet analysis would offer more information about the frequency content of the wave as a function of time. Finally, the bat audiogram and threshold shift data by Brittan-Powell et al. [106] could be combined with blast models scaled for firing of different weapons to determine the region of safe operation for each. Appendix

A Photographs from Blast Experiments

This appendix includes a microphone baﬄe diagram, photographs from several of the blast experiments, and a video of one of the blasts.

A.1 Microphone Baﬄe Diagram

Figure A.1. Diagram of front and side of microphone baﬄe (not to scale). Designed by Marston [81]. 1 in = 2.54 cm. 100

A.2 Photographs

Figure A.2. Front view of microphone baﬄe.

Figure A.3. Assembled microphone baﬄe. 101

Figure A.4. Microphone baﬄe plug and backplate.

Figure A.5. Microphone baﬄe with microphone. 102

Figure A.6. August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater.

Figure A.7. August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater. 103

Figure A.8. August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater.

Figure A.9. August 2004. Experiment site and microphone setup. Photo courtesy of L. L. Pater. 104

Figure A.10. August 2004. Microphone setup. Photo courtesy of L. L. Pater.

Figure A.11. August 2004. C-4 explosive. Photo courtesy of L. L. Pater. 105

Figure A.12. August 2004. Bat cage mounted with microphones. Photo courtesy of L. L. Pater.

Figure A.13. August 2004. Bat cage mounted with microphones. Photo courtesy of L. L. Pater. 106

Figure A.14. April 2005. Microphone setup. Photo courtesy of L. L. Pater.

Figure A.15. April 2005. Microphone setup. Photo courtesy of L. L. Pater. 107

Figure A.16. April 2005. Bat cage with bat. Photo courtesy of L. L. Pater.

Figure A.17. August 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater. 108

Figure A.18. August 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater.

Figure A.19. August 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater. 109

Figure A.20. August 2005. C-4 explosive. Photo courtesy of L. L. Pater.

Figure A.21. August 2005. C-4 explosive. Photo courtesy of L. L. Pater. 110

Figure A.22. August 2005. C-4 explosive. Photo courtesy of L. L. Pater.

Figure A.23. August 2005. C-4 explosive. Photo courtesy of L. L. Pater. 111

Figure A.24. November 2005. B & K power supply. Photo courtesy of L. L. Pater.

Figure A.25. November 2005. B & K power supply. Photo courtesy of L. L. Pater. 112

Figure A.26. November 2005. B & K power supply. Photo courtesy of L. L. Pater.

Figure A.27. November 2005. Closeup of microphone baﬄe. Photo courtesy of L. L. Pater. 113

Figure A.28. November 2005. Closeup of microphone baﬄe. Photo courtesy of L. L. Pater.

Figure A.29. November 2005. Closeup of microphone baﬄe. Photo courtesy of L. L. Pater. 114

Figure A.30. November 2005. Wide-band microphone and B & K power supply. Photo courtesy of L. L. Pater.

Figure A.31. November 2005. Wide-band microphone. Photo courtesy of L. L. Pater. 115

Figure A.32. November 2005. Wide-band microphone. Photo courtesy of L. L. Pater.

Figure A.33. November 2005. Wide-band and baﬄed microphones. Photo courtesy of L. L. Pater. 116

Figure A.34. November 2005. Wide-band and baﬄed microphones. Photo courtesy of L. L. Pater.

Figure A.35. November 2005. Wide-band and baﬄed microphones. Photo courtesy of L. L. Pater. 117

Figure A.36. November 2005. Experiment site and microphone setup. Photo courtesy of L. L. Pater.

Figure A.37. November 2005. Mounted baﬄed microphone. Photo courtesy of L. L. Pater. 118

Figure A.38. November 2005. Mounted baﬄed microphone. Photo courtesy of L. L. Pater.

Figure A.39. November 2005. C-4 explosive. Photo courtesy of L. L. Pater. 119

A.3 Blast Video

Figure A.40. August 2004 blast video. Video courtesy of L. L. Pater. Appendix

B Johnson and Hammerton Model MATLAB Code

B.1 Introduction

The Johnson and Hammerton (JH) numerical model was implemented in MAT- LAB, and the function is provided in Sec. B.2 for reference. The function is called by the following command,

[time,pdim,trise]=hammertonfn_numerical(TC,rh,ps,psh,blast,range); where there are 6 input parameters and 3 output variables. The input parameters appear in parentheses, where TC is the temperature in ◦C, rh is the relative humidity in %, ps is the atmospheric pressure in atm, psh is the shock overpressure in Pa, blast is the blast number, and range is the range in meters. The output variables appear in square brackets, where time is the time vector in s, pdim is the pressure vector in Pa, and trise is the rise time in s. The function begins by calculating atmospheric parameters based on the input parameters. Next, relaxation frequencies are calculated by a function call to absorption.m, which is included in Sec. C.2.2. Then the low-frequency equilibrium speed of sound and sound speed increments due to relaxation are calculated. Non-dimensional parameters needed to solve the JH model equation are calculated next. The root of Eq. 5.13 (Eq. 30 in Ref. [63]), lambda_1, is calculated using the MATLAB function fzero. This function requires an initial guess, fguess, which is found by evaluating Eq. 5.13 for a range of λ and calculating the minimum of the absolute value of the function. 121

Now the Runge-Kutta solution of Eq. 5.1 (Eq. 9 in Ref. [63]) begins. The step size, equation coefficients, solution range, and initial conditions are defined before beginning the Runge-Kutta loop, which calls the function myode.m (see Sec. B.2.2). This subroutine implements the system of 3 first-order ODEs in Eq. 5.8. The solution p is plotted as a function of xi. Finally, the results are dimensionalized, and the rise time is calculated.

B.2 MATLAB Code

B.2.1 JH Model Program function [time,pdim,trise]=... hammertonfn_numerical(TC,rh,ps,psh,blast,range) %Alexandra Loubeau %hammertonfn_numerical.m %Created 1/10/2006 %Last Modified 11/1/2006 %Perform Johnson/Hammerton numerical analysis for blast wave %Wave Motion 38: 229-240, 2003 %input: atmospheric parameters % (temp in C, relative humidity, atmospheric pressure in atm), %shock overpressure in Pa, blast number, and range in meters %output: dimensional pressure vs. time; trise

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Atmospheric Parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %fluid is dry air at 20 degrees C %source frequency f=33.33; %needed for absorption routine to run %ambient density rho=1.21; %shear viscosity mu=1.85*10^-5; 122

%bulk viscosity, deduced from acoustic absorption data by Greenspan %(Pierce p.553) mu_b=0.6*mu; %ratio of specific heats for air, =cp/cv gamma=1.402; %Prandtl number, =cp*mu/kappa Pr=0.710; %coefficient of nonlinearity beta=0.5*(gamma+1); %diffusivity, factor of 2 is for Kang/Pierce and JH diffusivity=1/(2*rho)*(4*mu/3+mu_b+mu*(gamma-1)/Pr); %reference pressure in Pa pref=20e-6;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute small signal speed of sound as a function of %temperature and humidity %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %temperature in K T=TC+273.15; %FrN and FrO = scaled relaxation frequencies %h = molar concentration of water vapor in atmosphere [FrN,FrO,alpha,disp,h]=absorption(T,rh,f,ps); %specific gas constant, J/(kg K) r=287.06; %small signal sound speed (frozen in high-freq limit) c=sqrt(gamma*r*T)*(1+0.16*h/100); %0

%relaxation frequencies fr_N=ps*FrN; fr_O=ps*FrO; %relaxation times t_N=1/(2*pi*fr_N); t_O=1/(2*pi*fr_O); %fraction of all molecules that are of species nu 123 n_Nratio=0.78084; n_Oratio=0.20946; %characteristic molecular-vibration temperatures, K Tstar_N=3352; Tstar_O=2239.1; %equilibrium temperature T_N=T; T_O=T; %specific heat coefficient at constant pressure c_p=1.01e3; %specific heat associated with internal vibrations of nu-type molecules c_vN=n_Nratio*r*(Tstar_N/T_N)^2*exp(-Tstar_N/T_N); c_vO=n_Oratio*r*(Tstar_O/T_O)^2*exp(-Tstar_O/T_O); %maximum absorption per wavelength associated with relaxation process % %Pierce p.558, typical values, dimensionless % alphalambda_N=0.0002; % alphalambda_O=0.0011; alphalambda_N=pi*(gamma-1)*c_vN/(2*c_p); alphalambda_O=pi*(gamma-1)*c_vO/(2*c_p); %low frequency equilibrium sound speed c0=c/(1+(alphalambda_N+alphalambda_O)/pi); %small-signal sound-speed increment, Pierce p.587 c_N=c*alphalambda_N/pi; c_O=c*alphalambda_O/pi;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %JH non-dimensional parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Psh=psh*beta/(c0^2*rho); V=0.5*Psh+1; tref=V*t_N; deltaN=c_N/c0/Psh; deltaO=c_O/c0/Psh; tauN=V*t_N/tref; tauO=V*t_O/tref; 124 d=diffusivity/(tref*c0^2)/Psh;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Find root of Equation 30 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Plot to guess the root lambda=[0:1e3:1e5]; thirty=d*lambda+deltaN*tauN*lambda./(1+tauN*lambda)+deltaO*tauO*... lambda./(1+tauO*lambda)-0.5; figure; plot(lambda,thirty); xlabel(’\lambda’); ylabel(’Eq. 30’);

%Guess root based on plot [minthirty,iminthirty]=min(abs(thirty)); fguess=lambda(iminthirty);

%Find root of Equation 30 f=@(lambda)d*lambda+deltaN*tauN*lambda./(1+tauN*lambda)+deltaO*tauO*... lambda./(1+tauO*lambda)-0.5; lambda_1=fzero(f,fguess);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Solve ODE Equation 9 %Fourth-order Runge-Kutta %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %step size h=-0.00001; %initial value at xi_0 epsilon=0.0001;

%DE coefficients A=d*tauN*tauO; B=d*(tauN+tauO)+tauN*tauO*(deltaN+deltaO-0.5); 125

C=0.5*(tauN+tauO)-(deltaN*tauN+deltaO*tauO+d); D=0.5; E=tauN*tauO; F=tauN+tauO;

%range xi=[0:h:-0.1];

%"initial" conditions at xi_0 p(:,1)=[epsilon;-lambda_1*epsilon;(lambda_1^2)*epsilon]; for i=1:length(xi)-1 k1=myode(xi(i),p(:,i),A,B,C,D,E,F); k2=myode(xi(i)+h/2,p(:,i)+k1.*h/2,A,B,C,D,E,F); k3=myode(xi(i)+h/2,p(:,i)+k2.*h/2,A,B,C,D,E,F); k4=myode(xi(i)+h,p(:,i)+k3.*h,A,B,C,D,E,F); p(:,i+1)=p(:,i)+(k1+2.*k2+2.*k3+k4).*h./6; end figure; plot(xi,p(1,:)); title(’Fourth-order Runge-Kutta solution for Eq. 9’); xlabel(’\xi’); ylabel(’solution p’); p=p(1,:);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Dimensionalize and compute rise time %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x=0; %dimensional time time=(x/(c0*tref)-xi)*tref/V; %dimensional pressure pdim=p*psh; 126

%Overpressure maximum blastmax=psh;

%rise time %time for amplitude to rise 10%-90% of peak for n=1:length(pdim), if pdim(n)>=blastmax*0.1 n10=n; break; end end for n=1:length(pdim), if pdim(n)>=blastmax*0.9 n90=n; trise=time(n90)-time(n10); break; end end

B.2.2 ODE Subroutine

%Alexandra Loubeau %myode.m %Subroutine of system of 3 first-order equations for JH model %Called by hammertonfn_numerical.m function dpdxi = myode(xi,p,A,B,C,D,E,F) dpdxi = [p(2);p(3);... (B+E*p(1))/A*p(3)+((C-F*p(1))/A+E/A*p(2))*p(2)-D/A*p(1)*(1-p(1))]; Appendix

C Anderson Model MATLAB Code

C.1 Introduction

The Anderson algorithm was implemented in MATLAB, and the code is provided in Sec. C.2. Included are the main program anderson_propagation.m and supporting subroutines, as well as an example plotting program. The main program, anderson_propagation.m, begins by loading an input waveform. The user must supply the .mat blast data file containing pressure and time vectors and select it by browsing when the dialog box “Open Input Blast Data File” appears. To speed up FFT processing, the data is then padded with zeros to make the length a power of 2. Atmospheric parameters are calculated next, and the user must select a .mat meteorological data file when the dialog box “Open Input Meteorological Data File” appears. This data file must contain the variables TC, rh, and ps, representing temperature in ◦C, relative humidity in %, and atmospheric pressure in atm, respectively. Next, relaxation frequencies, absorption, and dispersion are calculated by a function call to absorption.m, which is included in Sec. C.2.2. Then the complex absorption and low-frequency equilibrium speed of sound are calculated. Before starting the propagation routine, the step size factor (η) is set to 0.01, the output ranges are defined, and the output pressure array is initialized. The propagation loop, which runs until reaching the final output range, starts by calculating the range step size dr and incrementing the range position. The spherical spreading factor rfact is calculated, and the nonlinear time distortion based on 128 the Earnshaw solution is calculated. After transforming to the frequency domain, spherical spreading and complex absorption are applied, and then the data is transformed back into the time domain. Nonlinear distortion is applied by the function rsamp, which resamples the pressure on the distorted time scale back onto a uniform time scale. The pressure array is saved, and the loop begins again. After saving data at the final range position, the pressure waveforms are plotted and saved in a .mat file. The example plotting program, anderson_plot.m, loads the saved .mat file, calculates the rise times via a function call to trisecalc (see Sec. C.2.4), plots the 1 rise portion of the waveforms, computes the sound exposure level LE in 3 -octave bands via a function call to thirdoctaveLEfn.m (see Sec. C.2.5), and plots the

LE.

C.2 MATLAB Code

C.2.1 Main Propagation Code

%Alexandra Loubeau %anderson_propagation.m %Version 1.2 %Created 2/18/2005 %Last Modified 11/1/2006 %Propagate blast wave using Anderson algorithm %Input 25 m data and propagate to 200 m close all; clear all; clc; tic; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Load Input Data %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %load calibrated blast wave data (in Pa) 129

%pressure and time arrays [fname,dirpath]=uigetfile(’*.mat’,’Open Input Blast Data File’); myfile=[dirpath fname]; load(myfile); cd(dirpath);

%file parameters dt=t(2)-t(1); fs=1/dt; N=length(x);

%for faster FFTs, pad x to be a power of 2 Npad=2^nextpow2(N); x=[zeros((Npad-N)/2,1);x;zeros((Npad-N)/2,1)]; t=[(min(t)-(Npad-N)/2*dt):dt:(min(t)+(N-1+(Npad-N)/2)*dt)]’; N=Npad; df=fs/N; f=[0:df:fs/2];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Atmospheric Parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %ambient density rho=1.21; %shear viscosity mu=1.85*10^-5; %bulk viscosity, deduced from acoustic absorption data by Greenspan %(Pierce p.553) mu_b=0.6*mu; %ratio of specific heats for air, =cp/cv gamma=1.402; %Prandtl number, =cp*mu/kappa Pr=0.710; %coefficient of nonlinearity 130 beta=0.5*(gamma+1); %reference pressure in Pa pref=20e-6;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Load meteorological data %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TC = temperature in degrees Celsius %rh = relative humidity in % %ps = atmospheric pressure in atm [fname,dirpath]=... uigetfile(’*.mat’,’Open Input Meteorological Data File’); myfile=[dirpath fname]; load(myfile); %temperature in Kelvin T=TC+273.15;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute small signal speed of sound as a function of %temperature and humidity %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %FrN and FrO = scaled relaxation frequencies %alpha = absorption %disp = dispersion %h = molar concentration of water vapor in atmosphere [FrN,FrO,alpha,disp,h]=absorption(T,rh,f,ps); %complex absorption alphac=alpha-j*disp; %specific gas constant, J/(kg K) r=287.06; %small signal sound speed (frozen in high-freq limit) cbasic=sqrt(gamma*r*T); c=sqrt(gamma*r*T)*(1+0.16*h/100); %0

%relaxation frequencies 131 fr_N=ps*FrN; fr_O=ps*FrO; %dimensionless theta_v=w*t_v theta_N=f/fr_N; theta_O=f/fr_O; %fraction of all molecules that are of species nu n_Nratio=0.78084; n_Oratio=0.20946; %characteristic molecular-vibration temperatures, K Tstar_N=3352; Tstar_O=2239.1; %equilibrium temperature T_N=T; T_O=T; %specific heat coefficient at constant pressure c_p=1.01e3; %specific heat associated with internal vibrations of nu-type molecules c_vN=n_Nratio*r*(Tstar_N/T_N)^2*exp(-Tstar_N/T_N); c_vO=n_Oratio*r*(Tstar_O/T_O)^2*exp(-Tstar_O/T_O); %maximum absorption per wavelength associated with relaxation process % %Pierce p.558, typical values, dimensionless % alphalambda_N=0.0002; % alphalambda_O=0.0011; alphalambda_N=pi*(gamma-1)*c_vN/(2*c_p); alphalambda_O=pi*(gamma-1)*c_vO/(2*c_p); %low frequency equilibrium sound speed c0=c/(1+(alphalambda_N+alphalambda_O)/pi); %small-signal sound-speed increment, Pierce p.587 c_N=c*alphalambda_N/pi; c_O=c*alphalambda_O/pi;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Definition of Range Steps %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %initial distance 132 r(1)=25; %step fraction of local shock formation distance stepsizefrac=0.01;

%distances to save rsave=[25;50;100;200];

%create output pressure array xsave=zeros(length(x),length(rsave)); %pressure at initial distance xsave(:,1)=x;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Nonlinear Propagation Code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% savind=2; n=1; while r(n)

%calculate new range position %max step size for single-valued waveform (shock formation distance) drmax=rho*c0^3/(beta*max(diff(x)/dt)); dr=stepsizefrac*drmax; r(n)=r(n-1)+dr;

%make sure we hit rsave exactly if r(n)>=rsave(savind) r(n)=rsave(savind); dr=r(n)-r(n-1); end

%spherical spreading factor rfact=r(n-1)/r(n); 133

%distortion based on pre-transform waveform, Earnshaw solution tdistort=t-beta*dr*x/(rho*c0^3);

%Absorption %FFT to frequency domain P=fft(x,N); %apply spreading/complex absorption, single-sided Pss=rfact*exp(-alphac.’*dr).*P(1:N/2+1); %wrap about nyquist P=[Pss;conj(Pss(N/2:-1:2))];

%IFFT back to time domain x=real(ifft(P,N));

%repin endpoints (for finite interpolation) x(1)=0; x(N)=0;

%resample, apply nonlinearity xnew=rsamp(tdistort,x,t);

%update x x=xnew;

%save if r(n)==rsave(savind) xsave(:,savind)=x; savind=savind+1; end end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Plot and Save %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 134 plot(t,xsave); xlabel(’\tau (s)’); ylabel(’Pressure (Pa)’); legend(’25 m’,’50 m’,’100 m’,’200 m’); save([’anderson_nov2005_’,datestr(now,’mmddyyyy’),’.mat’],’t’,’xsave’); toc;

C.2.2 Absorption Subroutine

%Alexandra Loubeau %absorption.m %Created 4/14/2004 %Last modified 11/1/2006 %Subroutine to compute absorption and dispersion coefficients %using molecular relaxation %Adapted from code by Brian Tuttle, Penn State %Converted to MATLAB from original FORTRAN %SUBROUTINE absorption(alpha_np, alpha_dB, T, hr, f, p) %Called by anderson_propagation.m function [FrN,FrO,alpha_np,disp,h]=absorption(T,hr,f,ps)

T0 = 293.15; T01 = 273.16;

%Calculate temperature ratios ToverT01 = T / T01; T01overT = T01 / T; ToverT0 = T / T0; T0overT = T0 / T;

%Calculate the saturation vapor pressure per 1.0 atm. psat = 10^( 10.79586 * (1-T01overT) - 5.02808 * log10(ToverT01)... + 1.50474E-4 * (1-10^(-8.29692*(ToverT01-1)))... 135

- 4.2873E-4 * (1-10^(-4.76955*(T01overT-1))) - 2.2195983 ); h = hr * psat / ps;

%Calculate scaled relaxation frequency for Oxygen %(div. by 1.0 atm omitted) FrO = ( 24 + 4.04E4 * h * (h + 0.02) / (h + 0.391)); %oxygen relaxation frequency frO=ps*FrO;

%Calculate scaled relaxation frequency for Nitrogen %(div. by 1.0 atm omitted) FrN = T0overT^0.5 * ( 9.0 + 280.0*h*exp(-4.17*(T0overT^(1/3) - 1))); %nitrogen relaxation frequency frN=ps*FrN;

%Calculate scaled frequency squared F2 = (f./ ps).^2;

%Calculate absorption in nepers/m %(division by 1.0 atm is omitted) alpha_np = (1.84E-11 * ToverT0^0.5 +... ToverT0^(-2.5) * (0.01275 * exp(-2239.1/T)./(FrO+F2./FrO) +... 0.1068 * exp(-3352.0/T)./(FrN+F2./FrN))) * ps.*F2;

%Absorption due to nitrogen relaxation alphaN=(ToverT0^(-2.5) *... (0.1068 * exp(-3352.0/T)./(FrN+F2./FrN))) * ps.*F2;

%Absorption due to oxygen relaxation alphaO=(ToverT0^(-2.5) *... (0.01275 * exp(-2239.1/T)./(FrO+F2./FrO))) * ps.*F2;

%Calculate dispersion disp=f.*alphaN./frN+f.*alphaO./frO; 136

C.2.3 Example Plotting Program

%Alexandra Loubeau %anderson_plot.m %Version 1.1 %Created 1/10/2005 %Last Modified 10/6/2006 %Load waveforms from Anderson code (anderson_propagation.m) %Compute peak pressure, rise time, and frequency content close all; clear all; clc;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Load Anderson blast wave data (in Pa) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Pressure and time arrays [fname,dirpath]=uigetfile(’*.mat’,’Open Anderson Prediction Data File’); myfile=[dirpath fname]; load(myfile); cd(dirpath); time=t; x25=xsave(:,1); x50=xsave(:,2); x100=xsave(:,3); x200=xsave(:,4);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Calculate rise time %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [trise25,blastmax25,blastmaxindex25,tbegin25,time25,tinterp25,... n10_25,n90_25]=trisecalc(time,x25); 137

[trise50,blastmax50,blastmaxindex50,tbegin50,time50,tinterp50,... n10_50,n90_50]=trisecalc(time,x50); [trise100,blastmax100,blastmaxindex100,tbegin100,time100,tinterp100,... n10_100,n90_100]=trisecalc(time,x100); [trise200,blastmax200,blastmaxindex200,tbegin200,time200,tinterp200,... n10_200,n90_200]=trisecalc(time,x200);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Plot rise portion of waveforms %Display peak pressure and rise time %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Plot setup set(0,’DefaultAxesFontSize’,14); set(0,’DefaultAxesFontName’,’Times New Roman’); set(0,’DefaultLineLineWidth’,1.3); set(0,’DefaultLineMarkerSize’,8); set(0,’DefaultFigurePaperOrientation’,’landscape’); figure; subplot(221),plot(time25*1e6,x25) hold on; plot(time(blastmaxindex25)*1e6-time(tbegin25)*1e6,blastmax25,’k*’) hold on; plot(tinterp25(n10_25)*1e6-time(tbegin25)*1e6,0.1*blastmax25,’k*’) hold on; plot(tinterp25(n90_25)*1e6-time(tbegin25)*1e6,0.9*blastmax25,’k*’) axis([-2 50 -50 blastmax25+150]); title(’25 m’); ylabel(’Pressure (Pa)’); h=text(9,500,{[’$\Delta p$=’,num2str(blastmax25,’%1.1f’),’$\,$Pa’],... [’Measured $t_{\mathrm{rise}}$=’,num2str(trise25*1e6,’%1.1f’),... ’$\,\mu $s’]}); set(h,’Interpreter’,’latex’,’FontName’,’times new roman’,’FontSize’,14); subplot(222),plot(time50*1e6,x50) 138 hold on; plot(time(blastmaxindex50)*1e6-time(tbegin50)*1e6,blastmax50,’k*’) hold on; plot(tinterp50(n10_50)*1e6-time(tbegin50)*1e6,0.1*blastmax50,’k*’) hold on; plot(tinterp50(n90_50)*1e6-time(tbegin50)*1e6,0.9*blastmax50,’k*’) axis([-2 50 -50 blastmax50+120]); title(’50 m’); h=text(9,225,{[’$\Delta p$=’,num2str(blastmax50,’%1.1f’),’$\,$Pa’],... [’Predicted $t_{\mathrm{rise}}$=’,num2str(trise50*1e6,’%1.1f’),... ’$\,\mu $s’]}); set(h,’Interpreter’,’latex’,’FontName’,’times new roman’,’FontSize’,14); subplot(223),plot(time100*1e6,x100) hold on; plot(time(blastmaxindex100)*1e6-time(tbegin100)*1e6,blastmax100,’k*’) hold on; plot(tinterp100(n10_100)*1e6-time(tbegin100)*1e6,0.1*blastmax100,’k*’) hold on; plot(tinterp100(n90_100)*1e6-time(tbegin100)*1e6,0.9*blastmax100,’k*’) axis([-2 50 -25 blastmax100+50]); title(’100 m’); xlabel(’Time (\mus)’); ylabel(’Pressure (Pa)’); h=text(9,75,{[’$\Delta p$=’,num2str(blastmax100,’%1.1f’),’$\,$Pa’],... [’Predicted $t_{\mathrm{rise}}$=’,num2str(trise100*1e6,’%1.1f’),... ’$\,\mu $s’]}); set(h,’Interpreter’,’latex’,’FontName’,’times new roman’,’FontSize’,14); subplot(224),plot(time200*1e6,x200) hold on; plot(time(blastmaxindex200)*1e6-time(tbegin200)*1e6,blastmax200,’k*’) hold on; plot(tinterp200(n10_200)*1e6-time(tbegin200)*1e6,0.1*blastmax200,’k*’) hold on; 139 plot(tinterp200(n90_200)*1e6-time(tbegin200)*1e6,0.9*blastmax200,’k*’) axis([-2 50 -10 blastmax200+20]); title(’200 m’); xlabel(’Time (\mus)’); h=text(9,30,{[’$\Delta p$=’,num2str(blastmax200,’%1.1f’),’$\,$Pa’],... [’Predicted $t_{\mathrm{rise}}$=’,num2str(trise200*1e6,’%1.1f’),... ’$\,\mu $s’]}); set(h,’Interpreter’,’latex’,’FontName’,’times new roman’,’FontSize’,14);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute Sound Exposure Level (LE) in 1/3-octave bands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [fc,LE_spectral25]=thirdoctaveLEfn(time,x25,10); [fc,LE_spectral50]=thirdoctaveLEfn(time,x50,10); [fc,LE_spectral100]=thirdoctaveLEfn(time,x100,10); [fc,LE_spectral200]=thirdoctaveLEfn(time,x200,10); figure; semilogx(fc,LE_spectral25,fc,LE_spectral50,fc,LE_spectral100,... fc,LE_spectral200); xlabel(’Frequency (Hz)’); ylabel(’1/3-Octave L_E (dB re (20 \muPa)^2s)’); legend(’25 m Measurement’,’50 m Prediction’,’100 m Prediction’,... ’200 m Prediction’);

C.2.4 Rise Time Calculation Subroutine

%Alexandra Loubeau %trisecalc.m %Created 5/30/2006 %Last Modified 8/31/2006 %Subroutine to compute rise time for blast wave %Called by anderson_plot.m function [trise,blastmax,blastmaxindex,tbegin,timeplot,tnew,n10,n90]=... trisecalc(time,x); 140

dt=time(2)-time(1); %time resolution blastmax=0; %initialize blastmax variable

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute peak pressure at shock %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for i=1:length(x)-1 if x(i)>100 if diff(x(i:i+1))<0 & diff(x(i+1:i+2))<0 if x(i)>=max(x)/2 blastmax=x(i); blastmaxindex=i; break; end i=i+1; end end end if blastmax==0 [blastmax,blastmaxindex]=max(x); end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute tbegin for shock arrival time %(when derivative goes positive) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for i=1:length(x)-1 if diff(x(i:i+1))>5 & diff(x(i+1:i+2))>5 tbegin=i; break; end end 141

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Interpolate for better resolution at the shock (10X) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t=time(1:blastmaxindex); tnew=[min(t):dt/10:max(t)]; y=x(1:blastmaxindex,1); yinterp=interp1(t,y,tnew);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute rise time %(time for amplitude to rise 10%-90% of peak) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for n=1:length(yinterp), if yinterp(n)>=blastmax*0.1 n10=n; break; end end for n=1:length(yinterp), if yinterp(n)>=blastmax*0.9 n90=n; break; end end trise=tnew(n90)-tnew(n10);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Set shock arrival to timeplot=0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% timeplot=time-time(tbegin);

C.2.5 LE Subroutine

%Alexandra Loubeau %thirdoctaveLEfn.m %Created 8/25/2006 142

%Last Modified 8/31/2006 %Subroutine to compute 1/3-octave band sound exposure level (L_E) %Called by anderson_plot.m function [fc,LE]=thirdoctaveLEfn(t,x,base);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Calculate time and frequency resolution %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dt=t(2)-t(1); %time resolution fs=1/dt; %sampling frequency T=length(x)/fs; %period, record length N=length(x); %number of points, =T/dt df=fs/N; %frequency resolution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute PSD %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %FFT xfft=fft(x); %single-sided FFT xfftss=xfft(1:N/2+1); %frequency vector for single-sided FFT f=[0:df:N/2*df]; %compute narrowband single-sided PSD PSD=2*dt^2.*abs(xfftss).^2./T; %PSD at f(1) and f(N/2+1) does not double %because there are no redundant components in the %Fourier transform at these frequencies PSD(1)=dt^2.*abs(xfftss(1)).^2./T; PSD(end)=dt^2.*abs(xfftss(end)).^2./T;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %1/3-octave band frequencies %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 143

%ANSI S1.6-1984 and ANSI S1.11-2004 %Bands 11 through 50 if base==2 G=2; %octave ratio elseif base==10 G=10^(3/10); %octave ratio end

%bandwidth designator for 1/3-octave bands b=3; %reference frequency = 1000 kHz fr=1000; %band number band=[11:50]; %number of bands nbands=length(band); %Exact midband frequencies, for b odd fc=G.^((band-30)./b).*fr; %Lower bandedge frequencies fl=G.^(-1/(2*b)).*fc; %Upper bandedge frequencies fu=G.^(1/(2*b)).*fc; %Bandedge frequencies fedge=[fl fu(end)];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Average PSD over 1/3-octave bands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Combine FFT frequencies and 1/3-octave bandedge frequencies and sort f_all=sort([f fl fu(end)]);

%Use linear interpolation to calculate PSD at f_all frequencies PSDinterp=interp1(f,PSD,f_all);

%Initialize array of mean-squared values for each band 144 mean_sq=zeros(nbands,1);

%Integrate PSD over each band for i=1:nbands %For each frequency band %indexes of f_all that are within band inds=find(f_all>=fedge(i) & f_all<=fedge(i+1)); for j=1:length(inds)-1 mean_sq(i)=... mean_sq(i)+0.5*(PSDinterp(inds(j+1))+PSDinterp(inds(j)))*... (f_all(inds(j+1))-f_all(inds(j))); end end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Compute L_E in 1/3-octave bands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %reference pressure p0=20e-6; %reference time t0=1; %compute L_E in 1/3-octave bands LE=10.*log10(mean_sq.*T./(p0^2*t0)); Appendix

D Microphone Calibrations

In Chapter 4, the calibration of microphones for the experiments was mentioned. The purpose of this appendix is to briefly describe the calibration procedure used for all the experiments. Calibration signals from a Brüel& Kjær 4228 1 pistonphone [108] were recorded for each 8 -in. microphone at the beginning and end of each measurement day. The pistonphone produces a 250-Hz sinusoid at 124 ±0.2 dB re 20 µPa. First, the calibration signals were high-pass filtered to eliminate any possible low-frequency noise. Next, the root-mean-square (rms) voltage

Vrms for each calibration signal was calculated, and the calibration factors were calculated as p 10124/20 M = 0 , (D.1) Vrms where M is the calibration factor in units of Parms/Vrms and p0 is the reference pressure 20 µPa. Finally, the blast measurement time records were converted from V to Pa by multiplying by the corresponding calibration factors. Uncertainty in the measured pressure levels is estimated by examining the pistonphone’s tolerance and the diﬀerence between calibration factors for the same microphone. The manufacturer’s tolerance for the pistonphone is ±0.2 dB at 124 dB re 20 µPa, or 31.7 ±0.7 Parms. This translates into an error of approximately 2.3%. For the ﬁnite-amplitude blast wave pressures, a 2.3% error at 25, 50, and 100 m indicates an error of approximately 76, 34, and 15 Pa, respectively. This indicates that measured pressure values should be reported to the nearest 10 Pa. For the November 2005 measurements, the nominal calibration factor was

1000 Parms/Vrms, which corresponds to the manufacturer’s nominal sensitivity of 1 mV/Pa [76]. It was found that most of the diﬀerences between calibration factors 146

were within 1–6 Parms/Vrms, or less than 1%. In a couple instances, the diﬀerence in calibration factors was larger, up to 36 Parms/Vrms, or 3.6%. These diﬀerences are reasonable when considering the pistonphone tolerance of 2.3%. Thus measured pressure values in the range 600–4000 Pa are reported in this thesis to the nearest 10 Pa with an uncertainty of approximately 2.3%. Bibliography

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Alexandra Loubeau

Alexandra Loubeau was born on September 16, 1976 in Miami, FL and was raised in the Dominican Republic and Ecuador before returning to Miami. In May of 1998, she graduated Summa Cum Laude with a Bachelor of Music degree in Music Engineering Technology and a minor in Electrical Engineering from the University of Miami in Coral Gables, FL. After graduation, she became a Television Systems Engineer working on high-definition television research, and then she joined Via- com as an RF Systems Engineer, preparing and filing FCC licensing applications for radio and television stations. In August of 2001, she began graduate study in the Graduate Program in Acoustics at The Pennsylvania State University. She received a Master of Science degree in Acoustics in December of 2003 for her research on reducing low frequency sound pressure variability in small rooms using a source with frequency-independent radiated power. She then began her doctoral studies in the same Acoustics program. Alexandra has accepted a position with the Laboratoire de Modélisationen Mécaniquein Paris, France, where she will research sonic boom propagation from supersonic vehicles flying at high speeds and high altitudes.