ThesisFinal2

The Scattering of Waves from Randomly Rough
Surfaces
Clay Stanek
Darwin College
and
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
A dissertation submitted to the University of Cambridge for the degree
of Doctor of Philosophy
17 June 2002

The Scattering of Waves from Randomly Rough Surfaces
ii
Preface
This is the presentation of my work towards a Ph.D. thesis conducted between
April 1999 and June 2002.
Chapter 5 represents the collaboration of my supervisor, Barry J. Uscinski, and
me during the period of December 1999 and December 2000. All the other
chapters in this volume contain original work carried out by myself and not done
in collaboration with another. When an already existing result is used, this is
stated clearly in the text. The graphic results in this project were produced by a
combination of the Matlab programming environment, Digital Fortran 90,
Microsoft Visual C++, and IPLab software packages.
Clay Stanek

iii
Acknowledgements
My wife Terry has been my biggest advocate and friend to me during the past
three years. This thesis belongs as much to her as to me. Barry Uscinski has
mentored me with enthusiasm and care, treating me like I was his own son.
Mark Spivak has given his time to me on many occasions and offered me
technical advice and personal friendship. This work could not have been
completed without the encouragement and expertise of Steve Bottone.
I would like to acknowledge my immediate family members and thank them for
their encouragement: Valeria, Frank, Jan, Sherry, Donna, June, Teeter, Ralph,
Janet, John P and Donna P.
Finally, gratitude is extended to the members of ANZUS. When professional
colleagues are such a large part of one’s life, they become extended family.
Within this family, let me mention specifically Philip Y and Janet B, Steve D,
Robert T, and Patrick M. Thanks for your patience with me during the last
several years.

iv
Abstract
This thesis investigates acoustic wave scattering from rough surfaces using the
paraxial wave equation in both differential and integral forms. The principal
physical mechanism is forward scatter. Applications include the propagation of
sound over rough terrain subject to varying weather conditions including
temperature, humidity, and wind. This is of interest to those in city planning,
airport planning, meteorology, and munitions testing to name a few. Still other
applications include the propagation of sound in the ocean in deep and shallow
water. In all these cases, the slope of the surface is assumed to be gentle but the
depth of the surface relative to the wavelength of the radiation can be large.
In the first third of the work, we develop numerical solutions to acoustic fields in
both Cartesian and cylindrical coordinate systems using the differential form of
the paraxial wave equation. In the Cartesian case, we extend the numerical
method to handle a varying refractive index profile over the surface and a
varying, complex reflection coefficient. In the cylindrical formulation, we
propagate a true point source and compare this to the Gaussian source of the
Cartesian case.
The bulk of the thesis uses the integral equation method to find an analytic
solution for the mean acoustic field for randomly rough surfaces with Gaussian
statistics. The method of Laplace transforms plays a pivotal role. A solution to
the one-surface problem in an isovelocity medium is derived and shown to agree
with accepted solutions in the limiting case of a flat surface. The effect of surface
roughness on the field is also characterized. Finally, the technique is generalized
to address the mean field for the rough waveguide, or two-surface problem.

Symbol List
v
The symbols defined here are the ones used throughout the thesis and have the
meaning described here unless stated otherwise. The order of presentation is
alphabetical with Roman before Greek, lower case before upper case, vectors
before scalars, normal before calligraphic, symbols without subscripts before
symbols with subscripts.
f< > ensemble average of f
β (1) Fourier shift operator (Chapter 2)
β (2) linear medium parameter (Chapter 9)
c speed of sound
0
c reference speed of sound; usually 350 meters per second
ˆC covariance estimate
δ Dirac delta function
D differential operator
E paraxial acoustic field
0
E initial acoustic field (Chapter 2)
, imI
E E image acoustic source (marching technique, Chapter 2)
, reR
E E real acoustic source (marching technique, Chapter 2)
tot
E< > ensemble average of total field in scaled coordinates
ε wind speed slope parameter in [sec-1
]
ε refractive index slope parameter in [m-1
]
( )0
, ,ˆinc
Zλ γε Laplace transform of incident acoustic field
( ), ,ˆs Zλ γε Laplace transform of acoustic field
( )ˆ ,F λ γ Laplace transform of ( )( ),
E
x S x
z
∂
′ ′< >
′∂
Φ power spectrum function
G Green function for parabolic wave equation
G Green function for full wave equation
( )ˆ ,G λ γ Laplace transform of surface parabolic Green function
( )ˆ , ,G Zλ γ Laplace transform of parabolic Green function of medium
ˆ
jk
G Laplace transform of two-surface parabolic Green function
γ surface roughness parameter

Symbol List (Con’t)
vi
Γ Gamma function
H Hankel function
ˆH Laplace transform of derivative of parabolic Green function of
medium
ˆ
jk
H Laplace transform of derivative of two-surface parabolic
Green function
η coordinate in direction normal to rough surface at each point
i 1−
J surface current
0
k primary wavenumber [m-1
]
L horizontal scaling length [m]
λ Laplace transform variable
0
λ primary wavelength [m]
n total refractive index, including constant and varying
components
n′ varying component of refractive index
0
n constant component of refractive index
N (1) number of mesh points in z direction
( ),N x z (2) refractive index definition with real and image media
ν frequency [s-1
]
( ),p x z acoustic pressure field, includes envelope and rapidly varying
component
P covariance matrix
Q cumulative distribution function
r radial curvilinear coordinate in the direction of propagation [m]
R reflection coefficient
ρ correlation coefficient
( )S x rough surface [m]
2
σ dispersion of rough surface [m2
]
T transmission coefficient
( ),u r θ full acoustic field in curvilinear coordinates (Chapter 3)

Symbol List (Con’t)
vii
θ polar coordinate [rad], [deg]
( )v z wind speed profile [m-sec-1
]
ˆV sample variance
x direction of propagation [m]
X scaled direction of propagation [m-m-1
]
ξ Fourier transform of acoustic field
w source width [m]
ω angular frequency [rad-sec-1
]
z vertical direction [m]
0
z source position above or below surface [m]
Z scaled vertical direction [m-m-1
]

Table of Contents
Page
viii
1 INTRODUCTION _________________________________________________ 1-1
2 CARTESIAN WAVE PROPAGATION OVER ROUGH SURFACES
WITH VARYING REFRACTIVE INDEX PROFILES __________________ 2-1
2.1 THE MARCHING TECHNIQUE............................................................................ 2-2
2.2 2-D PROPAGATION IN CARTESIAN COORDINATES............................................. 2-4
2.2.1 Finite Difference Method for Propagation ............................................ 2-5
2.2.2 Fourier Method for Propagation............................................................ 2-8
2.3 2-D PHASE MODULATION IN CARTESIAN COORDINATES................................ 2-12
2.4 EXAMPLES OF CARTESIAN PROPAGATION USING THE
MARCHING TECHNIQUE ................................................................................. 2-13
2.4.1 Free Space Propagation with Flat, Absorbing Surface........................ 2-13
2.4.2 Source at 20 m with Reflecting and Absorbing Surface ..................... 2-16
2.5 CASES INVOLVING ARBITRARY PROPAGATION MEDIAAND
ROUGH SURFACES.......................................................................................... 2-19
2.5.1 Comparing Numerical Results in Varying Medium to Theory........... 2-20
2.6 VARYING REFLECTION COEFFICIENT OVER THE TERRAIN............................... 2-22
2.7 EXTENSION OF MARCHING TECHNIQUE TO 3-D.............................................. 2-24
3 CURVILINEAR WAVE PROPAGATION OVER ROUGH SURFACES
WITH VARYING REFRACTIVE INDEX PROFILES __________________ 3-1
3.1 2-D PROPAGATION IN CURVILINEAR COORDINATES ......................................... 3-2
3.2 2-D MODULATION IN CURVILINEAR COORDINATES.......................................... 3-4
3.2.1 Mapping a Wind Velocity Profile in Cartesian Coordinates to
Curvilinear Coordinates......................................................................... 3-5
3.3 EXAMPLES OF CURVILINEAR PROPAGATION USING THE
MARCHING TECHNIQUE ................................................................................... 3-8
3.3.1 Curvilinear Solutions in Partial Form.................................................... 3-9
3.3.2 Curvilinear Solutions in Full Form...................................................... 3-13
4 COMPARISON OF GAUSSIAN SOURCE TO POINT SOURCE _________ 4-1
4.1 COMPARISON FOR ε = .00134 ........................................................................... 4-5
4.2 COMPARISON FOR ε = 2.58E-3.......................................................................... 4-7
4.3 COMPARISON FOR ε = 5.795E-3........................................................................ 4-9
4.4 COMPARISON FOR ε = 9.86E-3 ....................................................................... 4-12
5 SOLUTIONS TO THE SCATTERED FIELD USING THE
INTEGRAL EQUATION METHOD _________________________________ 5-1
5.1 INTRODUCTION TO THE INTEGRAL EQUATION................................................... 5-2
5.2 THE PARAXIAL POINT SOURCE AND GREEN FUNCTION .................................... 5-4
5.3 VOLTERRA EQUATIONS OF THE FIRST AND SECOND KIND ................................ 5-6
5.3.1 Pressure-Release Surface....................................................................... 5-6
5.3.2 Reflecting (Hard) Surface...................................................................... 5-7
5.4 AN INTEGRAL EQUATION INVOLVING THE FIRST MOMENT OR MEAN FIELD .... 5-7
5.4.1 Gaussian Random Process..................................................................... 5-9

Table of Contents (Con’t.)
Page
ix
5.4.2 The Ensemble Averages...................................................................... 5-10
5.4.3 Scaling the Ensemble-Averaged Equations......................................... 5-11
5.5 THE LAPLACE TRANSFORM TECHNIQUE......................................................... 5-12
5.5.1 Definition of the Laplace Transform................................................... 5-12
5.5.2 Relation of the Laplace Transform to the Fourier Transform ............. 5-13
5.5.3 Laplace Transform Solution for Mean Field with Volterra Equation
of the First Kind, Isovelocity Profile, and Rough Surface .................. 5-13
6 ELEMENTS OF THE SOLUTIONS TO ENSEMBLE-AVERAGED
INTEGRAL EQUATIONS USING THE LAPLACE TRANSFORM_______ 6-1
6.1 THE POINT SOURCE REPRESENTATION AND ITS LAPLACE TRANSFORM............ 6-1
6.1.1 Visualizing the Point Source and Its Laplace Transform...................... 6-3
6.1.2 Asymptotic Results................................................................................ 6-5
6.2 LAPLACE TRANSFORM OF ENSEMBLE-AVERAGED GREEN FUNCTION
FOR THE SURFACE ............................................................................................ 6-8
6.3 THE LAPLACE TRANSFORM FOR THE GREEN FUNCTION OF THE MEDIUM ...... 6-11
6.4 SOLUTION TO THE MEAN FIELD PROBLEM VIA LAPLACE TRANSFORMS ......... 6-12
6.4.1 The Solution in the Laplace Domain................................................... 6-12
6.5 THE SOLUTION IN THE SPATIAL DOMAIN ........................................................ 6-13
6.5.1 Example with Simple Poles................................................................. 6-15
6.5.2 Example with Branch Cuts.................................................................. 6-16
6.5.3 The Scattered Field Inverse Transform ............................................... 6-17
7 ACOUSTIC SCATTERING FROM A ROUGH SEA AND BOTTOM
SURFACE. THE MEAN FIELD BY THE INTEGRAL EQUATION
METHOD FOR SHALLOW WATER ________________________________ 7-1
7.1 WAVE PROPAGATION IN SHALLOW WATER: THE MEAN FIELD BETWEEN
TWO SURFACES................................................................................................ 7-1
7.2 BOTH SURFACES ARE PRESSURE-RELEASE SURFACES...................................... 7-3
7.3 ONE PRESSURE-RELEASE SURFACE AND ONE HARD SURFACE......................... 7-4
7.4 ENSEMBLE AVERAGING OF INTEGRAL EQUATIONS ........................................... 7-5
7.4.1 Previous Ensemble Averages Performed .............................................. 7-6
7.4.2 New Ensemble Averages to Perform .................................................... 7-8
7.4.3 Ensemble Averaging ( )( );
G
x x z S x
z
∂
′ ′< − − >
′∂
............................... 7-8
7.4.4 Ensemble Averaging ( ) ( )( ); jk
G
x x S x S x
z
∂
′ ′< − − >
′∂
when k j= . 7-9
7.4.5 Ensemble Averaging ( ) ( )( ); jk
G x x S x S x′ ′< − − > and
( ) ( )( ); jk
G
x x S x S x
z
∂
′ ′< − − >
′∂
when k j≠ .................................. 7-10
7.5 NOMENCLATURE ............................................................................................ 7-12
7.6 THE SCALED INTEGRAL EQUATIONS FOR THE MEAN FIELD............................ 7-13

Page
x
7.7 THE SOLUTION TO THE TWO-SURFACE MEAN FIELD IN THE
LAPLACE DOMAIN ......................................................................................... 7-16
7.7.1 Laplace Representation for One Hard and One Pressure-Release
Surface................................................................................................. 7-17
7.7.2 Additional Laplace Transforms to Perform......................................... 7-19
7.8 LOOKING AHEAD ........................................................................................... 7-21
8 INDEPENDENCE OF THE FIELD DERIVATIVE AT THE SURFACE
FROM THE GREEN FUNCTION AFTER SEVERAL CORRELATION
LENGTHS _______________________________________________________ 8-1
8.1 ENSEMBLE AVERAGING THE FIELD DERIVATIVE AND THE GREEN FUNCTION ... 8-2
8.1.1 Independence......................................................................................... 8-3
8.1.2 Expectation and Averaging ................................................................... 8-3
8.1.3 Independence and Correlation of Normal Variables............................. 8-5
8.2 DEDUCING THE CORRELATION COEFFICIENT BETWEEN RANDOM VARIABLES . 8-7
8.2.1 Transforming Correlated Random Variables to Uncorrelated Ones ..... 8-9
8.2.2 Estimating the Correlation Coefficient Between the Surface and
the Field Derivative............................................................................. 8-11
8.2.3 Estimating the Correlation Coefficient Between the Green Function
and the Field Derivative at the Surface ............................................... 8-15
8.2.4 Estimating the Correlation Coefficient Between the Surface and
the Field at One Correlation Length Above the Surface ..................... 8-16
8.2.5 Distribution of the Sample Correlation Coefficient ............................ 8-17
8.3 CONCLUSIONS REGARDING THE ENSEMBLE AVERAGING OF GREEN
FUNCTION AND FIELD DERIVATIVE ................................................................ 8-18
8.4 GENERATION OF ROUGH SURFACES WITH KNOWN STATISTICS....................... 8-19
9 FINDING ANALYTIC SOLUTIONS TO ACOUSTIC SCATTERING
FROM A ROUGH SEA AND BOTTOM SURFACE AND OTHER
FOLLOW-ON WORK _____________________________________________ 9-1
9.1 FINDING ANALYTIC SOLUTIONS TO THE TWO-SURFACE PROBLEM: TWO
PRESSURE-RELEASE SURFACES WITH 0γ = ................................................... 9-1
9.1.1 A Flat-Surface Solution via Eigenfunction Expansion ......................... 9-2
9.1.2 The Laplace Transform Approach to Two Pressure-Release
Surfaces with 0γ = ............................................................................. 9-3
9.2 LAPLACE TRANSFORM APPROACH FOR NON-ISOVELOCITY MEDIUMS ............. 9-7
9.3 NUMERICAL COMPUTATION OF INTEGRAL EQUATION SOLUTIONS.................. 9-11
9.3.1 Solutions Via Wavelet Transforms ..................................................... 9-11
10 CONCLUSIONS _________________________________________________ 10-1
10.1 RECOMMENDATIONS FOR FOLLOW-ON WORK RELATED TO CHAPTERS 2-4.... 10-2
10.2 RECOMMENDATIONS FOR FOLLOW-ON WORK RELATED TO CHAPTERS 5-9.... 10-4

Page
xi
11 APPENDICES ___________________________________________________ 11-1
11.1 APPENDIX A THE PARABOLIC WAVE EQUATION IN CARTESIAN
COORDINATES ................................................................................................ 11-2
11.2 APPENDIX B THE IMAGE METHOD FOR PARAXIAL WAVE PROPAGATION
WITH 1R = ................................................................................................... 11-4
11.3 APPENDIX C THE PARABOLIC WAVE EQUATION IN CYLINDRICAL
COORDINATES ................................................................................................ 11-7
11.4 APPENDIX D THE INTEGRAL EQUATION ......................................................... 11-9
11.5 APPENDIX E RELATION OF THE LAPLACE TRANSFORM TO THE FOURIER
TRANSFORM................................................................................................. 11-11
11.6 APPENDIX F ................................................................................................. 11-13
11.7 APPENDIX G................................................................................................. 11-15
11.8 APPENDIX H................................................................................................. 11-17
11.9 APPENDIX I .................................................................................................. 11-20
12 BIBLIOGRAPHY ________________________________________________ 12-1
ENDNOTES _____________________________________________________ 12-6

Table of Figures
Page
xii
Figure 1 The image method requires the sources to be symmetric about
the surface..................................................................................................... 2-4
Figure 2 Results from absorbing, flat surface with constant refractive index
and source at 0
10z = ................................................................................. 2-14
Figure 3 Numerical to analytical solution comparison for the source height
at 10 m and absorbing surface. ................................................................... 2-16
Figure 4 Reflecting and absorbing surfaces with source at 20 m. ............................ 2-17
Figure 5 Multiple slit interference demonstrated by moving source to 200 m
of the ground............................................................................................... 2-17
Figure 6 Plot of intensity with distance from source for 0z = meters.................... 2-18
Figure 7 Acoustic intensity for flat, hard surface with wind speed profile of
( ) , .0105v z zε ε= = and wind direction of 90°................................................ 2-20
Figure 8 Acoustic intensity shown for isovelocity medium and rough surface........ 2-21
Figure 9 Flat, hard surface with constant reflection coefficient throughout
propagation. ................................................................................................ 2-23
Figure 10 Flat, mostly hard surface with reflection coefficient change 78%
through propagation.................................................................................... 2-23
Figure 11 Flat, mostly hard surface with change to absorbing surface 78%
through propagation.................................................................................... 2-23
Figure 12 Mild wind profile and resultant refractive index, .00258ε = ..................... 3-6
Figure 13 Strong wind profile and resultant refractive index with .00989ε = ........... 3-6
Figure 14 Linear wind profile results in θ asymmetry................................................. 3-8
Figure 15 Field intensity with progressively stronger refractive index profiles......... 3-10
Figure 16 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 1. ............. 3-11
Figure 17 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 0. ............ 3-11
Figure 18 Full curvilinear solutions with ε increasing from left to right,
top to bottom............................................................................................... 3-14
Figure 19 The field from a point source (left) and from a Gaussian source (right)
are shown in the double-half plane. The surface is flat and reflecting
and the source is positioned on the surface. ................................................. 4-1
Figure 20 Numerical results of curvilinear and Cartesian field intensities for
case ε = 1.34e-3 ............................................................................................ 4-5
Figure 21 Comparison of field intensities along ground for case ε = 1.34e-3.............. 4-6
Figure 22 Comparison of field intensities at 500 m for case ε = 1.34e-3. .................... 4-6
case ε =2.584e-3. .......................................................................................... 4-8
Figure 24 Comparison of field intensities along ground for case ε = 2.58e-3.............. 4-8
Figure 25 Comparison of field intensities at 500 m for case ε = 2.58e-3. .................... 4-9
case ε = 5.795-3.......................................................................................... 4-10
Figure 27 Comparison of field intensities along ground for case ε = 5.795e-3.......... 4-11
Figure 28 Comparison of field intensities at 500 m for case ε = 5.79e-3. .................. 4-11
case ε = 9.86e-3. ......................................................................................... 4-12

Table of Figures (Con’t.)
Page
xiii
Figure 32 The acoustic intensity along the ground in the curvilinear case is
compensated by
w
π
θ
.................................................................................... 4-14
Figure 33 The closed surface, s, includes the real surface, ( )S x , but excludes
the source at 0
z ............................................................................................. 5-3
Figure 34 Point source field, ( )0
, ,inc
E X Z γ , for several source positions................... 6-3
Figure 35 Point source field, ( )0
, ,inc
E X Z γ , for several source positions over
a smaller range 0
Z ........................................................................................ 6-3
Figure 36 Laplace transform integrand for point source at several source
positions with 1λ = and 2
1γ = ................................................................. 6-4
Figure 37 A plot of point source Laplace transform for several 0
Z and fixed 2
γ ....... 6-5
Figure 38 The Laplace transform of the point source agrees well with theory
for small and large λ . At the crossover 1λ ∼ , the real part of the
integral (Laplace transform) becomes negative, but close to zero. .............. 6-5
Figure 39 The complex error function exhibits Stokes phenomenon, which
implies different asymptotic representations depending on location
within the complex plane.............................................................................. 6-7
Figure 40 Exact expression and approximation as function of X . ............................ 6-10
Figure 41 Green function with exponential autocorrelation function......................... 6-10
Figure 42 Numeric versus analytic Laplace transform of surface Green function. .... 6-11
Figure 43 Several level curves for mean scattered field in Laplace domain
with fixed source depth, surface roughness, and desired depth.................. 6-13
Figure 44 A line integral in the complex plane with c right of all singularities. ....... 6-14
Figure 45 Bromwich contours closed to left and right in complex plane................... 6-14
Figure 46 Integration contour with branch points....................................................... 6-16
Figure 47 Searching for zeros of ( )ˆ ,G λ γ ................................................................... 6-18
Figure 48 Interpretation of the scattered field from a flat surface. ............................. 6-24
Figure 49 Total field as scattered off a flat surface for a source at 0
20Z = (left)
and 0
2Z = (right)...................................................................................... 6-25
Figure 50 Laplace inversion integrand real and imaginary parts above and
below the branch cut (corresponding to (6.68)).......................................... 6-26
Figure 51 Curves of ( ),sE X Z< >, the mean scattered field, as a function of
scaled range X at different scaled distances Z from a rough surface
with ACF (6.27) . A point source is situated at 0
2Z = . The effect of
increasing surface roughness is evident from the following different
values of surface roughness, 2
γ : 2
0.0,γ = [ ];
[ ]
2
1.0,γ = −−−− ; [ ]
2
5.0,γ = −⋅−⋅−⋅−⋅ . ................................. 6-27

Page
xiv
Figure 52 Effect of surface roughness on the fringe pattern of the mean total
field shown as ( ) ( )( )log , ,sinc
E X Z E X Z< > + < > . In all cases,
the source is at 0
2Z = . Red corresponds to higher field strengths.......... 6-28
Figure 53 The sea surface and bottom are represented along with an acoustic
source to the left............................................................................................ 7-2
Figure 54 Venn diagram explains possible combinations............................................. 8-6
Figure 55 Left: One realization of rough surface with 40 sample points marked
in red. First sample point value, 1
Y , is then subtracted from the
remaining 39 points to generate 2
Y at different separations. Right:
Correlation coefficient for 1
Y , 2
Y as a function of these separations. ........... 8-8
Figure 56 Gaussian probability density function in two variables. Uncorrelated
on left, correlated on right to 1
2
ρ = − with
2 1
2Y Y
σ σ= in both cases. ... 8-9
Figure 57 Field beneath reflecting rough surface for point source at –75 m
shown as Sound Pressure Level (related to decibels). The depth is
shown as negative here due to a plotting package technicality. We
still maintain a positive-down sense to our coordinate system................... 8-13
Figure 58 Instantiation of rough surface and corresponding field just above and
below the surface along with interpolated derivative at the surface........... 8-13
Figure 59 Family of 400 rough surfaces ( )S x with 0
5000, 100x dx= = ,
and 300nx = samples. If sample is in red, then the correlation
coefficient being nonzero is significant to 2.5%. Blue points on
the correlation coefficient plots imply we accept the hypothesis
that 0ρ = with 97.5% confidence. Here,
( )( ),E x S x
z
∂
′∂
is
correlated with ( )S x as a function of x .................................................... 8-15
Figure 60 Sample correlation coefficient between ( ) ( )( );G x x S x S x′ ′− −
and
( )( ),E x S x
z
′ ′∂
′∂
as a function of x x′− . Blue points on the
correlation coefficient plots imply we accept the hypothesis that
0ρ = with 97.5% confidence.................................................................... 8-16
Figure 61 Sample correlation coefficient between ( )( )0
, /E x S x L k+ and ( )S x
as a function of x . Blue points on the correlation coefficient plots
imply we accept the hypothesis that 0ρ = with 97.5% confidence. ........ 8-17
Figure 62 A uniform distribution mapped to a Rayleigh distribution using the
inverse cumulative distribution function. ................................................... 8-21
Figure 63 Real and imaginary random Fourier components and corresponding
surface......................................................................................................... 8-21
Figure 64 Unfiltered surface and three subsequent filtered surfaces with
different Gaussian correlation lengths........................................................ 8-24

Page
xv
Figure 65 Level curves of real part of Airy function exponent (left) and
exponent of equation (9.27) (right)............................................................... 9-9
Figure 66 Real and imaginary parts for Airy-like function. Different values
of λ correspond to different colored curves. β3
= 4.17e-8 (left)
β3
= 4.17e-6 (right)........................................................................................ 9-9
Figure 67 Example of wavelet transform matrix for DB2 (left), scaling and
wavelet functions for DB4 (right)............................................................... 9-13

The Scattering of Waves From Randomly Rough Surfaces
1-1
C h a p t e r 1
1 Introduction
From a propagation point of view, each encounter of light or sound with matter
can be viewed as an event where a wave interacts with an array of atoms. The
fact that both transverse and longitudinal fields share much in common in their
mathematical description is a tribute to the power of mathematical abstraction in
describing complex phenomena. For the electromagnetic wave, the journey of the
field through the matter determines the appearance of objects, the color of the
sky, the translucency of glass, and the reason snow is white and water clear. For
the acoustic wave, the sound generated from freeway and airport traffic and
propagated to a nearby village or the ability of whales to communicate over long
distances are described as well by this interaction with matter. The propagation
and scattering processes are fundamental.
Lord Rayleigh (1871) analyzed scattering in terms of molecular oscillators and
correctly concluded that the intensity of scattered light was proportional to the
fourth power of the wavelength of the light ( 4
λ
−
∼ ) in the upper atmosphere.
The red end of the spectrum is mostly undeviated whereas the blue, high-
frequency scattered light reaches the observer from many directions. Before this
work, it was widely believed that the sky was blue because of scattering from
dust particles. Today, Rayleigh’s treatment of the dipole is still a powerful tool
in understanding many aspects of scattering.
One rule of thumb in scattering states that the denser a substance through which
a field advances, the less the lateral scattering, and that applies to
electromagnetic propagation through much of the lower atmosphere.
M. Smoluchowski (1908) and A. Einstein (1910) independently provided the basic
ideas for the theory of this type of scattering as a result of the density
fluctuations on local scales. Their results are similar to those of Rayleigh [1].
In the Rayleigh theory, each molecule is independent and randomly arrayed in
space so that the phases of the secondary wavelets scattered off to the side have

1-2
no particular relationship to one another and no sustained pattern of interference.
This situation (such as a rarified gas like the upper atmosphere) occurs when the
spacing of the scatterers is roughly a wavelength or more. In the forward
direction, the scattered wavelets add constructively with each other.
In a denser medium, the scattered wavelets cannot be assumed to arrive at a
point P with random phases and interference will be important. Again,
scattered wavelets will interfere constructively in the forward direction, but
destructive interference dominates in all other directions and little or none of the
field ends up scattered laterally or backwards in a dense homogeneous medium.
Thus, in many acoustic and electromagnetic applications, the forward direction is
of prominent importance. Keeping this in mind, we will find great utility in
using the parabolic form of the wave equation. For volume propagation, this
form of the wave equation is natural for forward scattering. For interactions with
rough surfaces, the parabolic form of the wave equation limits us to the case of
forward scattering, when backscattering may occur in some situations.
Other effects of scattering include those that involve the irregularities of the
medium through which the waves propagate. The effect, termed scintillation,
explains (to name a few) the twinkling of stars, certain aspects of sonar operation
in the ocean, and the effect of turbulence on sensors including radar and
microwave devices. Scattering by random media is often undesirable, but can be
used to deduce properties about the medium itself. In this way, the intensity
patterns observed through an acoustic or electromagnetic measurement can be a
means of remote sensing. Deducing properties of the medium through indirect
measurement has been accomplished by Jakeman (1978) and Uscinski on several
occasions. Mostly recently, Uscinski used this technique to study vertical water
motion in the Greenland Sea [2].
Acoustic applications form the main area of study for this thesis, but many of the
results will prove useful in electromagnetic applications as well. Our study of the
acoustic applications fall into two main categories: (1) the propagation of sound
in the ocean, or underwater acoustics, and (2) the propagation of sound in the
atmosphere.

1-3
In the next three chapters, we examine the propagation of sound through varying
media and over rough terrain. We develop a numerical solution to the parabolic
wave equation and demonstrate how this technique can be used to solve for the
acoustic field over randomly rough terrain with arbitrary wind profiles over the
terrain. Here, the application is in the prediction of sound intensities along the
ground produced by quasi-point source explosions. One parameter of interest was
the wind profile when the explosion occurred. Everyday experience tells us that
when we are downwind of sound, we tend to notice more of the acoustic
disturbance. If testing munitions within several kilometers of habitat, a windy
day can mean the difference between harmless, acoustic background noise and the
destruction of property such as glass windows.
Chapter 2 begins with a discussion of the parabolic wave equation (PWE) [3] and
its numerical solution via the marching technique [4]. We solve the PWE in
Cartesian coordinates given an initial field condition of that due to a Gaussian
source. We discuss several numerical issues and formulate the solution using
spectral methods. After a fairly thorough discussion of the Fourier aspects of the
numerical solution, we provide examples of PWE solutions for different surface
roughness and wind profiles. The marching technique is extended to surfaces of
varying reflection coefficient and variable wind profiles along the surface. Finally,
we explain some of the difficulties in extending the marching technique to three
spatial dimensions.
Chapter 3 is parallel to Chapter 2 in many ways, except we formulate the
marching technique in curvilinear coordinates and we use a true point source in
the numerical solution for the acoustic field. We are unable to deal with
anything but flat surfaces, but are able to accommodate various refractive index
profiles.
Finally in Chapter 4, we are able to compare the numerical solutions using a
Gaussian source propagated in Cartesian coordinates and those using a true point
source propagated in curvilinear coordinates. In doing so, we are able to discuss
differences and explore what information we may lose when using a Gaussian

1-4
source. The Gaussian source only approximates that of the point source. Even
where it does approximate it reasonably well, it is only over a relatively small
field of regard. Of particular interest is the effect of these errors on acoustic
propagation in the atmosphere when there is some form of ducting caused by the
sound speed profile. The profile will be a function of wind, temperature and
humidity. All of the work in the first three chapters relies on the differential
form of the PWE.
In Chapter 5, we begin our study of scattering from randomly rough surfaces
using the integral form of the PWE. When the full wave equation is used, the
integral formulation begins as an exact solution to the surface scatter problem.
In the parabolic form, we will be well equipped to handle scattering at low
grazing angles ( )20< from a deeply modulated, rough surface. We approach the
problem in terms of an acoustic source with an incident field interacting with the
underside of the sea surface. However, the approach is applicable to other classes
of acoustic problems and electromagnetic problems too. The goal of this work is
to derive an expression for the mean acoustic field when the incident field is that
due to a point source and the surface(s) have certain statistical properties.
The problem of rough surface scattering has led to a variety of approximate
techniques including: Kirchhoff and perturbation theories, operator expansion
technique, ray theory, perturbation theory extensions such as the smoothing
method and phase perturbation [5], and composite methods that combine others
for various regions. All of these techniques have been examined with respect to
the exact theory. In general, approximations can be divided into two categories:
small surface slopes and small surface heights. The small surface height
approximation contains the variations of perturbation theory. If the surface
height variation is characterized by σ and the wavelength of the radiation is given
by 0
λ , then the theory is usually valid when 0
1σ
λ . We should note that this is
a guideline as the actual regime of validity also depends on angle of incidence and
surface slope to some degree [6]. The theory is accurate for backscatter, but can
only handle small surface modulations.

1-5
The Kirchhoff theory can be viewed as considering the surface to be locally flat at
each point and neglecting multiple scatter [7]. The field derivative at the surface
is found by replacing the surface with its tangent plane at each point. It is also
seen from the viewpoint of uniform illumination of the rough surface. This
method will break down for small angle incidence, or quasi-grazing incidence as
multiscatter effects become more important. Certainly, shadow regions do not
meet the uniform illumination condition [8]. The Kirchhoff theory is accurate
when applied to the condition 0
1L
λ ≥ where L is the correlation length scale on
the surface. It can also be applied in the case of deeply modulated surfaces so
long as the surface slope remains small. One can only have deeply modulated
surfaces with 0
1L
λ when the propagation is taken over many L . The
approach in this thesis handles deeply modulated rough surfaces and multiple
scattering, but is restricted to the case of forward scattering, which is implicit in
the PWE.
Chapters 5 through 9 contribute to the body of knowledge in scattering from
rough surfaces by offering a solution to the mean acoustic field given a surface of
known statistics. To date, no one has provided a useful, analytical expression for
the mean field. Current attempts at determining this quantity rely on performing
numerous simulations that are averaged to approximate the true ensemble
average. Numerically, we can only estimate the true population mean from the
sample mean; the variance on the estimate is a function of the number of
samples. Thus, to get an accurate estimate of the population mean, a large
number of samples are needed. Furthermore, new insight is often found in
analytic solutions that is missed from the pure simulation point-of-view.
Chapter 5 describes the specific problem of the scattering of an acoustic signal
incident from below at low angles on a rough sea surface. The problem is treated
by the integral equation method in the parabolic approximation. First, we obtain
equations allowing the mean scattered field to be calculated. We show how the
equations can be scaled into a more simplified form and then offer a general
solution via Laplace transforms.

1-6
In Chapter 6, we use the general solution of Chapter 5 to demonstrate the power
of the technique. Here, we develop expressions for the Laplace transform of a
point source and Green functions necessary to solve the set of integral equations.
After developing the transform expressions, we use them to express the full
solution to the mean field in the Laplace domain for a surface with Gaussian
statistics and exponential-like autocorrelation function (ACF). At this point, we
examine the inversion of this expression and discuss some of the difficulties.
Knowledge of branch cuts is necessary to develop a more simplified expression for
the mean field. We are able to show how this solution approaches the accepted
solution in the limiting case of a flat surface. Then we provide solutions in the
case of increasing surface roughness. The effect of surface roughness on the field
is clearly visible.
Chapter 7 further extends the analytic treatment for the mean acoustic field by
examining the two-surface problem. We examine the case where both surfaces
are pressure-release surfaces and the case where one surface is hard and the other
is pressure-release. The new Laplace transform expressions needed in the two-
surface problem are derived. Finally, the general solution to the two-surface
problem using the method of Laplace transforms is provided.
Chapter 8 is devoted to the study of statistical independence and correlation.
Central to the derivation of the analytic expression for the mean field is
performing ensemble averaging of various expressions. Chapter 8 examines, and
answers, how justified we are in our treatment of the ensemble averaging of the
Green functions and the derivative of the acoustic field at the surface.
Finally, Chapter 9 discusses the way ahead. We examine the full solution to the
two-surface problem, discuss using this technique in other than isovelocity media,
and then completely change topics to the wavelet solutions of integral equations.
The final topic of wavelets brings much promise to the fast solution of integral
equations and is an important topic of current research in many disciplines.

2-1
C h a p t e r 2
2 Cartesian Wave Propagation Over Rough Surfaces
with Varying Refractive Index Profiles
There has been great interest in the numerical solution to the propagation of
acoustic and electromagnetic sources over rough terrain in various media.
Terrain can include areas that are not earth covered, such as lakes and rivers.
Furthermore, terrain may contain different types of vegetation, including
coniferous and deciduous trees, cultivated land, natural grassland, and others.
The different types of surfaces lead to different reflection and transmission
coefficients for an acoustic or electromagnetic wave incident on the surface. In
the electromagnetic case, these coefficients, andR T , are determined by the
angle of incidence for the plane wave component at wavenumber 0
k , the two
refractive indices of the media at the interface, and the state of polarization of the
incoming plane wave.
In the acoustic case, the field is not transverse, but longitudinal. Ultimately, in
either case, the coefficients must be derived from the parameters above and the
conservation of energy at the interface. For an acoustic field, the reflection
coefficients for many types of surfaces are known empirically and in some cases
closely approximate certain familiar boundary conditions. For example, the
reflection coefficient for water is often taken as 1R = , which is complete
reflection. A simple model for other types of non-water surfaces might use a
completely absorbing surface with 0R = . Here, the implication of the term
“absorbing” is the interface acts as if it were an open window— the sound goes
straight through. At the other extreme, a pressure-release surface with 1R = −
implies the field goes to zero on the surface.
Other types of surfaces and their interface to the adjacent media [i] might be
represented with a complex reflection coefficient R i
R e φ
= . In this form, we can
[i] Usually air to some other medium, water to some other medium, or water to air in the domain of study
here.

2-2
take into account amplitude as well as phase changes between the incident and
reflected fields. Complex reflection coefficients can lead to evanescent waves [9].
We will be examining propagation cases when the paraxial form of the wave
equation applies:
( )
2
20
2
0
, 1
2 2
ikE i E
n x z E
x k z
∂ ∂  = + −  ∂ ∂
(2.1)
where ( ),E x z is the acoustic field, 0
k is the primary wavenumber, and ( ),n x z is
the refractive index of the medium. We provide a derivation of equation (2.1) in
Appendix A.
2.1 The Marching Technique
The technique has been described in detail in the work of Sheard [10] and
Hatziioannou [11]. A brief summary of the method is provided here. There are
three important elements to the marching technique:
1. The surface scatter, or boundary, problem can be mapped to an extended
medium and solved as a volume problem with real and image media.
2. The field can be solved numerically by considering the propagating part of the
equation to be independent of any scattering during a step, while the phase
modulation can be considered to occur in a single ‘screen’ before the next
propagation. This is known as the split-step method.
3. The initial condition, field ( )0 0
0,E x z= with source at ( )0
0,z , requires that
an auxiliary source exist in the image medium at ( )0 0 0
0,2 ( )E x S x z= − . The
field is propagated in both media, with the portion of the field in the real
medium representing the desired part. The location of the image source with
respect to the real source is symmetric about the surface 0
( )S x .
Thus, the marching technique is an adequate name for it captures how the effect
of distance and modulation are essentially decoupled and enacted separately as
the numerical scheme steps forward in the direction of propagation.

2-3
Consider a rough surface ( )S x . It has been shown that the rough surface can be
replaced by a medium in the space ( )z S x< with an artificial refractive index.
The refractive index is defined as
( ) ( ) ( )
( ) ( )( ) ( ) ( )[ ] ( )
1
1
, , ,
, ,2 2 ,
N x z n x z z S x
N x z n x S x z S x z S x z S x
= >
′′= − − − <
(2.2)
where
( ) ( )0 01
, , , 1n x z n n x z n= + = (2.3)
For the initial condition, the real source and an image source must be defined as
to satisfy
( ) ( )( )0, 0,2 0I R
E z RE S z= − (2.4)
R
E is the real source, I
E is the image source, and R is the reflection coefficient
at the surface. For the case of a Gaussian source, the initial condition becomes
( ) ( ) ( )
( ) ( )( ) ( ) ( )[ ]
( )
2 2
0
2 2
0 0
/21
2
2 0 /2 2 0 01
2
0, , 0
0, e , 0
z z w
w
S z z w ik S z S
w
E z e z S
E z R e z S
π
π
− −
′− − + −
= >
= <
(2.5)
where w is the source width. We give more detail in Appendix B.
Finally, this is subject to the parabolic wave equation (PWE):
( )
2
2 0
0
,
2
E i E
ik N x z E
x k z
∂ ∂
= +
∂ ∂
(2.6)
with the initial field condition, ( )0
,E x z . There is no fundamental difference
between (2.6) and (2.1). A plausible rough surface and location of the real and
image sources are shown in Figure 1.

2-4
Figure 1 The image method requires the sources to be symmetric about the surface.
Once the initial field is configured, the marching technique begins and the field is
propagated/modulated in a series of steps. The solution obtained in the real
medium is used and that of the image medium discarded.
2.2 2-D Propagation in Cartesian Coordinates
In the next few sections, we will examine some of the issues involved in
numerically solving the PWE in Cartesian coordinates. We examine some
standard techniques that might be used when solving a parabolic partial
differential equation and then describe the spectral method for propagation. The
latter technique offers several advantages and was used in the work presented
here.
The general expression for paraxial, rectilinear wave propagation in two
dimensions is given by
2
2
0
2
E i E
x k z
∂ − ∂
=
∂ ∂
(2.7)
There are two general procedures applicable to the numerical simulation of wave
propagation: one is to work in the spatial domain and use finite difference
approximations to the derivatives to propagate the field in discrete steps, the
other is to transform the equation into spatial frequency space via the Fourier
transform and perform the propagation in the Fourier domain.

2-5
2.2.1 Finite Difference Method for Propagation
The 2-D form of the PWE is identical in structure to the Schrödinger equation:
( )( )
2
2
i iV x
t x
ψ ψ
ψ
∂ ∂
= + −
∂ ∂
(2.8)
Both are parabolic partial differential equations. The difference between them is
the parabolic form of the Helmholtz equation in (2.6) relates two spatial
derivatives to each other (the propagation), which is balanced by scintillation of
the field (2nd
term, right-hand side), while (2.8) looks like a diffusion equation and
accompanying source term with time being the ‘propagated’.variable
An obvious numerical scheme might try to approximate the derivatives in x and
z by
( )
( )
1
2
1 1 2
2 2
0
0
2
k k
j j
k k k
j j j
k
j
E EE
O x
x x
E E EE
O z
z z
x x k x
z z j z
+
+ −
−∂
= + ∆
∂ ∆
− +∂
= + ∆
∂ ∆
= + ∆
= + ∆
(2.9)
such that the full, differenced equation looks like
( )
1
21 1 0
2
0
2
, 1
2 2
k k k k k
j j j j j k
k j j
E E E E Ei ik
n x z E
x k z
+
+ −− − +  = + −  ∆ ∆
(2.10)
Using a von Neumann stability analysis, we can test for unstable eigenmodes of
this difference scheme by allowing k k imj z
j
E eζ ∆
= . We find that
( )
( )
( )2 20
2
0
2
1 sin 1
2 2
ik xi x m z
m n
k z
ζ
∆ ∆ ∆ = − + −  ∆
(2.11)
Let’s assume that the refractive index of medium is unity: ( ), 1n x z ≡ . Then, the
stability criterion requires that all the modes meet ( ) 1 for allm mζ ≤ such that

2-6
( )
[ ]
2
22
0
2
1 1 sin 1m zi x
k z
∆∆
− ≤ − ≤
∆
(2.12)
Taking the nontrivial case, we must have
( )
2
0
1
2
x
k z
∆
≤
∆
for a numerically stable
scheme, as 2sin 1m z∆
≤ [12].
This shows some of the issues when a numerical scheme is conceived on a spatial
grid. The stability requirements are strenuous on the acceptable mesh fidelity as
we more closely approximate a true point source. For instance, when a Gaussian
source is used, the grid size must be fine to pick up the features on a narrow
source. This is often the case as the narrower the source, the more closely it
approximates a point source and the greater the angle of regard that contains
useful information. But the stability requirements insist that z∆ at least be
equal to
0
2 x
k
∆ . Arguably the most widely used differencing scheme is the
famed Crank-Nicholson method [13], but the variety and extent of research and
application into different differencing methods is enormous.
In the next section, we formulate derivative operators for numerical analysis. Of
key importance to the stability and truncation error of the method is the
approximation of the Laplacian term in (2.6).
2.2.1.1 Approximating the Laplacian Operator
Our first step will be to analyze the accuracy and fidelity of the spatial second
derivative. Define the following finite difference operators [14]
The shift operator, ( ) 1k k
z zβ +
= (2.13)
The forward difference operator, ( ) 1k kk
z z z+ +
∆ = − (2.14)
The backward difference operator, ( ) 1k kk
z z z− −
∆ = − (2.15)
The central difference operator, ( ) 1 1
2 2
0 k kk
z z z+ −
∆ = − (2.16)
The differential operator ( ) ( )k
Dz z kh′= (2.17)

2-7
In the above definitions, z is a real or complex sequence indexed by all the
integers, k . In the last definition, h refers to the sampling on equally spaced
points so that ( ); 0k
z z kh h= > .
Iserles shows that the differential operator D can be expressed as
( ) ( ) ( ) ( )
1 22 2 2 2 61 1
12 452 0 0 0
1
, 0
s s ss
s
D h h
h
+ + = ∆ − ∆ + ∆ + Ο →  
(2.18)
The definition above applies for even derivatives (e.g. second, fourth, etc.). Some
basic manipulation shows that 1/2 1/2
0
β β−
∆ = − and therefore 2 1
0
2β β−
∆ = − + .
Applying 2
D to z and retaining the first two terms on the right hand side
( ) ( )2 2
2 41 1
01 1 12
2k k k kh h k
D z z z z z+ −
 = − + − ∆   (2.19)
A more heuristic derivation involves the use of Taylor’s Theorem [15]. Suppose
that the function, ( )z k , has at least m continuous derivatives. Then by Taylor’s
Theorem
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 3 4 5
2 3 4 5
2! 3! 4! 5!
2! 3! 4! 5!
iv vh h h h
iv vh h h h
z k h z k h z k z k z k z k z
z k h z k h z k z k z k z k z
ζ
ζ
+
−
′ ′′ ′′′+ = + + + + +
′ ′′ ′′′− = − + − + −
(2.20)
The existence of ζ+ and ζ- is an extension of the mean value theorem; there must
exist a number on the interval in question that makes the expansion an equality.
By adding the two relations, we obtain an approximation for the second
derivative of ( )z k .
( )
( ) ( ) ( )
( )
2 6
122
2
,ivh
z k h z k z k h
z k z k h k
h
ζ ζ−+
+ − + −
′′ = − + Ο = = (2.21)
We now write a form of the parabolic propagation in z and x coordinates with a
discrete right-hand side:

2-8
( )
( ) ( )
2
0
2
0
1
2
0
2
2
2
EE i
x k z
EE i
x k z
β β−
∆∂
=
∂ ∆
 − +∂   =
∂ ∆
(2.22)
2.2.2 Fourier Method for Propagation
The second option is commonly used because it makes use of the discrete Fourier
Transform and Fast Fourier Transform (FFT) algorithm. This allows the field to
be propagated in a smaller number of computing cycles; results are almost
immediately available. It has the additional benefit of reducing the number of
previous iterations that must be accessible for propagation at the current
location. This brings large computational savings. The idea using the continuous
Fourier transform will be easily demonstrated.
One defines the continuous Fourier transform pair as
( ) ( )
( ) ( )
1
2
i x
i x
F f x e dx
f x F e d
ω
ω
ω
π
ω ω
∞
−∞
∞
−
−∞
=
=
∫
∫
(2.23)
and can apply this transform to the two-dimensional, rectilinear or curvilinear
propagation equation. In this definition, the variable ω is assumed to be angular
frequency with dimensions of radians per second.
A second definition, which can be more intuitive when working in a system where
time or space is the independent variable, is to define the continuous Fourier
transform pair as
( ) ( )
( ) ( )
2
2
i x
i x
F f x e dx
f x F e d
π ν
π ν
ν
ν ν
∞
−∞
∞
−
−∞
=
=
∫
∫

2-9
where ν has dimensions of cycles per meter or inverse wavelength. ν and ω are
related by
2ω πν= (2.24)
When implementing this scheme numerically, care must be given in applying this
technique. The typical approach is to discretize the problem onto a grid that
contains 2N
points and employ the Fast Fourier Transform (FFT) algorithm.
While variants of the Cooley-Tukey algorithm can accommodate non-powers-of-2
grid sizes, their performance requires more than the 2
logN N steps the
power-of-2 algorithm uses [16].
A few subtleties that arise in applying this technique are:
• Mapping the θ (polar) coordinate space to angular frequency
space, ω , or the z coordinate to wavenumber space, ν .
• Understanding the errors involved in the discrete Fourier
transform of the Laplacian operator.
2.2.2.1 Discrete Fourier Transform of the Laplacian Operator
To take the Fourier transform of this equation, we need to recognize a
fundamental property of the shift operator, β . That is ( )[ ]( ) ( )
i h
f x e Fω
β ω=F
when F denotes the Fourier transform and h the step size in x . We will now
define the Fourier transform of our basic operators:
The shift operator ( )( )[ ] ( )
i h
f x e Fω
β ω=F (2.25)
The forward difference operator ( ) ( ) ( )1i h
f x e Fω
ω+
 ∆ = − F (2.26)
The backward difference operator ( )[ ] ( ) ( )1 i h
f x e Fω
ω−
−∆ = −F (2.27)
The central difference operator
( ) ( ) ( ) ( ) ( )2 2
20
2 sin
i h i h
h
f x e e F i F
ω ω
ω
ω ω
−
 ∆ = − = F (2.28)
Applying the discrete Fourier transform to equation (2.22) and explicitly noting
the components with subscript n :

2-10
( )
( )
( ) ( )( )
( )
2
0
2
2
2
0
, 2
,
2
2 sin,
,
2
n n
n
i z i z
n
n
z
n
n
x i e e
x
x k z
ix i
x
x k z
ω ω
ω
ξ ω
ξ ω
ξ ω
ξ ω
∆ − ∆
∆
∂ − +
=
∂ ∆
∂
=
∂ ∆
(2.29)
The term,
( )( )
2
2
2
0
2sin
2
n z
i
k z
ω ∆
−
∆
, is part of a quantity known as the kernel for this
equation. ( ),xξ ω is the Fourier transform of the field, ( ),E x z .
A little more insight can be gathered by expanding each expression in a series:
( )
( )
( )
( ) ( ) ( ) ( )
( )( ) ( )
( ) ( )
2 4 2
2
2 2
12
2! 4! 2 !
2 4 2 6 41 1
12 360
2 cos 12
, ,
1 1 ,
... ,
n n
mm
n n n
i z i z
n
n n
z z z
nmz
n n n n
ze e
x x
z z
x
z z x
ω ω
ω ω ω
ω
ξ ω ξ ω
ξ ω
ω ω ω ξ ω
∆ − ∆
∆ ∆ − ∆
∆
 ∆ −− +  =
 ∆ ∆ 
 = − + − + −  
= − + ∆ − ∆ +
… (2.30)
or with the second expression
( )( )
( )
( ) ( ) ( )
( ) ( )( ) ( )
( ) ( )
2 1
2
2 4 4 6
2
2
2
23 5 2 114 1 1
2 2 2 23! 5! 2 1 !
2
12 360
2 sin
,
... ,
... ,
n
m
n n n n
n n
z
n
mz z z z
nmz
z z
n n
i
x
z
x
x
ω
ω ω ω ω
ω ω
ξ ω
ξ ω
ω ξ ω
−
∆
−−∆ ∆ ∆ ∆−
−∆
∆ ∆
∆
= − + −
= − + − +
(2.31)
In each case, we see that this is the Fourier transform with respect to z of
( ) ( ) ( ) ( ) ( )
2 42 4 6
2 4 6
, , ,
...
12 360
z zE x z E x z E x z
z z z
∆ ∆∂ ∂ ∂
− + −
∂ ∂ ∂
(2.32)
with an error of ( )
6
zΟ ∆ .

2-11
An excellent, discrete approximation to the Laplacian operator can be made by
retaining the fourth order term:
( ) ( )( ) ( )( ) ( )[ ]
2 4
2
2 2
2 2 2
2 sin 2 sin,
,
12
n nz z
i iE x z
E x z
z z z
ω ω∆ ∆  ∂    = +    ∂ ∆ ∆     
F F (2.33)
And this is the expression implemented in the simulations with an error of
( )
4
zΟ ∆ .
2.2.2.2 From the Spatial to the Frequency Domain in Discrete Systems
Consider a mesh in z upon which there are N samples of spacing z∆ . Then the
spatial frequency components will be defined as
n
n
N z
ν =
∆
(2.34)
with n varying between 0 and 1N − in the discrete Fourier transform. Strictly
speaking, the important point is that the samples in Fourier space will be periodic
in n with period N . The second definition ν of the Fourier transform with the
2 iπ factor in the exponential is explicitly saying that the principal range is 0 to
2π . However, there is no reason why one can’t consider the period [ ],π π− with
n ranging on [ ]/2, /2N N− with the Nyquist frequency occurring for 0n = .
How is ω Defined?
As an example, consider an N point mesh in the z direction with a grid spacing
of z∆ . Then the nω ’s will be
2
2 , 0,... 1n
n n N z n Nπ
ω πν ∆≡ ≡ = − (2.35)
Going back to the kernel, we can now see what the actual values to be evaluated
will be
( )( ) ( )( ) ( )
2
2 2 2
22
2 2 2
2 sin2 sin 2sin
, 0, 1,... 1
2 2
n
n z nz
N z N
ii
n N
z z z
π πω
∆
∆
∆ −
≡ = = −
∆ ∆ ∆
(2.36)

2-12
Here is the elegance and efficiency of the method. The Fourier transform has
essentially decoupled the original spatial relation into its constituent modes, each
orthogonal to the other. The propagation operation in the spatial domain now
becomes multiplication by the kernel for each constituent plane wave component
in the field. It should be noted that these pseudo-spectral methods work well
when the field does not have discontinuities. In the presence of discontinuities
the results from the method may not be as expected. However, the behavior of
the Gaussian source is well defined and has excellent convergence properties
towards infinity. This makes the method an excellent choice in propagating our
field numerically.
Now the propagation is a simple algorithm. To advance the field from one
position in nx to 1n
x +
, simply transform the field at nx with respect to z . For
each grid point in z , evaluate the kernel for that mode and multiply it by the
value of the transformed field for that n . Take the inverse Fourier transform of
this product, and the result is the new field at the next position, 1n
x +
for all z .
2.3 2-D Phase Modulation in Cartesian Coordinates
The general expression for paraxial, rectilinear phase modulation in two
( )0
,
E
ik N x z E
x
∂
=
∂
(2.37)
This is also known as the scattering component of the equation. The general
form of this equation is
E
AE
x
∂
=
∂
, which can be rewritten
as ( ) ( )
( , )
( , )
ln
E x x z x x
E x z x
d E d Ax
+∆ +∆
=∫ ∫ . The general solution is
( ) ( )0
, ,ik N x
E x x z e E x z∆
+ ∆ = (2.38)
Spivak and Uscinski have shown that for constant step size, x∆ , that small
scattering per step with this independent phase screen model is valid when the
scale size in the direction of forward propagation is less than x∆ [17].

2-13
2.4 Examples of Cartesian Propagation Using the
Marching Technique
In this section, we examine the use of the marching technique for numerically
solving various initial field conditions, rough surface types, and propagation
media. The purpose is to demonstrate that the numerical solutions appeal to
physical intuition. Also, in the cases where analytic solutions exist, we can
directly compare theory to numerical work to serve as a useful benchmark to the
technique’s accuracy. For instance, a flat surface with a constant refractive index
medium or a linear refractive index medium can be solved exactly and with the
marching technique as well.
2.4.1 Free Space Propagation with Flat, Absorbing Surface
In this particular example, the ground is taken to be flat, with no wind profile,
and the surface to be absorbing ( )0R = . The analytic solution for the field
intensity is
( ) ( ) ( ) ( )( )2
2
0
22 2 4
04 2 2
0
1
, , exp /
2 /
x
k
I x z E x z w z z w
w x kπ
= = − − +
+
(2.39)
when the initial field is defined as
( ) ( )( )2 2
02
1
0, exp /2
2
E z z z w
wπ
= − − (2.40)
The initial field condition is known as a Gaussian source. Notice that w is the
source width parameter and not ω . The result in (2.39) is derived by
considering the solution to be a summation of point sources along an aperture. In
this case, the aperture can be taken as along the 0x = axis. All points in the
aperture plane may be thought of as secondary point sources by the Huygens-
Fresnel principle. The aperture plane is divided into elementary segments, dz , so
small that each infinitesimal segment can be thought of as a secondary source.
Thus, we may write the contributing element to the acoustic field, dE , at P as

2-14
( ) ( )( )2
0
1
Cnst 0, exp /2dE E z dz ik z z x
x
′ ′ ′= ⋅ ⋅ − (2.41)
where the observation point P is at ( , )x z .
We may integrate over the entire aperture plane as shown in equation (2.42) to
produce the result of (2.39)
( ) ( ) ( )( )2
0
0
1 1
, 0, exp /2
2 2
i
E x z E z ik z z x dz
k xπ
∞
−∞
′ ′ ′= ⋅ −∫ (2.42)
Carrying out the integration proves that (2.42) is tantamount to (2.39).
The numerical solution was computed and is shown in Figure 2 with the key
parameters in Table 1. A color map was chosen so that areas of highest intensity
appear in red, followed then by orange, yellow, green, blue, and black.
Figure 2 Results from absorbing, flat surface with constant refractive index and source at
0
10z = .

2-15
Table 1 Absorbing surface input parameters for marching technique solution.
Parameter Result Parameter Result
dx 100 m Terrain flat
dz (initial) 0.1 m Refractive index n constant
nz 2048 Reflection coefficient R 0
Source Height 0
z 10 m Primary wavenumber 0
k 1 m-1
Source width w 10 m
Table 1 lists several of the important inputs into the numerical model. We have
not yet stated the equation that relates the refractive index of the medium to the
speed of sound profile through it, but it follows from the definition of refractive
index in terms of the ratio of the reference to local speed of sound. A constant
refractive index profile implies a constant wind or no wind conditions for all
heights. From standard diffraction theory, one can envision the acoustic source
impinging a slit and having a characteristic spreading angle. This angle is
proportional to 0
D
λ
where D is usually the diameter of the aperture and 0
λ the
wavelength. Here 0
λ is 0
1 k and D is ( )O w . Thus, one would expect θ to be
proportional to 0
1 k w . In fact, at any point x , the width of the field in z is
1
22
2 4
0
4
1
2
w
w x
z
k w
  = +   
. For x large, this becomes
0
2
w
x
z
k w
= (i.e.
0
2
,wz x
k w
θ θ= = ).
Figure 3 provides the accuracy of the results compared to the analytic solution.
For fixed step size, accuracy decreases locally as the wave propagates. Here the
numerical solution is well within 1% for the first 100 steps and continues to stay
within 4% over the full 200 steps of the simulation. Much of this can be
attributed to accumulation of round-off error in comparison to small field
intensities as we move farther and farther from the source. The round-off error
becomes, as a percentage, a larger part of the field intensity far from the source.

2-16
Figure 3 Numerical to analytical solution comparison for the source height at 10 m and
absorbing surface.
2.4.2 Source at 20 m with Reflecting and Absorbing Surface
The solution for the reflecting surface case in a constant index of refraction
medium is found by the superposition principle: the field for a point source at 0
z
is added to the solution to the field for a point source at 0
z− .
If the initial field is defined as ( )( )2 2
02
1
exp /2
2
z z w
wπ
− − , then the solution will
be
( ) ( )( ) ( )( )
( )( ) ( )( )
0
0
2 2 2
02
0 0
2 2 2
02
0 0
1
, exp /
2 2 /
1
exp /
2 2 /
ix
k
ix
k
E x z z z w
ik w ix k
z z w
ik w ix k
= − − + +
+
− + +
+
(2.43)
and the field intensity will be
( )
( )2 2
0
24 24
2
20
0
2
2
22 0
42 4 2 2 4
0 0
22
, exp cos
4 /
x
k
z z
w x
k
zz w
E x z
wk w x k
+
+
     = −      +  +  
(2.44)

2-17
Figure 4 Reflecting and absorbing surfaces with source at 20 m.
This result is understood in the context of Young’s fringes, or interference from
multiple slits. Diffraction theory tells us that the far-field pattern from two slits
will be the Fraunhofer diffraction pattern from one of the slits modulated by a
cosine term whose frequency depends on the separation of the two slits. We
demonstrate this in Figure 5. The plot on the right side shows the interference
term generated by showing the fringe intensity.
Figure 5 Multiple slit interference demonstrated by moving source to 200 m of the ground.
2.4.2.1 Forward Maximum on Ground
With the reflecting case (left side of Figure 4) and the source noticeably off the
ground, theory predicts the occurrence of a local maximum away from the line

2-18
0x = for 0z = . This local maximum will occur when
2
0
2
2
1
z
w
> where w is the
source width and 0z is the positional height of the source. This will occur on the
ground at
2
2 0
20
2
1Max
z
x k w
w
= − (2.45)
For this example, 0
1k = , the source height is 0
20z = meters, and the source
width 10w = meters. Evaluating (2.45) shows that Max
x should occur at
265 meters. In Figure 6, the plot shows a maximum occurring at step 26. With
10 meters between steps, the numerical solution agrees very well with the
analytical result.
Figure 6 Plot of intensity with distance from source for 0z = meters.
For the absorbing surface on the right side of Figure 4, we are no longer in the
multiple-slit situation and the fringe should disappear. The figure confirms this.

2-19
2.5 Cases Involving Arbitrary Propagation Media and
Rough Surfaces
The software to perform the marching technique was written for predicting the
acoustic fields due to a point source over rough terrain in the presence of various
meteorological conditions. One of the chief parameters that affect the sound
patterns away from the source is the wind profile. Common experience tells us
that we better hear things when we are so-called ‘downwind’ of them.
Consider an acoustic source at a height 0
z off the ground and let it be a Gaussian
source such as that in (2.40). We might imagine it to be an explosion of some
kind. Based upon the wind patterns around the source, can we predict what the
acoustic intensity will be along the ground away from the source? This same
type of approach can be used in urban planning when designing airports near
cities or habitats, motorway and road planning through villages, and others.
The marching technique offers a computationally efficient means to answer real-
world problems in a timely fashion. The software written here was used with real
terrain data and meteorology data to predict the acoustic intensities over a range
of 30 km from the source. In the next two figures, we provide examples of the
marching technique to handle a linear wind profile blowing in the x+ direction
with a flat surface and the same wind profile blowing over a rough surface.
In Figure 7, we demonstrate the effect of the medium on the acoustic field with a
flat surface. Just like a lens serves to refract an electromagnetic wave in a
manner governed by Snell’s law, the atmosphere can function similarly to a lens.
Here we see the effect of a linear wind profile. The field tends to bend in the
direction of higher index of refraction. In Chapter 3, we will explore the
generation of caustics caused by the bending of the sound field toward the ground
and the trapping of it there. This figure helps explain why we hear things so
much better ‘downwind’ — there is more of the sound energy along the ground.

2-20
Figure 7 Acoustic intensity for flat, hard surface with wind speed profile of ( ) , .0105v z zε ε= =
and wind direction of 90°.
2.5.1 Comparing Numerical Results in Varying Medium to Theory
For a Gaussian source propagating through a medium with refractive index
( )
0
1 ,n z z
c
ε
ε ε= − = , the Green function has been shown to be [18]:
( )
( )
( )
( )( ) ( )
2
2
30
0
, ; ,
1
exp
2 2 2 12
G x x z z
z ziki
z z x x x x
k x x x x
ε
ε
π
′ ′ =
   ′−    ′ ′ ′− + − − −    ′ ′ − −     
(2.46)
The subtlety between andε ε is discussed in Chapter 4. By using the Green
function and integrating over all source points along the 0x = aperture, the
acoustic field can be derived analytically. The intensity of the acoustic field is
found to be
( )
( )2
22
22
00
2 2
20
44
1
, exp
2
x
xx
kk
z z w
I x z
ww
ε
π
  − +  = − 
 ++    
(2.47)

2-21
when ( ) ( )( )2 2
02
1
0, exp /2
2
E z z z w
wπ
= − − and the surface is absorbing, e.g.
0R = . The theory was compared to the numerical results and showed that the
even after 200 steps, the error in intensity is less than 1.7% when following the
locus
2
2
x
z
ε
= − and 0
0z = . After 50 steps, the error in intensity is less than
1/1000th
of a percent [19].
In Figure 8, we give the example of a rough surface with constant reflection
coefficient of unity. In equation (2.2), notice how the curvature of the surface
affects the refractive index. With a surface of negative curvature, the refractive
index term for the image medium will tend to increase. At points of inflection of
the surface, the refractive index term in the image medium is symmetric with the
refractive index in the real medium.
Figure 8 Acoustic intensity shown for isovelocity medium and rough surface.

2-22
2.6 Varying Reflection Coefficient over the Terrain
The marching technique can be extended to handle variable terrain in a
straightforward manner. In fact, Hatziioannou did related work in dealing with
electromagnetic radiation incident on a rough surface [20]. Given the transverse
nature of the field, one must consider the horizontal and vertical polarisation
cases distinctly. He showed that for a horizontally polarised electromagnetic
plane wave incident on a rough surface at angle i
θ going from a non-conducting
to a conducting medium has a reflection coefficient of
( )
( )
2 2
2 2
cos 1 sin
cos 1 sin
n
ni i
n
ni i
n n
R
n n
θ θ
θ θ
′
′
′− −
=
′+ −
(2.48)
where ,n n′ represent the refractive indices of the two media. He went on to
show how a variable reflection coefficient could then be accommodated in the
marching technique for electromagnetic radiation. (2.48) is known as the Fresnel
equation for the interface of two media.
In this work, the marching technique was expanded to handle the same type of
situations for acoustic radiation. The propagation part of the marching technique
can be thought of as generating a new source at each step, which is then
propagated to the next step. When the reflection coefficient does not change, we
can retain the image field from step to step and just propagate it along with the
real field above the surface. However, when the reflection coefficient changes,
then we have to always satisfy (2.4). Thus, when the reflection coefficient is
variable, at each new step we discard the image field and reapply (2.4) with the
new value. This technique is demonstrated in Figure 9 through Figure 11 on the
next page.
In all three figures, we start with 1R = . In Figure 10 and Figure 11, we change
the reflection coefficient after about 230 steps. The effect on the field is clear
when compared to the perfectly reflecting case for all 300 steps. The right half of
each figure provides a plot of the complex reflection coefficient at each point
along the surface. The location of the change in reflection coefficient and change
in the field are in agreement.

2-23
Figure 9 Flat, hard surface with constant reflection coefficient throughout propagation.
Figure 10 Flat, mostly hard surface with reflection coefficient change 78% through propagation.
Figure 11 Flat, mostly hard surface with change to absorbing surface 78% through propagation.

2-24
Also, the technique was expanded to handle a three-dimensional refractive index
profile.
2.7 Extension of Marching Technique to 3-D
One potential prize in this work is the extension of the image method and
marching technique to three dimensions. The analogue seems to peer from the
page, yet the derivation is much more difficult than one would expect, and to
date has not been discovered.
The governing parabolic equation in three dimensions is
( )2 0
0
2 2
2
2 2
, ,
2 2T
T
ikE i
E N x y z E
x k
y z
∂
= ∇ +
∂
∂ ∂
∇ = +
∂ ∂
(2.49)
One might expect the image method to cleanly map to the three-dimensional case
with the initial field defined as
( ) ( )
( )( ) ( )( ) ( )[ ] ( ){ }
0 0 0
0 0 0 0
0,0, 0,0,
ˆ0,0,2 0,0 0,2 0,0 exp 2 0,0 0,0
R
R
E z E z
E S z RE S z ik z S S
=
− = − − ∇ ⋅ u
(2.50)
with
( ),
ˆ
dS x y
S
ds
= ∇ ⋅ u being the derivative in the direction of the wavefront at
( ),x y with the refractive index defined as
( ) ( ) ( )
( ) ( )( ) ( ) ( )[ ] ( )
1
2
21
, , , , ; ,
, , , ,2 , 2 , , ; ,
N x y z n x y z z S x y
d S
N x y z n x y S x y z x y z S x y z S x y
ds
= >
= − − − >
(2.51)
and
2
2
2 22 2 2
2 2 2
2xx x yy y xy
d S d S x S y
ds ds x s y s
d S x x y y y x
S S S S S
ds s s s s s s
 ∂ ∂ ∂ ∂ = +  ∂ ∂ ∂ ∂ 
   ∂ ∂ ∂ ∂ ∂ ∂  = + + + +       ∂ ∂ ∂ ∂ ∂ ∂
(2.52)

2-25
However, the derivation quickly leads to a conundrum: the technique used in two
dimensions to replace R
E
x
∂
∂
with the right hand side of the 2-D paraxial equation,
in this case can not be done for both the x and y components of the field. It is
almost as though the physics of the 3-D paraxial equation is resisting this
multidirectional approach.
Recall the defining equation on the image medium: ( ) ( ), , , ,reim
E X Y Z RE x y z= ,
with ( ); ; 2 ,X x Y y Z S x y z= = = − the coordinates defined for the image
medium. We have let im I
E E≡ and re R
E E≡ for clarity in notation. The image
field differential can be written two ways:
ˆ ˆ ˆ ˆ ˆ ˆ;
r reim
r
dE R E d
d dx dy dz
x y z
= ∇ ⋅
∂ ∂ ∂
= + + ∇ = + +
∂ ∂ ∂
r
r i j k i j k
(2.53)
or
ˆ ˆ ˆ ˆ ˆ ˆ;
im imR
R
dE E d
d dX dY dZ
X Y Z
= ∇ ⋅
∂ ∂ ∂
= + + ∇ = + +
∂ ∂ ∂
R
R i j k i j k
(2.54)
Relating dR to dr via the chain rule: ˆ ˆ ˆ2 1 2 1
S S
d d
X Y
    ∂ ∂    = + + + − ⋅          ∂ ∂
r i j k R ,
one can show that in the first case the equation becomes
ˆ ˆ ˆ2 2re re re re re
im
E S E E S E E
dE R d
x X z y Y z z
   ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + − ⋅    ∂ ∂ ∂ ∂ ∂ ∂ ∂  
i j k R (2.55)
Using the fact that reim
E E
R
Z z
∂ ∂
= −
∂ ∂
and equating the differential components
from the two relations one can get an expression for im
E directly:
2
2
reim im
reim im
E ES E
R
X X Z x
E ES E
R
Y Y Z y
∂ ∂∂ ∂
= − +
∂ ∂ ∂ ∂
∂ ∂∂ ∂
= − +
∂ ∂ ∂ ∂
(2.56)

2-26
At this point in the 2-D derivation, one would only have the x component
equation. One could then substitute the original paraxial equation for the real
source in place of reE
x
∂
∂
and the derivation is nearly completed except for an
integrating factor.
But attempting the same approach in three dimensions leaves two uncoupled
equations with no means to directly substitute for reE
y
∂
∂
and there is no obvious
way to close the equations.
In the next chapter, we formulate the marching technique in cylindrical
coordinates and propagate a point source. We will be unable to handle rough
terrain, but can accommodate arbitrary refractive index profiles.

3-1
C h a p t e r 3
3 Curvilinear Wave Propagation Over Rough Surfaces
with Varying Refractive Index Profiles
In this section, we extend the marching technique to curvilinear coordinates and
in doing so perform some simulations of wave propagation with a varying
refractive index.
The numerical solution to the PWE in curvilinear coordinates has several useful
applications. For example, the study of the moments of the field (such as the 4th
moment) for a medium containing weak variations in refractive index and the
study of ocean sound ribbons have both been carried out in curvilinear
coordinates. Its utility comes from the ability to more adequately describe a true
point source in the numerical solution. As we stated in the last section, the use
of a Gaussian source requires a very narrow source if it is to approach a point
source over an appreciable range of angles. This is the method proposed by
Tappert [21].
In this work, the curvilinear marching technique has not been able to handle
varying terrain, but flat terrain with varying refractive index profiles can be
accommodated. Thus, in the Cartesian cases when the surface slopes of curvature
were important, all these terms will be zero for the cylindrical cases presented
here. The conditions on the refractive index will reduce to
( ) ( )
( ) ( )
1
1
, 1 , ; 0
, 1 , ; 2
N r n r
N r n r
θ θ θ π
θ θ π θ π
= + < <
= + − < <
(3.1)
And the initial field will be taken as a delta function in the real medium,
0 θ π< < , with the usual definition in the image medium. Any reflection
coefficient is still valid, but the examples will be restricted to reflecting and
absorbing surfaces.

3-2
( ) ( ) ( )
( ) ( )
0
0
0, 0, ; 0
0, 0, ; 2
R
R
E E r
E RE
θ θ δ θ π
θ θ π θ π
= = < <
= − < <
(3.2)
The first section provides a brief outline of the derivation of the paraxial wave
equation in circular coordinates.
3.1 2-D Propagation in Curvilinear Coordinates
Uscinski [22] has shown that the Helmholtz form of the 2-D wave equation in
curvilinear coordinates
( )( )
( ) ( ) ( )
2
2
2 2 0
1 1
,
, , exp
u u
r k n r u
r r r r
U r u r i t
∂ ∂ ∂
θ
∂ ∂ ∂θ
θ θ ω
  + = −  
=
(3.3)
can be reduced to an approximate, paraxial form. The approach is similar to
that for the rectilinear case as the solution is assumed to be separable into the
product of a field term and a function, ( ),E r θ . Said another way, the rapidly
varying phase term and geometrical, amplitude fall-off necessary for energy
conservation are implicit in the solution to the field. Therefore
( ) ( )
( )0
exp
, ,
ik r
u r E r
r
θ θ= (3.4)
The paraxial approximation implies that ( ),E r θ is a slowly varying function that
can only change over the distance on order of the scale size of the medium.
Namely
2
20
E E
k
r r
∂ ∂
>>
∂ ∂
, which implies the second order term can be neglected.
The final approximate equation for E(r, θ) is
( )
2
2 2 0 1
0
,
2
E i E
ik n r E
r k r
θ
θ
∂ ∂
= +
∂ ∂
(3.5)
with ( ) ( )1
, 1 ,n r n rθ θ= + . We derive (3.5) in Appendix C.

3-3
Focusing on the propagation term in the equation above, we will examine
methods for the numerical simulation of this part of the equation. We start with
2
2 2
0
2
E i E
r k r θ
∂ ∂
=
∂ ∂
(3.6)
and begin by examining the Fourier technique applied to equation (3.6).
One can apply the Fourier transform to the two-dimensional, curvilinear
propagation equation above.
2
2 2
0
1
2 2
i E i E
e d
r k r
ωθ
θ
π θ
∞
−∞
 ∂ ∂ = ∂ ∂  
∫ (3.7)
As examined in Chapter 2, this technique can be used to avoid the direct
computation of the second derivative (or Laplacian operator in multiple
dimensions) on the right side of the equation. Instead, the rule for Fourier
transforms of derivatives is used. Namely,
( ) ( )
2
2
2
f
i F
x
∂
ω ω
∂
= (3.8)
Denoting the Fourier Transform of ( ),E r θ with respect to θ as
1
( , ) ( , )
2
i
r e E r dωθ
ξ ω θ θ
π
∞
−∞
= ∫ (3.9)
Then in Fourier space the propagation equation is
( )
( ) ( )2
2
0
,
,
2
r i
i r
r k r
∂ξ ω
ω ξ ω
∂
  =   
(3.10)
Separating terms and rearranging
( )( )
2
0
1
ln
2
i
d d
k r
ω
ξ
 =   
(3.11)

3-4
Finally, evaluating the integrands yield
( )( )( )
1 1
2
0 1
2
0
1 1
2
1
1
ln ,
2
( , ) ( , )
n n
n n
nn
r r
r r
i
k r r
nn
i
d r d
k r
r e r
ω
ω
ξ ω
ξ ω ξ ω
+ +
+
   −     
+
 =   
=
∫ ∫
(3.12)
In (3.12), we see how the propagation between any two concentric rings of radius
1n
r +
and nr can be done in the Fourier domain. The last step is to take the
inverse Fourier transform of the equation so that the result is in ( ),r θ
coordinates.
2
1
0 1
1 1
( , ) exp ( , )
2
i
nn
nn
i
E r e r d
k r r
ωθ ω
θ ξ ω ω
∞
−
+
+−∞
      = −           
∫ (3.13)
Therefore, the technique used for the Cartesian approach has a close analogue in
curvilinear coordinates. The major difference is in the kernel term,
2
0 1
1 1
exp
2 nn
i
k r r
ω
+
      −         
.
3.2 2-D Modulation in Curvilinear Coordinates
The general expression for paraxial, curvilinear wave modulation in two
( )0 1
,
E
ik n r E
r
θ
∂
=
∂
(3.14)
This is analogous to the Cartesian case. Furthermore, the general form of this
equation,
E
AE
r
∂
=
∂
, has the solution:
( ) { } ( )0 1
, exp ,E r r ik n r E rθ θ+ ∆ = ∆ (3.15)

3-5
3.2.1 Mapping a Wind Velocity Profile in Cartesian Coordinates to
Curvilinear Coordinates
In section 2.4.1, we stated that the acoustic refractive index is always calculated
with respect to a reference speed of sound and in section 2.5 we showed an
example of a Cartesian propagation with a linear wind profile. In this section, we
complete the definition of the refractive index for acoustic applications.
The refractive index with regard to sound is usually calculated with respect to a
stationary frame:
( )
( )
0
0
sin
c
n z
c v z φ
=
+
(3.16)
where, in general, 00 z zc c == is the ambient speed of sound (which can be a
function of temperature or humidity), 0
z taken as representing sea level and
standard temperature and pressure conditions, v is the wind speed, and φ is the
direction of the wind. The convention on wind direction is:
Wind blowing from the north in a southerly direction is 0 degrees.
Wind blowing from the west in an easterly direction is 90 degrees.
In Figure 12 and Figure 13, a linear wind speed profile was imposed and
zε=v (3.17)
where the wind is expressed in vector notation in (3.17). In the top half, the
resultant refractive index profile was computed with a wind direction of –90°
degrees. In the bottom half, the same wind speed profile was used, but blowing
in the opposite direction. The same calculations were performed for Figure 13, a
stronger profile. Even though the wind speed profile is linear, only when ε is
very small is the resultant refractive index profile also very linear.

3-6
Figure 12 Mild wind profile and resultant refractive index, .00258ε = .
Figure 13 Strong wind profile and resultant refractive index with .00989ε = .
This general result is true, independently of the coordinate system of choice.
However, the situation is further complicated in polar coordinates.

3-7
First imagine a ray coming from the source with angle θ . After traveling a
distance r , the height above the ground will be sinz r θ= . Based upon a linear
profile of wind speed, the ray will experience a wind speed
( )sinz rε ε θ= (3.18)
Now, the ray will only experience the portion of the wind speed at that height
projected into the direction of propagation, ˆr . So the effective refractive index
the ray will experience at any distance r from the origin at any angle θ is
( ) ( ) ( )sin cosrv z rε θ θ= (3.19)
when a linear wind profile is used. A plot of (3.19) can be seen in Figure 14.
The radial axis is labeled in meters, but the curves in blue are scaled to show
what refractive index the field would experience at that radial for all θ . For
example, take the curve at 10 km. If the refractive index were unity at all
positions, the curve would coincide with the 10000r = contour. But instead, we
see that the refractive index is unity at precisely 4 positions:
3
0, , ,
2 2
π π
π . In the
first and fourth quadrants, the refractive index is greater than unity; for the
second and third quadrants, it is less than unity. The general shape of the
refractive index curve at any particular radial distance is that of a cardoid.
Recognize that the cardoid in Figure 14 arises from imposing (3.1) in conjunction
with (3.19) and (3.16). We show the real and image media in Figure 14 through
Figure 18, the image medium on the bottom.

3-8
Figure 14 Linear wind profile results in θ asymmetry.
3.3 Examples of Curvilinear Propagation Using the
Marching Technique
In this section we present solutions of the marching technique in polar
coordinates with flat terrain and varying strengths of linear wind speed profiles.
For solutions demonstrated in partial form, that is, without the geometrical radial
fall-off factor necessary for energy conservation, we will be displaying images
where the mean paraxial field is 1.0. This is to be expected as the δ function
representing the point source on the surface is initialized to one there. As this
field propagates, the energy that is bent away from one region must be accounted
for in another region. Bear in mind that the colormap in the partial solution
figures is highly contrasted to provide richer detail and the field variations for
most of the image may only be a few percent, except for close to the ground.
In the full solution cases, the radial decay dominates the detail in many instances.
But this has the advantage of displaying only those features where a substantial
change in field intensity is occurring.

3-9
3.3.1 Curvilinear Solutions in Partial Form
By partial, we mean the exclusion of the field fall-off by r so that we may gain
insight that may otherwise be dominated by geometrical factors.
In the following figures, the scale size is a radius of 10000 m. Physical data for a
reflecting flat surface lay in the 0 < θ < π region. The numerical simulation has
been performed with 2048 samples in θ and radial step size of r∆ = 100 m.
The primary wavenumber is taken as unity for convenience.
The initial value of the point source was taken as 1 with a radius of .01 m to
avoid the singularity at the origin. Finally, a reflection coefficient of 1R = was
used.
The major feature of the following figures is the tendency of the field to bend
‘toward the normal’ when traversing from a region of lower to higher index of
refraction. In Figure 15, the wind is blowing right to left in the images, or
90ϕ = − . Therefore, the refractive index is increasing from the surface up, for
θ between 0 and 45 in the ( )0x > region. Similarly, the refractive index is
decreasing from the surface up for theta between 135 and 180 degrees ( )0x < .
As the refractive index profile in z becomes stronger, the breaks toward the
normal are more severe. One feature unobserved before is this tendency for
ducting to occur along the 4
π
θ = radial. The ducting becomes more and more
focused with a stronger refractive index profile, but does not change radials as
1r >> .

3-10
Figure 15 Field intensity with progressively stronger refractive index profiles.
To get a better look at potential caustics, the curvilinear data was spline fit to a
Cartesian grid, and the scale in z magnified by a factor of 5. The images in
Figure 16 and Figure 17 correspond to 20 km across and 4 km in the vertical, or
2000 m above the ground.
In Figure 16 and Figure 17 (left-hand images) the refractive index profile is
strong enough to duct the field along the ground, but not strong enough to
introduce many reflections off the surface ( 0x < ). However, the right-hand side
images show multiple bounces for 0x < . There appear to be asymptotes in the
field along
3
,
4 4
π π
θ θ= = . The explanation is as follows: imagine the rays
emanating near the
3
,
4 4
π π
θ θ= = radials. Examining Figure 16, one sees that a

3-11
local maximum or minimum in refractive index occur along these radials. Rays
will tend to bend toward this radial for the set of rays
4
π
θ ≈ . Once they
attempt to bend past this radial, they encounter a smaller index of refraction at
the next step and therefore tend to converge along this radial. The rays may
very well oscillate about the
4
π
θ ≈ but cannot diverge from it.
Figure 16 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 1.
Figure 17 ε= 1.34e-3 and ε= 5.795e-3 case with 5x scale in vertical, R = 0.

3-12
For the
3
4
π
θ = radial, a local minimum in refractive index is encountered for
those rays near it and will always tend to break from it. But notice from Figure
14 that the refractive index profile is also locally very flat along
3
4
π
θ = . Those
rays will not see a local change in refractive index from one step to the next.
3.3.1.1 The Presence of Caustics
The presence of caustics when the field is bending towards the ground can be
derived from three main principles:
1. When a ray strikes the ground at position 1
x , its next contact with the
ground occurs at 1
3x , etc.
2. A ray’s trajectory is unique and depends only on the launch angle from the
source.
3. As the angle of launch is increased, each subsequent ray will intersect the
preceding ray on its second arc.
Consider a beam centered on the source position
2
0
2
x
z z
ε
= − . The ray
intersects the ground where 0z = or at 0
1
2z
x
ε
 =   
and subsequent intersections
with the surface occur at ( ) 1
2 1n x+ where n is a whole number.
In terms of launch angle, 0 1
0x
z
x
x
∂
θ ε
∂ =
= = such that the first maximum in peak
height after a reflection occurs when
2
0
0
2
z z
θ
ε
= = . One can build a series of these
trajectories and show that for small θ , the caustic has the form
2
18
c
x
z
ε
= . This
compares to the full solution of
( )
2
1
tan
3
2
c
c
x
z
ε
ε
− 
 
  = and
3
tancx θ
ε
= [23].

3-13
3.3.2 Curvilinear Solutions in Full Form
Here the full form of the solution is used by returning to the equation:
( ) ( )
( )0
exp
, ,
ik r
u r E r
r
θ θ= (3.20)
That is, we now include the geometric radial term in the denominator and the
rapidly varying phase term. In the following figures we plot
( )
( ) ( )*
2 , ,
,
E r E r
u r
r
θ θ
θ = (3.21)
This is the intensity of the acoustic field in a vertical cross section of r and θ .
In the four subplots of Figure 18, the scale size is a radius of 10000 m. Physical
data for a reflecting flat surface obviously lay in the 0 θ π< < region. The
numerical simulation has been performed with 2048 samples in theta and radial
step size of 100m. The primary wavenumber is taken as unity for convenience.
As explained in section 3.2.1, the wind speed is stipulated by the equation (3.17)
and then the refractive index is calculated after a number of geometric factors are
accounted for.
Radial attenuation tends to dominate much of the structure displayed in earlier
pictures. But the most noticeable characteristics are very much present: bending
toward or away from the ground and regions of focusing.
In the next section, we will compare some of the curvilinear results to the results
achieved with the Cartesian approach.

3-14
Figure 18 Full curvilinear solutions with ε increasing from left to right, top to bottom.

4-1
C h a p t e r 4
4 Comparison of Gaussian Source to Point Source
In this chapter, we compare the methods of using a Gaussian source and using a
point source in numerical simulations. If accurate simulations are required over a
wide range of angles then the Gaussian source must have a very small width.
Even with relevant parameters identical, such as wavenumber, step size,
refractive index profile, reflection coefficient, etc., there will be differences. Many
of these differences can be traced to this statement: a Gaussian source, no matter
how narrow, is not a true point source.
In Figure 19, we show a Gaussian source and point source for the same quadrant.
The color scaling is over different intensity ranges from the left plot to the right
plot as it was chosen to convey the difference in angular information that will be
available from the two different sources.
Figure 19 The field from a point source (left) and from a Gaussian source (right) are shown in the
double-half plane. The surface is flat and reflecting and the source is positioned on the
surface.

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