Root finding in the complex plane for seismo-acoustic
propagation scenarios with Green’s function solutions
Brittany A. McColloma) and Jon M. Collis
Department of Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois Street,
Golden, Colorado 80401
(Received 17 June 2013; revised 19 July 2014; accepted 30 July 2014)
A normal mode solution to the ocean acoustic problem of the Pekeris waveguide with an elastic
bottom using a Green’s function formulation for a compressional wave point source is considered.
Analytic solutions to these types of waveguide propagation problems are strongly dependent on the
eigenvalues of the problem; these eigenvalues represent horizontal wavenumbers, corresponding to
propagating modes of energy. The eigenvalues arise as singularities in the inverse Hankel transform
integral and are specified by roots to a characteristic equation. These roots manifest themselves as
poles in the inverse transform integral and can be both subtle and difficult to determine. Following
methods previously developed [S. Ivansson et al., J. Sound Vib. 161 (1993)], a root finding routine
has been implemented using the argument principle. Using the roots to the characteristic equation
in the Green’s function formulation, full-field solutions are calculated for scenarios where an
acoustic source lies in either the water column or elastic half space. Solutions are benchmarked
C 2014 Acoustical Society of America.
against laboratory data and existing numerical solutions. V
[http://dx.doi.org/10.1121/1.4892789]
PACS number(s): 43.30.Bp, 43.30.Ma, 43.30.Zk [TFD]
I. INTRODUCTION
The primary focus of this work is the waveguide propagation problem of a point source that emits a compressional
wave within a range-independent environment featuring a
water column overlying an elastic sediment; this environment is termed the elastic Pekeris waveguide. The water
column is assumed to have a pressure release surface above
and to be bounded below by a semi-infinite isospeed elastic
half space. Problems of this type and their solution formulation have a basis in the original work of Pekeris in which a
separable solution to the elliptic wave equation is assumed
and solutions are summed over all possible wave numbers,1
where individual terms in the summation are referred to as
modes. The original work of Pekeris considered the bottom
to be a fluid half space. The elastic Pekeris waveguide problem was subsequently considered by Press and Ewing (EP),2
and their solution was recently benchmarked against laboratory data to high accuracy.3 To treat the point source singularity, the EP solution assumes an artificial interface at the
source, and applies continuity conditions across the interface. An alternative to this source treatment is to use a delta
function as a forcing term, resulting in a Green’s function
source representation.4 More recently, solutions have been
derived for a Green’s function formulation for the problem
of a compressional wave point source in the water column.5
Either solution approach leads to a complex valued equation,
referred to as the characteristic equation, the roots of which
are the horizontal wave numbers to the waveguide propagation problem. The characteristic equation is transcendental
a)
Author to whom correspondence should be addressed. Electronic mail:
b.mccollom1@gmail.com
1036
J. Acoust. Soc. Am. 136 (3), September 2014
Pages: 1036–1045
and its roots are complex valued; this motivates the use of a
numerical root finding routine in the complex plane.
In Sec. II, solutions are developed for the cases where a
compressional wave point source is assumed in either the
water or the sediment in the aforementioned environment.
An algorithm for root finding within the complex plane is
presented in Sec. III, based on methods previously developed
by S. Ivansson et al.6 The algorithm uses the argument principle, Romberg integration, a rectangle halving strategy, and
the Newton-Raphson method. With a routine capable of
determining horizontal wave numbers, the problem of the
source lying within the water column is benchmarked
against experimental data in Sec. IV. The solution of Press
and Ewing is also compared against the experimental data to
compare the different solution approaches. In Sec. IV, the
related problem of a compressional wave point source
assumed in the elastic medium is considered; solutions are
benchmarked against a wavenumber integration solution.
II. GREEN’S FUNCTION FORMULATIONS
Consider the problem of a time-harmonic acoustic point
source in a seismo-acoustic environment, that of the elastic
Pekeris waveguide, where a compressional wave point
source is allowed to be anywhere within the environment.
The point source is represented using a spatial delta function
in the frequency domain. Assume a water column with compressional wave speed c1 and constant density q1 overlies a
semi-infinite elastic half space of constant density q2 , with
compressional wave speed c2 and shear wave speed cs . A
pressure release boundary above the water column is
assumed at z ¼ 0, and a fluid-elastic interface at z ¼ H. An
azimuthally symmetric cylindrical geometry is assumed,
with depth z oriented positively downward, r the radial distance from the z-axis, and the boundaries planar and parallel;
0001-4966/2014/136(3)/1036/10/$30.00
C 2014 Acoustical Society of America
V
a range-independent environment. It is assumed that the
point source has angular frequency x and strength Sx and
lies on the z-axis at ðr; zÞ ¼ ð0; zs Þ, with time dependence of
exp ðixtÞ.
Solutions for the compressional and shear displacement
potentials, / and w, are given in terms of inverse Hankel
transformations
ð
1 1 0
ð Þ
/ ðkr ; zÞH01 ðkr r Þkr dkr ;
(1)
/1 ðr; zÞ ¼
2 1 1
ð
1 1 0
ð Þ
/ ðkr ; zÞH01 ðkr r Þkr dkr ;
(2)
/2 ðr; zÞ ¼
2 1 2
ð
1 1 0
ð Þ
w ðkr ; zÞH01 ðkr r Þkr dkr ;
(3)
w2 ðr; zÞ ¼
2 1 2
for j ¼ 2, s, the subscript (s) denoting a quantity associated
with the shear wave field.
The relationships between the horizontal and vertical
displacements and the range and depth-dependent displacement potentials are given by7
u2 ¼
@/2 @ 2 w2
þ
;
@r
@r@z
@/2 @ 2 w2
þ
þ ks2 w2 :
@z
@z2
/0 2 ðkr; zÞ ¼ Sx
1
B
B ikz;1 eikz;1 H
M¼B
B q x2 eikz;1 H
@ 1
0
1
A
BBC
B C
x ¼B C ;
@CA
D
(10)
(11)
(12)
(13)
where A, B, C, and D are horizontal wavenumber-dependent
amplitude coefficients, and the elastic medium vertical
wavenumbers are defined by Eq. (4).
Applying the boundary conditions, Eqs. (7)–(10), to the
potentials given by Eqs. (11)–(13) yields a system of equations of the form Mx ¼ b, where
Boundary conditions are pressure release at the sea surface;
and continuity of vertical displacement, normal stress, and
tangential stress at the bottom interface. Boundary conditions are expressed as
0
eikz;2jZZsj
þ Ceikz;2 ðzHÞ ;
4pikz;2
w02 ðkr; zÞ ¼ Deikz;s ðzHÞ ;
(6)
/1 ¼ 0; z ¼ 0;
(9)
/0 1 ðkr; zÞ ¼ Aeikz;1 z þ Beikz;1 z ;
(5)
w2 ¼
ðrzz Þ1 ¼ ðrzz Þ2 ; z ¼ H;
for the normal and tangential stresses rzz ¼ kr2 / þ 2l@w=
@z and rzr ¼ lð@u=@z þ @w=@rÞ, with Lame constants k and
l. The compressional and shear wave speeds are related to
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
the Lame constants by cj ¼ ðkj þ 2lj Þ=qj , for j ¼ 1, 2,
pffiffiffiffiffiffiffiffiffiffiffiffi
and cs ¼ l2 =q2 . Loss in the bottom is included by using
and
complex
wave
speeds
C2 ¼ c2 =ð1 þ igap Þ
Cs ¼ cs =ð1 þ igas Þ,8 where ap and as are the compressional
and shear wave attenuations in decibels per wavelength,
and g ¼ ð40p log10 eÞ1 . The compressional and shear wave
fields have corresponding medium wavenumbers,
k1 ¼ x=c1 , k2 ¼ x=C2 , and ks ¼ x=Cs . A Green’s function
representation is used to treat the source singularity. Normal
mode solutions for /1 ðr; zÞ, /2 ðr; zÞ, and w2 ðr; zÞ are derived
in the following sections.
Two scenarios are considered for benchmarking: a compressional wave point source in either the water column or the
elastic half space. The case of a source in the water column has
been considered before,2,3,5 and fluid layer seafloor compressional sources have also been considered previously.7 Suppose
that a compressional wave point source lies in the elastic seafloor at depth zs ¼ H þ d, with H < zs < 1 and d the depth
of the source below the interface. The depth-dependent potentials /0 1 ðkr ; zÞ, /0 2 ðkr ; zÞ, and w0 2 ðkr ; zÞ are defined as
ð1Þ
@/1
;
w1 ¼
@z
(8)
ðrzz Þ2 ¼ 0; z ¼ H;
for kr the horizontal wavenumber, H0 the Hankel function
of the first kind, and subscripts (1) and (2) denoting solutions
in the water column and bottom, respectively. The potentials
/01 , /02 , and w02 are defined based on the location of the
source. Vertical wavenumbers are given
in terms
ffi of medium
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
and horizontal wavenumbers: kz;1 ¼ k12 kr2 ; and, to satisfy radiation conditions in the half space,
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
< kj2 kr2 ; jkr j<jkj j
(4)
kz;j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
: i kr2 kj2 ; jkr j>jkj j
@/1
;
u1 ¼
@r
w1 ¼ w2 ; z ¼ H;
(7)
1
0
0
ikz;1 eikz;1 H
q1 x2 eikz;1 H
ikz;2
ð2l2 kr2 q2 x2 Þ
kr2
2il2 kr2 kz;s
0
2ikz;2
2
ðkr2 kz;s
Þ
1
C
C
C;
C
A
0
0
and
1
0
C
Sx eikz;2 d B
ikz;2
B
C
b¼
B
C:
4pikz;2 @ 2l2 kr2 q2 x2 A
2ikz;2
(14)
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
B. A. McCollom and J. M. Collis: Normal Mode Green’s Function Solutions
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1037
vertical wavenumbers all depend on kr . Despite the differing
source treatments, this equation is the same characteristic
equation as was found by Ewing, Jardetsky, and Press
(EJP).7
By row reducing the augmented matrix ½Mjb, the coefficients A, B, C, and D are found, and the potentials
/0 1 ðkr ; zÞ, /0 2 ðkr ; zÞ, and w0 2 ðkr ; zÞ are determined to be
Sx iq2 x2 2kr2 ks2 eikz;2 d sinðkz;1 zÞ
0
;
(17)
/ 1 ðkr; zÞ ¼
2pf ðkr Þ
The solution to this linear system has singularities when the
determinant of the coefficient matrix M disappears, giving rise
to poles in the inverse Hankel transform.4 This occurs when
q1 ks4 kz;2
tanð Hkz;1 Þ
iq2 kz;1
h
2 i
þ 4kr2 kz;2 kz;s þ 2kr2 ks2
¼ 0:
det ðMÞ ¼
(16)
Equation (16) is referred to as the characteristic equation
for this problem, and it is transcendental in kr , where the
/0 2 ðkr; zÞ ¼
2
Sx eikz;2 jzzz j Sx kz;1 c2s q2 2kr2 ks2 cosðkz;1 H Þ Sx f ðkr Þ ikz;2 ðdHÞ ikz;2 z
e
e ;
2pkz;2 f ðkr Þ
4pikz;2
(18)
and
w02 ðkr; zÞ
Sx iq2 c2s kz;1 2kr2 ks2 eiðkz;2 dkz;s HÞ cosðkz;1 H Þeikz;s z
;
¼
pf ðkr Þ
f ðkr Þ :¼
q1 x2 ks2 kz;2 sinðkz;1 HÞ þ icðkr Þ cosðkz;1 HÞ;
with
(20)
and cðkr Þ ¼ q2 c2s kz;1 ½4kr2 kz;2 kz;s þ ð2kr2 ks2 Þ2 . The function in (20) is the right hand side of the characteristic
equation multiplied by iq2 c2s kz;1 cos Hkz;1 so that it does
(19)
not have poles, and it is this function that is used to
determine horizontal wavenumbers in the complex
plane.
Using the residue theorem, and the EJP integration contour and branch cuts,4,9 modal sum representations for
/1 ðr; zÞ, /2 ðr; zÞ, and w2 ðr; zÞ are found to be
h i
ð nÞ 2
ð Þ
ðnÞ ðnÞ
1 2 k ð nÞ
X
ks2 kr eikz;2 d H01 rkr sinðkz;1 zÞ
r
ð nÞ ;
/1 ðr; zÞ ¼ 4p
f 0 kr
n¼1
/2 ðr; zÞ ¼
n
1 kð Þ
4pi X
z;1
ks2
n¼1
h i2
ðnÞ ðnÞ
ðnÞ 2
ðnÞ
ðnÞ ðnÞ ð
Þ ð Þ
2
2 kr
ks cos kz;1 H kr eikz;2 dH H01 rkr eikz;2 z
ð nÞ ;
ð nÞ
kz;2 f 0 kr
(21)
(22)
and
h i ðnÞ ðnÞ n ðnÞ
n
n 2
ð nÞ
ð nÞ ð Þ
ð Þ
2 i kz;2 dkz;s H
1 kð Þ 2 krð Þ
k
H
kr H01 rkr eikz;s z
cos
k
e
X
s
8p
z;1
z;1
ð nÞ ;
w2 ðr; zÞ ¼ 2
ks n¼1
f 0 kr
ðnÞ
ðnÞ
ðnÞ
for f 0 ðkr Þ, the first derivative of f ðkr Þ, and kz;1 , kz;2 , and kz;s
the vertical wave numbers in terms of the medium wave
ðnÞ
number and the nth horizontal wave number kr satisfying
the characteristic equation, Eq. (20). Note that in developing
this solution, branch line integrals that would make the integral approximation exact, have been ignored; these
pffiffiintegrals
decay as 1=r2 , whereas the residues decay as 1= r, so the
only contribution to the field would be near to the source.7
1038
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
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Note that Aki and Richards, in their discussion on leaky
modes, show that the branch line integrals can also be
expressed as a sum of residues. However, as differs from the
normal mode components, these leaky mode residues will
decay exponentially in the half space.10
In contrast, when the compressional wave point source
lies within the water column, i.e., 0 < zs < H, the potentials
are found to be
B. A. McCollom and J. M. Collis: Normal Mode Green’s Function Solutions
n
n
n
n
n
1 a krð Þ sin kð Þ zs sin kð Þ z krð Þ H ð1Þ rkrð Þ
X
iSx
0
z;1
z;1
ð nÞ /1 ðr; zÞ ¼ ;
ð nÞ
2 n¼1
k f 0 kr
(24)
z;1
/2 ðr; zÞ ¼ and
1
Sx q1 x2 X
2
n¼1
ðnÞ 2
kr
2 n ðnÞ
ðnÞ
ðnÞ
ð nÞ ð Þ
ð Þ
ð
Þ
kz;s
sin kz;1 zs kr H01 rkr eikz;2 zH
ðnÞ ;
f 0 kr
n ðnÞ
n
n
n
1 k ð Þ sin k ð Þ zs krð Þ H ð1Þ rkrð Þ eikz;s ðzH Þ
X
0
z;2
z;1
ðnÞ ;
w2 ðr; zÞ ¼ 2iSx q1 x2
0
f kr
n¼1
where
aðkr Þ :¼ q1 x2 ks2 kz;2 cos ðkz;1 HÞ þ icðkr Þsin ðkz;1 HÞ:
(27)
These potential solutions also depend on the roots to Eq. (20).
These roots lie in the complex plane and to find these analytically is an arduous task, if not impossible. Further, there is a
possibility of repeated roots to the characteristic equation
depending on the amount of attenuation present in the waveguide.10,11 In order to realize all of these roots, the horizontal
wavenumbers, a numerical root finding routine capable of
finding roots within the complex plane is required. In the next
section, a root finding algorithm capable of finding every
root, even when repeated, is presented.
III. ROOT FINDING IN THE COMPLEX PLANE
A hybrid root finding algorithm is presented, based on a
procedure developed by S. Ivansson et al., with improvements based on procedures developed by T. Johnson et al.,
and M. Dellnitz et al. 6,12,13 Routines for root finding in the
complex plane are generally based on the use of the argument principle. Consider a function f ðzÞ that is analytic
within a simple, closed, positively oriented contour C that
lies in the complex plane C (a meromorphic function). Let
N and P be the number of complex roots and poles, respectively, of f ðzÞ within C. The argument principle states that
ð 0
1
f ðzÞ
dz ¼ N P;
(28)
2pi C f ðzÞ
where each zero and pole is counted as many times as its
multiplicity.9 The integral in Eq. (28) is referred to as the
winding integral.
For the problems considered in this work, f , given by
Eq. (20), has been arranged so that there are not any poles of
f within C, i.e., P ¼ 0. With such an arrangement, the winding integral gives the number of complex roots within C.
Using this fact, the number of roots within a given contour
can be determined, and the contour manipulated until each
root is isolated. Once there is a single root within a contour
there are many methods that can be used to locate the root,
e.g., the Newton-Raphson or secant methods. The winding
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
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integral-based method determines repeated roots and their
multiplicity, which may arise physically within certain geoacoustic parameter regimes.11 For the problems considered in
this work repeated roots were not encountered.
A first step in the root finding routine is to define the contour C. Any simple closed contour can be chosen, and rectangles and discs are the most common;6,12–15 in the current work
rectangular contours are used. Given a contour, the winding integral can be evaluated either by directly applying any acceptable quadrature method, or by first integrating by parts once
and then numerically integrating. Once the number of roots
have been computed, the contour can be dissected into smaller
sub-contours until only one root lies within each sub-contour.
At that point, any root finding routine that works within the
complex plane can be used to find the root within the contour.
The root finding algorithm begins with a single rectangular contour containing all of the desired roots, where rectangle size is determined by geophysical parameters. Using
Romberg’s method for numerical quadrature, the winding integral is evaluated in order to determine the number of roots
within the rectangle. Upon iterating, if a rectangle contains
no roots, it is discarded. If the rectangle contains only one
root, then the Newton-Raphson method is used to find the
root. If the rectangle contains more than one root, n say, then
the length of the rectangle’s diagonal is compared with some
prescribed accuracy tolerance . If the diagonal is smaller
than , then the midpoint of the rectangle is accepted as the
root, with multiplicity n, otherwise, the rectangle is split in
half along its longest sides and the procedure described
above is performed on each sub-rectangle.
IV. EXAMPLES
The accuracy of the derived Green’s function solutions
are demonstrated through benchmark comparisons for two
scenarios, those of a compressional wave point source in the
water or in the sediment. After benchmarking, characteristics
of the waveguide are discussed in terms of eigenvalues for
two cases termed soft and hard bottom propagation scenarios. First, the Green’s function and EJP solutions for the
source in the water are compared against laboratory experimental data.16 Second, the Green’s function solution for a
compressional source in the sediment is benchmarked
B. A. McCollom and J. M. Collis: Normal Mode Green’s Function Solutions
1039
TABLE I. Geometric and geoacoustic values used in experimental data
comparisons.
Parameter
Value
Solid Depth, H (cm)
Liquid Density, q1 (g/cm3)
Solid Density, q2 (g/cm3)
Compressional Speed Liquid, c1 (m/s)
Compressional Speed Solid, c2 (m/s)
Shear speed, cs (m/s)
Compressional attenuation, ap (dB/m/kHz)
Shear attenuation, as (dB/m/kHz)
14.5
1.0
1.378
1482
2290
1050
0.33
1.00
against a solution produced by the wavenumber integration
model OASES.17 When evaluating modal sum solutions the
root finding algorithm described in Sec. III is used to
numerically compute the roots to f ðkr Þ ¼ 0, where f ðkr Þ is
given by Eq. (20). The roots to Eq. (20) are then used in the
solutions presented in the Sec. II, and also in the EJP solutions given in Ref. 7 (pp. 175–176).
In acoustics, a standard metric of comparison is the
transmission loss (TL), defined by
pðr; zÞ
;
TLðr; zÞ ¼ 20 log
p0 where p0 is the reference pressure 1 m from the source,
and pðr; zÞ is the acoustic pressure. For comparisons,
the source is normalized so that the reference pressure
amplitude is unity by assuming the source strength to be
Sx ¼ 4p=ðx2 qÞ. The units of this particular source
strength are m2 =Pa, representing the volume injection amplitude necessary to produce a pressure amplitude of 1 Pa at
1 m from the source.
A. The NRL experiment
In order to test the validity of various seismo-acoustic
models, a series of scale-model tank experiments were
performed at the U.S. Naval Research Laboratory in
Washington D.C. in 2004.16 A large fresh water tank was
used to represent the ocean, and an elastic bottom was modeled using a PVC slab (122 cm 122 cm 10 cm) suspended
in the water by cables attached to the corners. A robotic
apparatus was used to position the acoustic source and
hydrophone receiver for accurate positioning. The source
was fixed, while the receiver was moved in 2 mm increments
away from the source to produce a virtual aperture.
Experimental data was found to be of extremely high quality. A detailed explanation of the model and experiment can
be found in the reference.
FIG. 1. Transmission loss vs range for propagation in a soft bottom ðcs < c1 Þ range-independent environment at a near-bottom receiver depth of zr ¼ 13:71
cm. Comparisons show data (solid curve) and calculations from the elastic Pekeris waveguide Green’s function solution (dashed curve) for source frequencies
and positions: (a), (b) 180 kHz, and (c), (d) 280 kHz; (a), (c) mid-depth at 6.91 cm, and (b), (d) deep at 13.69 cm.
1040
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
B. A. McCollom and J. M. Collis: Normal Mode Green’s Function Solutions
B. Source in the water
Consider a compressional wave point source in the
water column. Using the physical parameters given in
Table I, the horizontal wavenumbers are computed using
the algorithm presented in Sec. III. This case is referred to
as the soft bottom case, where cs < c1 . The horizontal
wavenumbers are then used to compute modal sums for the
Green’s function and EJP solutions. Figure 1 displays comparisons of the transmission loss of experimental data (solid
curve) versus the Green’s function solution’s loss (dashed
curve) at a receiver depth of zr ¼ 13:71 cm. Figure 2 displays an analogous comparison, but for the EJP solution
against the experimental data. Comparisons are given for
shallow and deep source depths of zs ¼ 6:91 cm [Figs. 1(a),
1(c), 2(a), and 2(c)] and zs ¼ 13:69 cm [Fig. 1(b), 1(d), 2(b),
and 2(d)] at frequencies of 180 Hz [Figs. 1(a), 1(b), 2(a),
and 2(b)] and 280 Hz [Figs. 1(c), 1(d), 2(c), and 2(d)]. There
is very good agreement between the two analytic solutions
and the data. At both frequencies, solutions provide a close
match in both pattern phase and amplitude to the experimental data. The fields do differ between the two source
depths, with a more complicated pattern for the deep source
case and slightly poorer, though still good agreement. There
are points at which the solutions and data disagree, however
agreement is generally good with neither producing a
clearly better result. Note that the propagation track
extended beyond the edge of the PVC slab, at 1.05 m, and
this can be seen in the comparisons as differences are more
pronounced.
C. Source in the sediment
Consider a compressional point source in the sediment
bottom. Figure 3 shows the solution from the Green’s function formulation (dashed curve) compared to a solution computed using a wave number integration model, OASES
(solid curve),17 that is known to give accurate results for
range-independent seismo-acoustic problems, including
those with compressional sources.18 Assuming the bottom
depth is H ¼ 500 m, the source depth is zs ¼ H þ d ¼ 600
m (i.e., d ¼ 100 m), and using the geoacoustic parameters
given in Table II, normal mode solutions are determined. In
Figs. 3(a) and 3(c) the two solutions are compared at a frequency of 15 Hz, for receiver depths of zr ¼ 200 m and
zr ¼ 499 m; and in Figs. 3(b) and 3(d) analogous results are
presented for a 50 Hz source. The comparison plots of the
Green’s function solution and OASES transmission loss all
exhibit the same behaviors: Near the source, the shape of the
curves are similar; however, the numerical values of the
FIG. 2. Transmission loss vs range for propagation in a soft bottom ðcs < c1 Þ range-independent environment at a near-bottom receiver depth of zr ¼ 13:71
cm. Comparisons show data (solid curve) and calculations from the elastic Pekeris waveguide EJP solution (dashed curve) for source frequencies and positions: (a), (b) 180 kHz, and (c), (d) 280 kHz; (a), (c) mid-depth at 6.91 cm, and (b), (d) deep at 13.69 cm.
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
B. A. McCollom and J. M. Collis: Normal Mode Green’s Function Solutions
1041
FIG. 3. Transmission loss curves from OASES (solid curve) and the elastic Pekeris waveguide Green’s function solution (dashed curve) for a compressional
wave point source located in the elastic medium 100 m below the fluid-elastic interface z ¼ 600 m. Curves are shown for a receiver depth of z ¼ 200 m for (a)
15 Hz and (b) 50 Hz sources, as well as near the interface at depth zr ¼ 499 m for (c) 15 Hz and (d) 50 Hz sources.
losses differ (with greater differences when the receiver
depths are closer to the interface). At ranges farther from the
source, the shapes of the curves and computed losses move
closer to each other and become in excellent agreement.
Differences near the source are most likely due to either the
neglect of the continuous spectrum or the branch line integrals. The curves demonstrate excellent agreement across
frequencies and receiver depths, in particular for a receiver
located near to the interface.
D. Eigenmodes for soft and hard ocean bottoms
The propagation scenarios in the previous examples
were for problems in which the shear wavespeed in the
TABLE II. Geoacoustic values for comparisons against OASES.
seafloor was less than the speed of sound in the overlying
water layer, giving rise to so-called “Leaky modes.”10
Consider a scenario, e.g., a basalt seafloor in the Western
Atlantic, in which the shear wave speed is greater than the
acoustic wave speed in the water. This scenario is referred to
as the hard bottom case, where cs > c1 . Using the geophysical parameters given in Table III, and assuming the same
geometric parameters as in the experimental data comparison for the deep source case, horizontal wave numbers are
found from Eq. (20). Horizontal wave numbers are given in
Fig. 4 for both the soft, Fig. 4(a), and hard, Fig. 4(b), bottom
scenarios. Wave number lines corresponding to the medium
wave numbers are plotted vertically, representing the
TABLE III. Geoacoustic values used in hard-ocean bottom simulations.
Parameter
Value
Parameter
Value
Liquid Density, q1 (g/cm3)
Solid Density, q2 (g/cm3)
Compressional Speed Liquid, c1 (m/s)
Compressional Speed Solid, c2 (m/s)
Shear speed, cs (m/s)
Compressional attenuation, ap (dB/m/kHz)
Shear attenuation, as (dB/m/kHz)
1.0
1.378
1500
1700
800
0.06
0.25
Liquid Density, q1 (g/cm3)
Solid Density, q2 (g/cm3)
Compressional Speed Liquid, c1 (m/s)
Compressional Speed Solid, c2 (m/s)
Shear speed, cs (m/s)
Compressional attenuation, ap (dB/m/kHz)
Shear attenuation, as (dB/m/kHz)
1.0
2.58
1482
4500
2400
0.03
0.07
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J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
B. A. McCollom and J. M. Collis: Normal Mode Green’s Function Solutions
FIG. 4. Horizontal wave numbers plotted in the complex plane for the soft
and hard ocean bottom cases: (a) cs < c1 ; and (b) cs > c1 .
compressional wave numbers in the water and sediment, and
the shear wave number, denoting regimes of propagation.
Transmission loss curves are given in Fig. 5 for three different source depths for a 180 kHz deep sound source at
13.69 cm. The curves are noticeably different than those
obtained for the soft-bottom case. For the soft and hard bottom examples, 36 and 37 modes were found. Modes are
counted from the right of the figure, the first mode referred
to as mode zero, representing the Scholte mode. In either
case, the Scholte mode is found to have a real component
of the horizontal wavenumber greater than that of the
wavenumber in the water column. Also in both cases, mode
one is seen to be very close to the water wave number with
subsequent modes following to the left. In terms of percentage of shear wave speeds, for the soft and hard bottom examples, the Scholte wave speeds, determined from horizontal
wave number values, are found to be 82% and 61% of the
shear wave speeds.
Mode shape functions in terms of displacement potentials are plotted for select modes in Figs. 6 and 7. Shape
functions correspond to the soft, Fig. 6, and hard, Fig. 7, bottom scenarios considered in the previous paragraph. In
Fig. 6, the Scholte mode shape function (corresponding to
mode zero), is plotted along with the final mode shape function in the water, representing a water-borne mode, and the
first shape function in the elastic bottom, representing energy
propagated in the elastic medium. Note that the amplitude of
the shape function is non-negligible in the sediment for
modes 28 through 35, explaining why, for propagation
scenarios of this type, significant energy is propagated in the
ocean’s sediment. In Fig. 7, again the mode zero shape function is plotted along with the final mode shape function for
the water and the first shape function for the elastic bottom.
Of interest, the final shape function in the water has significant amplitude nearer to the ocean bottom interface, in fact
more than double the amplitude of all other modes, due to
significant imaginary component of the horizontal wave
J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
FIG. 5. Transmission loss vs range for propagation in a range-independent
environment over a hard ocean bottom ðcs > c1 Þ for a 180 kHz sound
source. Plots show the elastic Pekeris waveguide Green’s function solution
for a deep source at z ¼ 13:69 cm, and receiver depths: (a) 0.5 cm; (b)
7.5 cm; and (c) 13.71 cm.
number. The final two modes in the sediment have significant amplitude, and so field contributions.
V. DISCUSSION
Normal mode solutions to the elastic Pekeris waveguide problem using a Green’s function formulation for a
compressional wave point source were derived and benchmarked for scenarios featuring an acoustic source within
the water column or within the elastic sediment. The derivation and implementation of a normal mode solution
using a Green’s function formulation to the elastic Pekeris
waveguide problem with the source assumed in the
B. A. McCollom and J. M. Collis: Normal Mode Green’s Function Solutions
1043
FIG. 6. Select mode shape functions for the soft bottom case: (a) mode 0
(Scholte mode); (b) mode 27; and (c) mode 28.
sediment is newly implemented work. The Green’s function solution gives promising results, which is apparent
when benchmarked against the wavenumber integration
solution produced by OASES. Although there are differences in the comparison curves near the source, there is
excellent agreement in the far field. The cause of differences near the source is probably a result of the evanescent
field or boundary line integrals that are neglected by the
normal mode solution. These results demonstrate the validity of a Green’s function representation for a compressional source within the sediment.
FIG. 7. Select mode shape functions for the hard bottom case: (a) mode 0
(Scholte mode); (b) mode 34; and (c) mode 35.
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J. Acoust. Soc. Am., Vol. 136, No. 3, September 2014
An algorithm for root finding within the complex plane
was detailed, and used to compute the roots of the characteristic equation for the elastic Pekeris waveguide problem.
The root finding algorithm presented in Sec. III gives a
method for finding the roots to a meromorphic function
within the complex plane; this method has been numerically
validated through application to benchmark problems.
Additionally, the root finding algorithm and related normal
mode solutions demonstrate the approaches’ ability to find
the wavenumber associated with the Scholte interface wave,
with potential applications to classifying this poorly understood phenomenon. Application of the root finding algorithm
presents a potentially powerful tool for characterizing seismic phenomena, certainly giving more information than horizontal wavenumber spectra from the Hankel transform of
pressure data would give.
The simplicity of the derivation of the Green’s function solutions for the elastic Pekeris waveguide problems,
and great results from comparisons against the experimental data and OASES indicate the Green’s function
representation of a compressional source is accurate for
representing a compressional wave point source. This
approach could be applied when considering multiple
layers of different material, as well as multiple sources. A
Green’s function formulation for a shear wave source is
also a scenario worthy of investigation, which combined
with the representation of the compressional wave source
could provide a realistic representation for a generic geophysical source. Although the implementation of the root
finding routine gives desirable results, there is room
for improvement: A robust implementation would require
root detection on contours among other numerical
considerations.
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