Acoustic scattering by prolate and oblate liquid spheroids

Acoustical Oceanography: Paper ICA2016-298
Acoustic scattering by prolate
and oblate liquid spheroids
Juan D. Gonzalez(a) , Edmundo F. Lavia(a) , Silvia Blanc(a) , Igor Prario(a)
(a) Argentinean
Navy Research Office and UNIDEF ( National Council of Scientific and
Technological Research/Ministry of Defence), Underwater Sound Division, Argentina,
[email protected]
Abstract
The problem of scattering of harmonic plane acoustic waves by liquid spheroids (prolate and
oblate) is addressed from an analytical approach. Numerical implementation of the exact solution of the scalar wave Helmholtz equation in an unbounded domain with Sommerfeld radiation
condition at infinity and boundary conditions for the interface medium-immersed liquid spheroid
is presented. A general solution which is an expansion on prolate and oblate spheroidal functions is computed with no limiting assumption regarding neither eccentricity nor sound speed and
density ratios values. The implemented code is basically derived from software recently released
to public use. A high level solver layer code has been developed in Julia programming language.
Predicted results have been verified for far-field and near-field results in agreement with previously reported approximate solutions. The software can be downloaded from authors’ web site.
This numerical implementation of the exact solution is applied in order to extend some benchmarks models of acoustic backscattering used in aquatic ecosystem research, namely, rigid-fixed,
pressure-release, gas-filled, and weakly scattering.
Keywords: scattering, prolate and oblate spheroids, liquids
Acoustic scattering by prolate and oblate liquid
spheroids.
1
Introduction
The interaction of harmonic plane sound waves with prolate/oblate spheroids has been widely
and increasingly investigated, mainly for soft and hard or rigid scatterers (i.e. Dirichlet’s and
Neumann’s boundary conditions, respectively), along the last six decades in different branches
of acoustics [1, 2, 3]. In particular, underwater acoustics can be considered quite proliferous
in reported articles on different aspects of this topic due to its relevance in applications to
fisheries management and marine ecosystem research.
Many developed models provide exact or approximate solutions with harmonic time dependence for the general problem of acoustic scattering by spheroids which have been extensively
applied to several aquatic organisms and objects immersed in the ocean [4, 5, 6]. Computational implementations of approximate solutions have been published for either near or far-field
conditions for non-penetrable scatterers (soft and rigid ones). However, less attention has received the case of penetrable spheroidal scatterers (fluids, often and not accurately refered as
liquid ones; including the gas-filled scatterer and the case of weakly scattering) till 1964 when
the corresponding analytical solution for the oblate case was published for the first time [7].
Three years later numerically computed results of the scattered wave radiation pattern were
reported only for a particular case of sound speed ratio, c1 /c0 = 1, being c0 and c1 the sound
speed in the medium and the penetrable prolate spheroid, respectively [8]. In 1988 Furusawa [4] numerically computed an approximate solution for the acoustic backscattering pattern
by liquid prolate spheroids for specific cases when the sound speed and density ratios verify
c1 /c0 ≈ 1 and ρ1 /ρ0 ≈ 1.055. Several authors have used approximate solutions in the last years
[9, 10]. More recently, another reference [11] considers the general case c0 6= c1 and ρ0 6= ρ1
but only for the case of spheroids with low eccentricity. In contrast with such quite widespread
approximate solutions, this paper addresses an analytical approach; it provides a numerical
implementation of an exact solution for a liquid spheroidal scatterer.
Mathematically, the problem of acoustic scattering of plane waves with harmonic dependence
consists in solving Helmholtz equation in an unbounded domain with Sommerfeld radiation
condition at infinity. The domain where propagation takes place is characterised by density
and sound speed values ρ0 and c0 , respectively, while ρ1 and c1 are the corresponding density
and sound speed values of the object responsible of the scattered field. Different boundary
conditions must be verified for the medium-object interface according to the type of scatterer
considered, namely, soft object (Dirichlet’s boundary conditions), rigid or hard object (Neumann
b.c.) or penetrable object (that includes liquid, gas-filled and weakly scattering). The solution of
the scalar wave equation strongly depends on the object’s shape and there exist exact solutions
for a few cases only. When the object is a sphere, the exact solution is based on an expansion
in spherical wave functions [12]. For scatterers of spheroidal shape, the exact solution can
be expanded on prolate/oblate spheroidal wave functions as a result of applying separation of
variables to Helmholtz equation expressed in spheroidal coordinates [13, 14].
2
In this paper a new computer implementation based on the exact solution for penetrable spheroids
valid for arbitrary c0 , c1 , ρ0 , ρ1 , is provided. The implementation of this solution was developed
using recently free available computational routines by Adelman et al. [15] (from now on abbreviated to AGD) together with a high level layer code implemented in the Julia programming
language [16].
2
Analytical exact solution
When acoustic scattering by spheroids is considered, it is convenient to use a spheroidal curvilinear coordinate system (ξ , η, ϕ) [13]. The relationship between the prolate/oblate spheroidal
coordinates and the Cartesian coordinates, is given by the following transformation [15]:

d


x = [(ξ 2 ∓ 1) (1 − η 2 )]1/2 cos ϕ


2





d
y = [(ξ 2 ∓ 1) (1 − η 2 )]1/2 sin ϕ

2






d


z = ξ η,
2
where d is the interfocal distance of the ellipse of major semi-axis a = (d/2) ξ and minor semiaxis b = (d/2) (ξ 2 − 1)1/2 (See Figure 1).
In the prolate case, the coordinates verify 1 ≤ ξ , −1 ≤ η ≤ 1, 0 ≤ ϕ < 2π. The ξ -range of variation is directly associated with the shape of the spheroid, ranging from ξ = 1 (corresponding
to the segment of length d between both spheroid foci) to ξ → ∞ (corresponding to a sphere of
radius a = b). The surface of the prolate spheroid coincides with the coordinate surface given
by equation ξ = ξ0 = (1 − (b/a)2 )−1/2 with d = 2(a2 − b2 )1/2 .
In the oblate case, the coordinates verify ξ ≥ 0, −1 ≤ η ≤ 1, 0 ≤ ϕ < 2π, and the surface of
the oblate spheroid coincides with the coordinate surface given by equation ξ = ξ0 = ((a/b)2 −
1)−1/2 .
Figure 1: Prolate (left) and oblate (right) coordinate system and related conventions.
For harmonic wave fields propagating in an unbounded medium, the acoustic pressure exterior
3
and interior to a prolate/oblate spheroidal surface of a liquid scatterer immersed in that medium
is governed by Helmholtz equation which is separable in prolate/oblate spheroidal coordinates.
Thus, the problem of solving a partial differential equation in an unbounded domain with the
Sommerfeld radiation at infinity is reduced to solving two ordinary differential equations for radial and angular prolate/oblate spheroidal functions, R(ξ ) and S(η), respectively [1, 7, 15, 18].
At this point it is worthy to note that ξ → iξ and h0 = (d/2)k0 → −ih0 provides a direct transformation from the prolate to the oblate case [15, 18]. Therefore, from now on all mathematical
expressions are valid for the liquid prolate spheroid but can be easily transformed to the oblate
spheroid through the given conversion rule.
An incident plane acoustic wave on a prolate/oblate spheroid, with angular frequency ω and
wave number k0 = ω/c0 , is schematically illustrated in Figure 1. Without loss of generality,
due to the symmetry of revolution around the z axis, the acoustic pressure corresponding to
an incident plane wave can be written as pi = p0 exp(ik~0 ·~r), where k~0 = k0 (sin θi , 0, cos θi ), the
amplitude p0 is a real number and θi is the incident angle. This expression can be expanded
on prolate spheroidal functions [19] so that at an arbitrary field point ~r =~r(ξ , η, ϕ) ,
pi (~r) = 2p0
εm
(1)
∑ ∑ in Nmn Smn (h0 , cos(θi ))Smn (h0 , η)Rmn (h0 , ξ ) cos(mϕ)
(1)
m≥0 n≥m
where the subscripts m, n are natural numbers, Smn are the angular spheroidal wave functions,
(1)
Rmn are the radial spheroidal wave functions of first kind, i is the imaginary complex unit, εm is
the Neumann factor, defined as εm = 2 if m 6= 0 and εm = 1 if m = 0, the parameter h0 is defined
as h0 = (d/2)k0 and Nmn = kSmn k2L2 [−1,1] are the norms. The analytical solutions for the respective
scattered and transmitted acoustic pressure, ps (~r) and pt (~r), can be expressed as,
ps (~r) = 2p0
εm
(3)
(2)
εm
(1)
(3)
∑ ∑ in Amn Nmn Smn (h0 , cos(θi ))Smn (h0 , η)Rmn (h0 , ξ ) cos(mϕ),
m≥0 n≥m
pt (~r) = 2p0
∑ ∑ in Bmn Nmn Smn (h0 , cos(θi ))Smn (h1 , η)Rmn (h1 , ξ ) cos(mϕ).
m≥0 n≥m
The subscripts 0 and 1 refers to the surrounding medium and to the interior of the penetrable
prolate spheroid, respectively, and Amn and Bmn are the unknown expansion coefficients. Precisely, the key point of this work is to determine them applying appropriate boundary conditions
at the medium-spheroid interface, that means to require the continuity of both the pressure and
the normal component of media particle velocity at the boundary, given by ξ = ξ0 , i.e.
1 ∂ pt 1 ∂ (pi + ps ) (pi + ps )|ξ =ξ0 = pt |ξ =ξ0
=
.
(4)
ρ0
∂ξ
ρ1 ∂ ξ ξ =ξ0
ξ =ξ0
Using orthogonality properties of the family {Smn (h1 , η) cos(mϕ) : m ≥ 0, n ≥ m} and some algebraic manipulation, a matricial system involving the Amn and Bmn coefficients is obtained. It can
be shown that Amn , with (n = m, m + 1, ..), verify
∞
(m)
∑ An
(m)
(m)
Qn` = f`
` = m, m + 1, ...
(5)
n=m
4
(m)
(m)
where Qn` and f`
are defined as
"
(1)
(3)0
ρ1 Rmn (h0 , ξ0 )Rm` (h1 , ξ0 )
1
(m)
(3)
n
α Smn (h0 , cos(θi ))
− Rmn (h0 , ξ0 ) ,
=i
(1)0
Nmn n`
ρ0
R (h1 , ξ0 )
(6)
"
(1)0
(1)
ρ1 Rmn (h0 , ξ0 )Rm` (h1 , ξ0 )
1
(1)
(m)
=−∑i
− Rmn (h0 , ξ0 ) ,
αn` Smn (h0 , cos(θi ))
(1)0
ρ0
R (h1 , ξ0 )
n=0 Nmn
(7)
(m)
Qn`
m`
(m)
f`
∞
n
m`
with
(m)
αn`
R1
=
−1 Smn (h0 , η)Sm` (h1 , η)dη
,
R1 2
−1 Sm` (h1 , η)dη
(8)
where the ξ derivative has been indicated with a prime. Note that m is indicated as superscript
in Eq. (5) to emphasize the fact that for each fixed m, the LHS is a product of a row vector
(m) (m)
(m)
(m)
(Am , Am+1 , . . . , An , . . . ) and a matrix Qn` while the RHS is a row indexed by `. Once the Amn
are calculated through Eq.(5), it follows that the Bmn coefficients associated to the transmitted
waves, can also be calculated.
In the far-field limit it can be shown [7] that ps (r, θi , ϕi = 0, θ , ϕ) ≈ p0 (exp ik0 r/r) f∞ (θi , ϕi = 0, θ , ϕ),
where r, θ , ϕ are the spherical polar coordinates of the observation point and the two latter
are also the corresponding scattered angles
f∞ (θi , ϕi = 0, θ , ϕ) =
2
εm
Smn (h0 , cos(θi ))Smn (h0 , cos(θ )) cos(mϕ)
Amn
∑
ik0 m≥0n≥m
Nmn
(9)
is the so called far-field scattering amplitude function which is widespread used in different
acoustic scattering applications.
3
Algorithmic procedure
Numerical computation of the Amn and Bmn coefficients starts with a truncation procedure. The
series in Eq. (5) is truncated at M, so that for each value of m, there are M − m + 1 unknown
(m)
(m)
An , where index n takes natural values in [m, M]. Thus, An satisfies
M
(m)
∑ An
(m)
(m)
Qn` = f`
` = m, m + 1, .., M
(10)
n=m
where A(m) and f (m) are row arrays ∈ C1×(M−m+1) and Q(m) is a matrix ∈ C(M−m+1)×(M−m+1) .
Essentially, the algorithm calculates the Q(m) matrix and f (m) array using the AGD software at
each step (indexed by m) and solves the linear system induced by them at the Julia layer level
(m)
in order to get the An coefficients.
4
Numerical results
Algorithm verification was conducted via evaluation of the exact solution for | f∞ |, obtained
through the implemented codes for a penetrable prolate/oblate spheroidal scatterer, against
5
known results valid in some extreme cases, namely, spheroid tending to a sphere [20] and
situations of non-penetrable scatterers such as the case of Dirichlet b. c. at the mediumspheroid interface (soft case) [4, 5, 15] and Neumann b. c. (rigid-case) [1, 15]. In all cases the
implemented codes reproduced results used as benchmark [21].
Furthermore, a boundary continuity check in the near-field range of the spheroid was performed. The total acoustic pressure at a field-point ~r is equal to the the sum of incident and
scattered acoustic pressures if ~r is outside the liquid spheroid, or to the transmitted field if ~r is
inside it, i. e.
pt (~r)
if ξ (~r) ≤ ξ0
ptotal (~r) =
pinc (~r) + ps (~r)
if ξ (~r) > ξ0
In Figure 2 different planes of the near-field surrounding a spheroid with semiaxes a = 2, b = 1,
density ratio ρ1 /ρ0 = 1.5 and wave numbers k1 = 1, k0 = 1.5, can be visualized. Computed total
acoustic pressure verifies the continuity required by the b.c. at the interface ξ = ξ0 .
Figure 2: (a) Real part of incident acoustic pressure, Re(pi ), at the Cartesian planes z = 0 and y = 0. (b)
Absolute value of total acoustic pressure, |ptotal |, at z = 0. (c) Absolute value of total acoustic pressure,
|ptotal |, at y = 0. Assumed values: a = 2, b = 1, density ratio ρ1 /ρ0 = 1.5, wave numbers k1 = 1, k0 = 1.5 and
incidence angle θi = π/2.
4.1 Applications to acoustic backscattering in aquatic ecosystem research
In acoustical oceanography, the Prolate Spheroidal Model (PSM) [1, 4, 5], provides a useful
tool for determining target strength (TS) of individual fish and volume scattering strength (SV ) of
zoo and phytoplankton. In this research field, it is useful to estimate their numerical abundance
as well as to improve morphological and anatomical representations of the volume scatterers
present at sea.
In fisheries acoustics, as well as in other SONAR applications, sound scattering by an object is analyzed in the logarithmic scale using the target strength parameter, T S, that can be
expressed as
T Sbs (re. 1 m) = 10 log(| f∞ (θi , ϕi , π − θi , ϕi + π)|2 ),
√
where | f∞ | = σbs , has length dimensions and σbs is the backscattering cross-section. In particular, for the prolate spheroidal scatterer, f∞ is defined in Eq. (9).
6
Previously, other investigators using the PSM have reported solutions on scattering by fluid
spheroids for some cases of interest such as the so-called weakly scattering (i.e. c1 /c0 ≈ 1
and ρ1 /ρ0 ≈ 1) and the gas-filled case [24]. More precisely, Furusawa [4] worked with 1.01 ≤
ρ1 /ρ0 ≤ 1.07. Other authors limited their calculations to certain range of frequencies [6, 22].
In this work, an implementation of the exact analytical solution for a penetrable spheroid with
no limitations regarding sound speed and density contrasts values, is provided.
4.1.1 Weakly-Scattering by a prolate fluid spheroid
When the medium and the prolate spheroid have similar physical properties, i.e. ρ1 /ρ0 ≈ 1 and
c1 /c0 ≈ 1, a usual approximation consists in assuming that the non-diagonal matrix elements
(m)
αn` defined in Eq. (8) can be neglected [4, 5, 6], thus
(m)
(m)
αn` ≈ αnn δn,` ,
(11)
whereas the matrix system given by Eq. (10) becomes almost trivial so that straightforward
calculations lead to the approximate coefficients,
m(1)
Aapprox
mn = −
where
Enn
m(3)
.
(12)
Enn
(1)
m(i)
En
(i)
= Rmn (h0 , ξ0 ) −
ρ1 Rmn (h1 , ξ0 )
(i)0
Rmn (h0 , ξ0 ).
(1)0
ρ0 Rmn (h1 , ξ0 )
In order to compare results in the case of weakly scattering, the value of TS derived from the
exact coefficients’ computation through the solution (Eq. (10)) with the one derived from the
approximate solution (Eq. (12)), a prolate spheroid insonified at 38 kHz, has been considered.
The input data have been ρ1 = 1028.9 kg m−3 and c1 = 1480.3 m s−1 , a = 0.1 m, b = 0.01 m for
the spheroid immersed in seawater characterised by ρ0 = 1026.8 kg m−3 and c0 = 1477.4 m s−1 .
Comparison between both solutions is shown in Figure 3 (A) for the whole 0o to 90o incidence
angle-range. It can be observed that both curves do not fit. This is caused by the fact that
using the approximation (Eq. (11)) when computing the matrix elements Q defined in Eq. (6),
nearly no differences are detected between the exact and the approximate solutions for TS. On
the other hand, using the approximation (Eq. (11)) when computing the vector components f
(Eq. (7)), generates very significant differences when the density contrast verifies ρ1 /ρ0 . 1.002
like in this case. However, if the sound speed contrast is c1 /c0 ≈ 1 but the density contrast is for
example ρ1 /ρ0 = 1.06, the differences between the approximate and exact solutions is roughly
0.6 dB.
Additionally, in the frame of the approximate solution, another achievement of this work has
been the extension of the maximum frequency range previously reported [6] to compute TS
using Eq. (12) till h0 = 762.08. Results are shown in Figure 3 (B) in the frequency interval 25
kHz ≤ f ≤ 1800 kHz for θi = 90o . A previously published algorithm suitable for high values of h
was used in order to compute prolate spheroidal wave functions [23] .
7
Figure 3: (A) TS vs. incident angle, comparison between exact and approximate solutions for a prolate
spheroid. ρ0 = 1026.8kg m−3 , ρ1 = 1028.9kg m−3 , c0 = 1477.4m s−1 , c1 = 1480.3m s−1 , a = 0.1m, b = 0.01m,
f = 38kHz. (B) TS vs. frequency at θi = 90o for a spheroid with same parameters as (A) but ρ1 = 1088.4 kg
m−3
4.1.2 Gas-Filled Prolate spheroid
A gas-filled spheroid is a penetrable spheroid (fluid) with density and sound speed contrasts
similar to the ones corresponding to the water-air interface (i.e. c1 /c0 < 1 and ρ1 /ρ0 1). This
case has a wide spectrum of applications in acoustical oceanography. In particular, at intermediate frequencies, the fish swimbladder may be considered as a gas-filled or a soft-scatterer
when the PSM is used. At low frequency (k0 a < 1), it is possible to use an approximation similar
Figure 4: TS vs. incident angle for a gas-filled.ρ0 = 1026.8 kg m−3 , ρ1 = 1.24kg m−3 , c0 = 1477.4 m s−1 ,
c1 = 345.0 m s−1 , a = 0.07 m, b = 0.01 m and f=38 kHz.
to Eq. (11) instead of solving the exact Eq. (10) [7]. Scattering by a gas-filled prolate spheroid
for k0 a < 1 has already been theoretically investigated under the approach of the PSM, and
some numerical results were published [24]. However, at intermediate and high frequencies,
the gas-filled prolate spheroid has not been modelled with the PSM because of the problems
8
associated to the calculation of the spheroidal wave functions [6, 22]. Hence, other approximate alternatives were used such as the Boundary Element Method. More precisely, Jech et
al. [6] reported that convergence of the prolate spheroidal functions could not be obtained for
the following input data: ρ0 = 1026.8 kg m−3 , ρ1 = 1.24kg m−3 , c0 = 1477.4 m s−1 , c1 = 345.0 m
s−1 , a = 0.07 m, b = 0.01 m and f=38 kHz. The same spheroid has been treated in this work,
and convergence has been achieved. Results are shown in Figure 4, where it can be seen that
the soft-scatterer and the exact solution have the same behavior. On the other hand, it can be
observed that the approximate solution given by Eq. (12) clearly cannot be used in this case.
5
Conclusions
A novel implementation of the exact analytical solution for the problem of acoustic scattering by
penetrable (fluid) prolate/oblate spheroids governed by Helmholtz equation is presented. A set
of computational routines to calculate the expansion coefficients appearing in the mathematical
expressions of the scattered and transmitted acoustic pressures is provided. These routines are
freely available from the authors’ GitHub repository. The coefficients’ computation is precisely
the most cumbersome task related to numerical calculation of the mathematical expressions
above mentioned.
Results were tested for the geometrical limit when a prolate/oblate spheroid tends to the sphere
and for the physical limit when it tends to a soft or rigid spheroidal scatterers, taking low and
high values of densities ratios, respectively. In the latter two cases predicted results turn out to
be compatible with expected ones for Dirichlet and Neumann boundary conditions. Additionally,
it has been qualitatively verified, trough the calculation for near-field conditions, that the numerical solution satisfies the boundary condition at the surface of the spheroids. The implemented
exact solution has been successfully compared against a previously reported approximate solution
With the aim of generating the provided computational codes, the efficient software by AGD
was used in addition to the numerical capabilities of the new scientific programming language
Julia. Thus, it was possible to take advantage of the arbitrary floating point precision support
in both pieces of code.
The developed codes are released intending to contribute with researchers working in acoustic
scattering by simplifying the tedious algorithmic task involved in the resolution of the cumbersome matricial system when calculating the exact solution for a liquid spheroid scatterer.
Acknowledgements
This work is part of the project PIDDEF 13-2014. The authors acknowledge R. Adelman and his collaborators who have freely provided the available software as well as the developers of Julia language.
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