Ray representation of sound scattering by weakly scattering
deformed fluid cylinders: Simple physics and application
to zooplankton
Timothy K. Stanton
Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution,
Woods Hole, Massachusetts 02543
Clarence S. Clay
Department of Geology and Geophysics, University of Wisconsin, Madison, Wisconsin 53706
Dezhang Chu
Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution,
Woods Hole, Massachusetts 02543
(Received 28 April 1992; accepted for publication 2 July 1993)
Data indicate that certain important types of marine organisms behave acoustically like weakly
scattering fluid bodies (i.e., their material properties appear fluidlike and similar to those of the
surrounding fluid medium). Use of this boundary condition, along with certain assumptions,
allows reduction of what is a very complex scattering problem to a relatively simple,
approximate ray-based solution. Because of the diversity of this problem, the formulation is
presented in two articles: this first one in which the basic physics of the scattering process is
described where the incident sound wave is nearly normally incident upon a single target (i.e.,
the region in which the scattering amplitude is typically at or near a maximum value for the
individual) and the second one [Stanton etal., J. Acoust. Soc. Am. 94, 3463-3472 (1993)]
where the formulation is heuristically extended to all angles of incidence and then statistically
averaged over a range of angles and target sizes to produce a collective echo involving an
aggregation of randomly oriented different sized scatterers. In this article, a simple ray model is
employed in the deformed cylinder formulation [Stanton, J. Acoust. Soc. Am. 86, 691-705
( 1989)] to describe the scattering by finite length deformed fluid bodies in the general shape of
elongated organisms, The work involves single realizations of the length and angle of
orientation. Straight and bent finite cylinders and prolate spheroids are treated in separate
examples. There is reasonable qualitative comparison between the structure of the data collected
by Chu et al. [ICES J. Mar. Sci. 49, 97-106 (1992)] involving two decapod shrimp and this
single-target normal-incidence theory. This analysis forms the basis for successful comparison
(presented in the companion article) between the extended formulation that is averaged over an
ensemble of realizations of length and angle of orientation and scattering data involving
aggregations of up to 100's of animals.
PACS numbers: 43.20.Fn, 43.30.Gv, 43.30.Xm
LIST OF SYMBOLS
Pscat scattered pressure
•/j plane-wave/plane interface reflection coefficient i
where wave is initially in medium i and in-
cidcut upon medium j [=(pjcj/p•i-1)/ f(k•a)
( pjcj/p•+ 1 )]
Tiy transmission coefficient for planar interface due to
plane wave initially in medium i and in-
cident upon medium j [Tij=2(pjcj/p•ci)/ rpos
(1 -t- (pjcj/Pi½i) )]
1,2 subscripts indicating medium "1" (surrounding rs
fluid) and medium "2" (body medium)
k acoustic wave number ( = 2rr/A) ap
a radius of cylinder cross section Bp
P0 amplitude of incident plane wave r/p
b m coefficient determined from boundary conditions
azimuthal angle (•b=•' is the backscatter direc-
tion)
form function for infinitely long cylinder
scattering amplitude for finite-sized objects (sub-
scripts se, ps, and be refer to straight cylinder,
prolate spheroid, and bent cylinder)
diatancc from axi• of cylinder to the field point
(applied to infinite and finite bodies)
position vector from the origin of the coordinate
system to a point on the axis of the cylinder
distance from a point on the axis (at rpo s) to the
field point (receiver)
term in divergence factor for scattered ray
reflection-transmission factor
propagation phase delay of scattered ray
phase advance associated with crossing of caustics
3454 J. Acoust. Sec. Am. 94 (6), December 1993 0001-4966/93/94(6)/3454/9/$6.00 @ 1993 Acoustical Society of America 3454
q
•i,•r
L
Lebc
phase shift associated with external caustics a 0
distance between the point on the axis (at rpo s) Pc
and the plane that both contains the origin and is
perpendicular to the direction of the incident p
plane wave y
k•L sin 0 •'max
apparent volume flow per unit length of scattered
field due to infinitely long cylinder x
angle between direction of incident plane wave
and plane whose normal is the tangent to the axis •/
at each point r• ("angle of incidence" to local /•
tangent) •
unit vectors describing direction of incident and C,S
received (scattered) waves c
length of straight finite object; arc length (or pro- g
jetted length, depending on context) of uniformly h
bent cylinder RTS
effective length of bent cylinder
length of semi-minor axis of prolate spheroid
radius of curvature of axis of uniformly bent cyl-
inder
mass density
position angle of bent cylinder
angle that subtends portion of bent cylinder be-
tween midpoint and end
position of bent cylinder axis that has been pro-
jected onto x axis
(maximum deflection of bent cylinder ) /a
(length of cylinder)/a
Fresnel integral parameter
Fresnel integrals
compressional speed of sound
P2/Pt
c2/c•
target strength -- 10 log L 2 (reduced target
strength)
INTRODUCTION
Quantitative remote sensing of marine organisms with
sonars requires detailed knowledge of their scattering prop-
erties. In a series of papers, the deformed cylinder formu-
lation has demonstrated promise for the description of the
scattering by certain elongated zooplankton and
micronekton •-• and fish. 7 In particular, straight, bent, and
rough cylinder evaluations of the formulation have de-
scribed some or most of the scattering characteristics of
shrimplike (crustacean) zooplankton--an abundant class
of animals. The analyses suggest that most of these ani-
mals, whose bodies tend to be bent, behave acoustically like
bent cylinders 2• although some data imply that certain
animals may behave like straight cylinders. t The rough
cylinder formulation s'9 has described the statistical nature
of the scattering by moving, flexing animals. 3 Clay has
successfully described the scattering of sound by fish by
modeling the swimbladders as bent gas-filled cylinders and
the fleshy part of the fish as weakly scattering bent fluid
cylinders. ? In that paper, he showed that the apparent
"damping" effect at resonance hypothesized by others can
be explained by the elongated shape of the nonspherical
swimbladder.
To date, most of the uses of the deformed cylinder
formulation have involved the modal series solution which
requires many terms to converge in the high-frequency re-
gion and, in general, is very cumbersome, if not impossible,
to manipulate algebraically. Manipulation is essential when
deriving formulas (usually approximate) that not only ex-
plicitly illustrate the physics of the scattering process but
also describe the average scattering properties. For exam-
ple, in order to estimate the scattering by rough cylinders,
an approximate ray solution was used in the formulation. 8.9
The results were used to describe the general trend of the
statistics of the echo from zooplankton 3 although since
that solution involved dense elastic objects, it was not ap-
propriate to compare the predicted levels to those mea-
sured from the animals. In addition to the rough cylinder
problem, simple solutions are necessary to describe average
echo levels from aggregations of animals of random orien-
tation and length.
Data collected over the past several years provide in-
formation allowing us to develop a simple model. In par-
ticular, field data collected by Pieper et el., m and labora-
tory data collected by Stanton et el. • involving average
echoes from aggregations show a dip in the backscatter
versus ka curve toward the lower end of the geometric
scattering region. Laboratory data from Chu et el. 5 involv-
ing echoes from each of two individuals show a deep null in
the same region. The laboratory data presented in the latter
two articles showed the dip or null to consistently lie in the
region surrounding ka = 2 where a is the equivalent cylin-
drical radius of the elongated animals (decapod shrimp
whose bodies are representative of a broad class of marine
organisms). The dip or null is indicative of strong inter-
ference between the acoustic wave or "ray" reflected from
the front interface of the animal and a ray that has pene-
trated the interior with little loss and reflected off the back
interface. Complementing these data are direct measure-
ments of density and sound speed of various zooplankton
that indicate that those properties are very similar (to
within several percent) to those of the surrounding
seawater. 12-14
All of the above (direct measurement and scattering)
data viewed collectively suggest that the animals tend to
behave acoustically as weakly scattering bodies. For a body
to be weakly scattering, the material properties must be
similar to those of the surrounding medium which, as
stated above, is the case for these zooplankton. Also, as
will be illustrated in the theoretical and numerical portions
of this article, weakly scattering bodies of certain classes of
shapes (cylindrical in this case) can be described acousti-
cally by a simple two-ray model at broadside incidence.
Such a formulation gives rise to a deep null near ka = 2 for
a single target which is consistent with the data collected
by Chu et el. 5 Note that even the average echo from ag-
gregations of randomly oriented animals contains a dip
3455 J. Acoust. Soc. Am., Vol. 94, No. 6, December 1993 Stanton el al.: Scattering by deformed fluid cylinders 3455
that appears to be a "smeared" version of the null. 10,11 The
dip is strong enough (of the order 5-10 dB below sur-
rounding levels) that a model needs to be developed that
predicts it.
With this significant evidence that the animals behave
as weakly scattering bodies, we formulate a simple analyt-
ical approximate model that estimates the scattering levels
and illustrates the fundamentals of the scattering process.
Choice of this model was based, in part, on the desire for
one that can be easily manipulated algebraically and is
computationally efficient. These last three desirable fea-
tures are not possible with the modal-series-based model
described in Ref. 2 or other numerically oriented models
although note that, while this current model is limited to
weakly scattering materials, the modal-series-based model
can be used for a broad range of materials.
The analysis is divided into two articles: this present
one which involves formulating the basic physics of the
scattering process for a single realization of length and
angle of orientation in the region where the scattering am-
plitude tends to be at a maximum (near normal incidence)
and a companion paper where the work is heuristically
extended to include all angles of incidence for the purpose
of averaging over angle.•l The resultant extended model is
averaged in Ref. 11 over a range of angles and sizes to
produce a model describing an "aggregation echo." The
aggregation model is successfully compared in Ref. 11 with
data involving aggregations of animals ranging in numbers
from six to many hundreds. The data show the same 5- to
10-dB dip as predicted by the model, hence further vali-
dating the ray-based approach described in this present
paper.
In this present article, a simple existing ray model is
modified and incorporated in the deformed cylinder
formulation. 2 The modification involves heuristically alter-
ing the phase of the rays so that the ray description, which
is normally valid only in the geometric or ka• 1 region, can
be used accurately for values of ka well below unity. While
the derivation is motivated by the animal problem, the
results are general and not specific to scattering by animals.
Examples involving several shapes, comparison with the
modal-series-based solution that involves fewer mathemat-
ical approximations, and comparison with data from single
animals are given.
I. THEORY
A. General ray-based solution
The far-field solution to the scattering by an infinitely
long cylinder is given a• ]5
Pscat-* Po (x/-'•eit•rF ( øø ), kr•, 1, (1)
where the form function is
F(ø• )--•2 •-i•r/4 Z m•. (2)
= • m=0 bm cos
The modal series representation in Eq. (2) is exact for
all ka and material composition (axisymmetric profile).
For karl, however, many terms are required for mathe-
(a)
[ncident field
receiver
rs
oufer boundory
of deformed
cylinder ,',
rr
origin
cylinder Gxls
(b) incidenl pl .......
FIG. 1. (a) Weak scatterer geometry showing two types of rays scattered:
"specular" echo (fp=0) reflected off front interface and "transmitted"
echo (fp=2) reflected off hack interface after having traveled through the
body. (b) Deformed cylinder and general bistatic sonar.
matical convergence and, while modern computers can ef-
ficiently calculate the series to high precision, the series is
quite cumbersome to manipulate algebraically. Use of ray
solutions in this geometric scattering region has proven to
facilitate the manipulations while at the same time making
the physics of the scattering process more explicit in the
mathematics. 7-9'•6-25 The ray theories are based on a vari-
ety of methods including the Sommerfeld-Watson trans-
formation, heuristically derived formulations, and the
Kirchhoff method. The success of using ray methods to
describe the scattering by marine organisms is illustrated
by Foote 26 where the Kirchhoff method was used to nu-
merically calculate the scattering by the front interface
alone from (gas-filled) swimbladders of fish.
The types of rays that need to be taken into account
depend upon the material properties. In the weak fluid
scatterer case analyzed in this article (g,h • 1 ), it is shown
that only two rays--the ones reflected from the front and
back interfaces--need to be taken into account for reason-
able (although not exact) description of the scattering in
the transitional and geometric region [Fig. 1 (a)]. The form
function associated with each ray can be calculated from
the general (geometric scattering) formula for thepth scat-
tered ray from Marston: 2•
F• (øø)= (27a)]/21%--al•/2B•i(nv+t•v+t'*), (3)
3456 d. Acoust. Sec. Am., Vol. 94, No. 6, December 1993 Stanton et al.: Scattering by deformed fluid cylinders 3456
where the p = 0 and p = 2 rays correspond to the reflections
off the front and (first-order reflection) back interfaces,
respectively. The values of p> 2 correspond to higher-
order internal reflections. According to the formulas in
Marston, ap=o=a/2, B=.•t2, */p=o =--2k•a, and
/•=o=0 for the p=0 "specular" reflection off the front
interface while a r = 2 •- (3/2) a (weak scatterer approxima-
tion), B=-TnT2• m %=2-2k•a(2h-•- 1), and
/•=:=-rr/2 for the p=2 "transmitted wave" reflection
off the back interface. Here,/•e=0 for all rays in the case
presented in this article where c: > c•.
Although Eq. (3) is generally valid only in the geo-
metric optics limit, we have found that its usefulness can be
extended down to the Rayleigh/geometric transitional re-
gion (0.1 <• ka <• 1 ) by heuristically removing the phase ad-
vance/zr=2 that is due to a caustic. This caustic is related
to the curvature of the cross section of the body. Phase
shifts due to the caustic naturally reduce to zero in the long
wavelength limit. While the variation of phase with respect
to wavelength can be derived, we remove the caustic effect
gradually by replacing the constant value/•=2 = -rr/2 by
the convenient function /.tp=2(kla) = -- (rr/2)kla/
(kla+0.4). The high-frequency (k•a•l) and low fre-
quency (kla<l) limits oftzr=2(k•a} are -•r/2 and 0, re-
spectively. The constant 0.4 was determined empirically
and limits the errors to within about 2 dB for the values
ka>0.1 and (g,h) < ( 1.1,1.1 ) for the straight cylinder.
Evaluating Eq. (3) with these parameters gives the
approximate form function for the weakly scattering infi-
nitely long fluid cylinder:
F(•O)
• •=o•-' •=•, (4)
=•2e-a'•a(1-- T12T21einl•2aeit¾=2(kla)), (5)
where the general form for the scattering is given in Eq.
(4) and an explicit approximate form is in Eq. (5). The
assumptions for these equations are ( 1 ) the incident plane
wave is traveling in a direction normal to the axis of the
cylinder, (2) the cylinder is surrounded by a fluid, and (3)
the scattering is sufficiently weak that (a) the first reflec-
tion off the back interface dominates the higher-order in-
ternal reflections and (b) bending of the rays at the inter-
face (Snell's law) is negligible. Simulations later in this
article verify the validity of assumption (3).
A useful property of the scattered field from elongated,
deformed, sometimes finite bodies is the apparent volume
flow per unit length q of the field. •a This term can be
integrated over the length of the body to provide estimates
of the scattered field, and hence backscattering cross sec-
tion and target strength. The accuracy of the estimates
improve as the aspect ratio (length/diameter) increases.
The volume flow per unit length will be used in predicting
scattering by the finite objects in this article.
The far-field-scattered pressure due to a plane-wave
incident upon an infinitely long cylinder is written in terms
ofq as
Pscat-' ( elk'r/•)q e-i*r/4 x/•P•C•/( 2 2•) (k•r>> 1),
(6)
where
4Pø •. b.• cos md).
q=k•p•c• m=0
The volume flow per unit length can be related to the
form function with the following expression:
q = 2p ø ( ½rax/• ) ei•r/4F( oo )/p l C I' ( 8 )
Inserting this equation in Eqs. (4) and (5) gives an ap-
proximate expression for the volume flow per unit length
due to the scattering from a weakly scattering fluid cylin-
der:
q--•q•=o+qp=2, (9)
= 2Po• ne-ak•a( 1 -- Tl•T•lei4k2•eit¾ =•(k•) )
X ei*r / 4 X/-• / ( p l C I X•11) . (10)
The above volume flow term is incorporated into the
deformed cylinder formulation so that effects due to defor-
mities such as bend, taper, and roughness can be estimated.
The approximate formulation allows for calculation of fi-
nite or infinitely long bodies. The general solution for bod-
ies of any length is given as
--i f r elks(rs+Es)
Pscat=•-'-• plClkl q cos 0 -- I drpos
• r•
X (any length), ( 11 )
where some of the terms are illustrated in Fig. 1 (b). All
wavenumbers in q are modified by the factor cos 0. Note
that in Eq. (4) of Ref. 2 there is no explicit cos 0 term that
corresponds to the one that appears in the above expres-
sion. The misleading omission in Reft 2 made no difference
in the results because from Eq. (5) and beyond, the modal
series representation of q was used and a (cos 0) -• term in
q cancelled out the would-be cos 0 factor. In the ray rep-
resentation, there is no such cancellation hence the cos 0 is
written explicitly. The solution for bodies of finite length
(i.e., the length is much less than the first Fresnel zone of
the source/receiver combination) simplifies to
Pscat=Po( eitq•/r) f( k•a), (12a)
where
f(k,a)_•(Po)-•(-ik'p'cl) ; r qcos0
4rr
Xexp[ik•rpos(?i-?,) ' ?p os] larpol
(finite length). (12b)
The details of these equations are discussed in Refs. 2
and 8. Note that the implicit Po in q cancels the explicit
(P0) -• in F-xl. (12b).
B. Deformed smooth bodies
The ray formulation can readily be incorporated into
the above deformed cylinder equations by direct substitu-
tion to obtain approximate solutions to the scattering by a
3457 J. Acoust. Soc. Am., Vol. 94, No. 6, December 1993 Stanton et aL: Scattering by deformed fluid cylinders 3457
variety of bodies such as the straight finite cylinder, prolate
spheroid (high aspect ratio), and uniformly bent cylinder.
Adaptation of the integral formulation to each shape is
described in Refs. 2 and 8 and includes (1) allowing the
"cylindrical radius" of the prolate spheroid to vary along
the lengthwise axis, (2) taking into account the phase
shifts induced by the bend of the axis of the bent cylinder
(this takes place in the exponent in the integrand), and (3)
solving the integral along the length of the cylinders for the
prolate spheroid and bent cylinder by use of the method of
stationary phase or Fresnel integrals. Performing the above
manipulations gives the following approximate solutions
for those shapes in the geometric scattering region:
I. Straight cylinder (various angles of incidence):
-- i sin A
,e •isr/4•-- t2kl a cos Or
.• sc=W-• c c • •la cos 0•2 A
X ( 1 - Tx2T2•e i4k2a •os Oei•p=2(kta) ).
Z Prolate spheroid (broadside incidence):
(13)
fps= (1/4) L• •2e-a•'•o( 1 + iT12T•lei41*2aoeit•v =2(Iqa) ).
(14)
3. Bent cylinder (uniformly bent or nearly so; bent
symmetrically away from sonar).
a. Arbitrary deflections. For values of deflection of the
bent cylinder that are arbitrary with respect to wavelengths
of sound, one obtains
--i
f t•= •--• k•la Lebc•212e-alqa(1
-- T12 T 21ei4k2aeil•a =ø( kta) )e i•r/4, (15a)
= Lebcfsc(O=Oø)/L, (15b)
where the approximations k2=k • and cos 7• 1 were made
in amplitude terms and the limit pc•a was used (which is
reasonable for elongated zooplankton). The effective
length Let • can be expressed in terms of either the radius of
curvature p, of the cylinder axis or deflection
Z(X) :4Ax2/[•2a of the axis from a straight line:
frmax eiklpeY2dy (y,•l) (16a)
Lebc = Pc a - Yrnax
and
fjL/;/2 e,•kt•d
Le•c = dx. (16b)
The term y in Eq. (16a) denotes the position angle on
the uniformly bent cylinder and 2pc•'rnax = L (see Ref. 2 for
geometry). The terms chosen in the exponent in Eq. (16b)
relate the curvature of the axis to a parabolic curve. Here,
z(L/2)=Aa, L=Ba, and x is the position along the cyl-
inder projected onto a straight line (see Ref. 7 for geome-
try). O and A are parameters indicating the length and
deflection in terms of the cylindrical radius. These equa-
tions have the form of Fresnel integrals. As was done in
Ref. 7, a change in the variables to the following parame-
ters is performed for Eqs. (16a) and (16b), respectively:
and
(rr/2)•2=k,p•y 2 and •= •](2/•r)k,p•rm•, (17a)
(•/2)•2=8k•Ax2/B2a and •2,J•Aa/•, (17b)
which results in the compact formulas:
L½•= x/pc•l [C(•l) +iS(•,) ] (18a)
and
Le•c= L Xl•X/•--•[C(•i) +iS(•,)], (18b)
for Eqs. (16a) and (16b), respectively, where Aa is the
maximum deflection of the cylinder, •]p•/2 is the radius
of the first Fresnel zone of the bent cylinder, and the
Fresnel integrals C and S are given as
C(•)= cos • d• and
•(•1,: f:l sin(; •2)dg ' (19,
The Appendix contains expressions for C and S for
arbitra• bends (i.e., deflections are not restricted with re-
spect to wavelength).
b. Small deflections. In the limit of deflections of the
bent cylinder that are small compared with a wavelength,
the above equations reduce to the straight cylinder case
and L• = L.
c. Large deflections. For deflections of the bent cylin-
der much greater than a ce•ain fraction of a wavelength,
or more precisely, k•pc•m•}t or 8k•AL2/•za• 1, one can
use the method of stationau phase or asymptotic limits of
the Fresnel integrals (the Appendix) to solve for Le• in
•s. (16a) and (16b):
L• • •p•l/2e i•/4 (20a)
and
L• • ( L/4 ) •/A ae i•/4, ( 20b )
respectively. Substituting these expressions into •. (15b)
gives
f•= 4p•/2ei•4f•(O=O')/L, (21a)
1
f•=• A•e•'/4A½(0= 0•), (2lb)
where the scattering amplitude is shown to be equM to
within a phase factor to the product of the radius of the
first Fresnel zone of the bent cylinder and the nomalized
scattering amplitude of the straight cylinder at nomal in-
cidence in Eq. (21 a) and propo•ional to the product. of the
square root of the ratio of the wavelength to maximum
deflection of the bent cylinder and str•ght cylinder scat-
tering amplitude in •. (2lb).
II. NUMERICAL SIMULATIONS
In this section, the ray solution is explored under a
variety of conditions. The material properties and shape
3458 d. Acoust. Soc. Am., Vol. 94, No. 6, December 1993 Stanton et al.: Scattering by deformed fluid cylinders 3458