resonance scattering and radiation force calculations for an elastic cylinder using the translational addition theorem for cylindrical wave functions

Resonance scattering and radiation force calculationsfor an elastic cylinder using the translational addition theorem for cylindrical wave functions F.. Mitria Chevron, Area 52 Technolog

the translational addition theorem for cylindrical wave functions

F G Mitri

Citation: AIP Advances 5, 097205 (2015); doi: 10.1063/1.4931916

View online: http://dx.doi.org/10.1063/1.4931916

View Table of Contents: http://aip.scitation.org/toc/adv/5/9

Published by the American Institute of Physics

Resonance scattering and radiation force calculations

for an elastic cylinder using the translational addition

theorem for cylindrical wave functions

F G Mitria

Chevron, Area 52 Technology – ETC, Santa Fe, NM 87508, United States

(Received 27 July 2015; accepted 16 September 2015; published online 23 September 2015)

The standard Resonance Scattering Theory (RST) of plane waves is extended for the case of any two-dimensional (2D) arbitrarily-shaped monochromatic beam inci-dent upon an elastic cylinder with arbitrary location using an exact methodology based on Graf’s translational addition theorem for the cylindrical wave functions The analysis is exact as it does not require numerical integration procedures The formulation is valid for any cylinder of finite size and material that is immersed

in a nonviscous fluid Partial-wave series expansions (PWSEs) for the incident, internal and scattered linear pressure fields are derived, and the analysis is further extended to obtain generalized expressions for the on-axis and off-axis acoustic radiation force components The wave-fields are expressed using generalized PWSEs involving the beam-shape coefficients (BSCs) and the scattering coefficients of the cylinder The off-axial BSCs are expressed analytically in terms of an infinite PWSE with emphasis on the translational offset distance d Numerical computations are considered for a zeroth-order quasi-Gaussian beam chosen as an example to illustrate the analysis Acoustic resonance scattering directivity diagrams are calculated by subtracting an appropriate background from the expression of the scattered pressure field In addition, computations for the radiation force exerted on an elastic cylinder centered on the axis of wave propagation of the beam, and shifted off-axially are analyzed and discussed C 2015 Author(s) All article content, except where other-wise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4931916]

I INTRODUCTION

The present study concerns the extension of the standard formalism of the acoustic Resonance Scattering Theory (RST) for plane wave incident upon an elastic cylinder,1 3 to the case of 2D acoustical beams of arbitrary character The extension of the classical formalism is necessary, since the fundamental features of the results which are being analyzed here will be present in a large variety of applications involving scattering processes, ranging from the nondestructive evaluation and imaging of elastic cylinders, to biomedical and biophysical applications using finite beams

as opposed to plane waves of infinite extent It is of some importance to develop a formalism which accounts for the nature and finite character of the incident field, since the resulting resonance scattering process may be enhanced or suppressed depending on the beam’s parameters, as recently demonstrated for the case of an elastic sphere (or a shell) placed arbitrarily in an acoustical beam

of arbitrary wave-front.4 , 5Nevertheless, analyzing the arbitrary scattering with a cylinder using the formalism devoted to spheres4,5 as a first approximation, leads to significant inaccuracies in the numerical predictions Therefore, it is of particular importance to develop a method applicable to cylinders in order to numerically predict and compute the arbitrary scattering, and other phenomena such as the resulting acoustic radiation force for various applications

a Electronic mail: F.G.Mitri@ieee.org

Despite the extensive analyses related to plane wave scattering by an infinitely-long cylin-der,2 , 6 9earlier works considered the case of beams, based on the angular spectrum decomposition

of plane waves, requiring the numerical evaluation of indefinite integrals.10,11 On the other hand, the present analysis is based on the partial-wave series expansion (PWSE) method (known also as normal-mode decomposition in Fourier series) in cylindrical coordinates, and the evaluation of the beam-shape coefficients (BSCs) stemming from Graf’s additional theorem for the cylindrical wave functions, without the need of numerical integration procedures, used previously in the method of the angular spectrum decomposition into plane waves, or in the computation of the acoustic radia-tion force on a rigid (sound impenetrable) cylinder.12It is also worth mentioning that the formalisms for the electromagnetic scattering of optical beams by (dielectric or perfectly conducting) cylin-drical objects have used the angular spectrum and PWSE methods with numerical integration in several studies.13 – 18Nonetheless, the extension to predict and numerically compute the resonance scattering as well as the resulting acoustic radiation force using an exact method such as Graf’s additional theorem, remains to be accomplished, which is developed here in this investigation The analysis is based on the PWSE method in cylindrical coordinates and the evaluation

of the on- and off-axis beam-shape coefficients (BSCs) using the translational addition theorem Normalized form functions are derived and acoustic (resonance) scattering directivity diagrams are calculated by subtracting an appropriate background to isolate the pure resonances of an aluminum cylinder placed arbitrarily in the field of an acoustical zeroth-order quasi-Gaussian beam, chosen as

an example Numerical predictions for the axial and transverse acoustic radiation force components are also provided Particular emphasis is given on the effect of shifting the cylinder arbitrarily with respect to the incident waves, as well as the focusing properties of the illuminating beam

II GENERALIZED THEORY OF RESONANCE SCATTERING FOR A CYLINDER

IN A NON-VISCOUS FLUID

Consider an acoustical beam propagating in a nonviscous fluid of density ρ and a speed c, and incident upon an infinitely-long cylinder of radius a and density ρc The center of the cylinder coincides with the origin of a cylindrical coordinate system (r, θ, z), and the incident beam is of arbitrary shape (Fig.1) The acoustic field is described by its incident pressure field Pincthat is a solution of the Helmholtz wave equation,

where the wave number k= ω/c, ω is the angular frequency, and c is the compressional speed of sound in the surrounding fluid

For the case of a monochromatic beam, the most general separation of variables (non-singular) solution of (1) in cylindrical coordinates is (p 430 in Ref.19),

Pinc(r, θ, z)= P0

+∞



n =−∞

bnJn(krr) ei(kz z +nθ), (2)

where P0is the pressure amplitude in the absence of the cylinder, bn are the BSCs that will be determined subsequently, Jn(·) is the cylindrical Bessel function of the first kind, krand kzare the radial and axial wave-numbers, respectively, such that k2= k2+ k2.20A time-harmonic variation in the form of e−iωt is assumed, but suppressed for convenience from (2) since the space-dependent pressure field is only considered It is important to emphasize that (2) (and the subsequent equa-tions) is only applicable to 2D beams, with a discrete (single-valued) kz For the more general case

of beams bounded also in the z-direction, the mathematical expression for the incident wave-field must include an integration over all possible values of kz(i.e., continuous spectrum) (See Eq.(4.46)

in Ref.20)

Regarding the determination of the BSCs bn, there exist two different methods that could be used for this purpose In the first method, b can be obtained after integrating both sides of (2) using

FIG 1 The schematic describing the interaction of a 2D arbitrary-shaped acoustical beam with a cylinder of radius a of arbitrary location immersed in a non-viscous fluid The cylindrical coordinate system (r, θ, z) is referenced to the center of the cylinder, while the system of coordinates (r 0 , θ 0 , z 0 ) is referenced to the center of the beam.

the orthogonality relationship

2π

 0

where δi j is the Kronecker delta function, so that the BSCs are expressed after some algebraic manipulation as,

bn= e−ikzz 2πP0Jn(krr)

2π

 0

Since the BSCs describe the incident beam’s characteristics in the cylindrical coordinate system independently of the presence of the cylinder and its size, it is important to note from Eq (4) that the axial wave number with discrete value kzis solely determined by the form of the incident pressure field in order to cancel out any z-dependence in the BSCs Moreover, the integral must be proportional to Jn(krr) to eliminate any r-dependence as well From a numerical analysis stand-point, however, the apparent r-dependence in the denominator in Eq (4) may lead to mathematical indeterminacies if krr is not adequately chosen so as to avoid the zeros of Jn(krr) A way to circumvent this difficulty requires evaluating the BSCs over a virtual control cylindrical surface

of radius Rcthat encloses the cylinder of radius a such that Rc≫ a This approach may only be used for beams that satisfy (1) and propagate in an isotropic fluid medium in which attenuation and diffraction effects in the space between the virtual cylindrical surface and the cylinder are negligible Therefore, the BSCs evaluated at r = Rcare equivalent to those evaluated at r= a Note that evaluation of the BSCs as given by Eq (4) requires numerical integration21for a given incident pressure field Pinc(r, θ, z)

In the second method, which will be used here, evaluation of the BSCs is accomplished using the exact method of Graf’s translational addition theorem for the cylindrical wave functions (Ch 9

in Ref.22) The method requires deriving the off-axial BSCs b in the coordinates system centered

on the cylinder from a known expression of the BSCs (denoted by bp) in the coordinates system centered on the incident beam’s axis At point M (see Fig 1), the incident field, written in the system of coordinates (r0, θ0, z0), can be expressed as,

Pinc(r0, θ0, z0)= P0

+∞



p =−∞

bpJp(krr0) ei(kz z0+pθ 0 ) (5) Based on Graf’s addition theorem (Ch 9 in Ref.22) and the geometry for the distances and angles as shown in Fig.1, the function Jp(krr0) ei pθ0in (5) can be expanded as a PWSE as,

Jp(krr0) ei pθ0= +∞

n =−∞

Jn−p(krd) Jn(krr) ei(n−p)(θ−φ)ei pθ (6) Substituting (6) into (5), and equating the result with (2), leads to the expression for the

off-axial BSCs bnin terms of the axial BSCs bpas,

bn= +∞

p =−∞

bpJn−p(krd) e−i(n−p)φ, (7) where d is the distance that separates the two coordinates systems, φ is termed the shift-angle, which is defined as φ= tan−1(y0−yoff

x0−xoff

) , and d=x2

o ff+ y2

o ff, where xo ff and yo ff determine the amount of offset in the axial and lateral directions, respectively

Eq (7) constitutes the generalized result obtained without any approximations and without numerical integration It can be applicable to any 2D beam for which a closed-form expression for the on-axis BSCs is known, in order to evaluate the off-axis BSCs in the translated system of coordinates

Wave propagation in the cylinder’s core material is represented by suitable solutions of the equation of motion of a solid elastic medium, which may be written as23

(λc+ 2µc) ∇2Uc+ µc∇ ×(∇ × Uc)= ρc

∂2Uc

where λc and µc are the Lamé coefficients The parameter Uc represents the vector displacement that is expressed as a sum of the gradient of a scalar potential Φc representing longitudinal waves and the curl of a vector potential Ψcrepresenting shear waves as follows,

The displacement equations are satisfied if the potentials Φc and Ψc satisfy the Helmholtz equations for the solid medium,

∇2+ k2

∇2+ k2

where kL= ω/cL= ω/[(λc+ 2µc) /ρc]1/2, and kS= ω/cS= ω/[µc/ρc]1/2, refer to the longitudinal and shear wave numbers in the cylinder, respectively

Generally, the vector potential Ψc(ψr,ψθ,ψz) (describing horizontally polarized (SH) and vertically polarized (SV) shear-waves) has three scalar components,24 , 25but to ensure the unique-ness of the solution of (8), the vector potential has to satisfy the gauge invariance ∇ · Ψc= 0,26 , 27

thus, it is a solenoidal vector field This allows defining appropriate potentials related to the longitu-dinal and shear waves inside the cylinder, which are represented in cylindrical coordinates by,

Φc(r, θ, z)= P0

ρcω2

+∞



n =−∞

bnAnΩn(κLr) ei(kz z +nθ), (12)

ψr(r, θ, z)= P0

iρcω2

+∞



n =−∞

bnBnΩn+1(κSr) ei(kz z +nθ), (13)

ψθ(r, θ, z)= − P0

ρcω2

+∞



n =−∞

bnBnΩn+1(κSr) ei(kz z +nθ), (14)

ψz(r, θ, z)= P0

iρcω2

+∞



n =−∞

bnDnΩn(κSr) ei(kz z +nθ), (15)

where κLand κSare defined in TableI, and the functions Ωn(·) represent the Bessel functions of the first kind Jn(·), or the modified Bessel functions of the first kind In(·) respectively, according to the values of α= arctan(kz/kr)

Eqs (13) and (14) show that both potentials ψr and ψθ depend on the coefficients Bn, which

is anticipated from the property of the gauge invariance Note also that instead of finding the three components of the vector potential Ψc, one may express it in terms of a pair of scalar (Debye) potentials (Ch 13 in Ref.28),23,29to obtain directly two scalar potentials representing the SH and

SV waves, respectively.30This method has been employed in the generalized formalism developed for spheres.4It is emphasized that both methods are commensurate with the same result as long

as an elastic isotropic cylinder is considered Nevertheless, for an anisotropic cylinder, different equations have been established when decomposing the vector potential in terms of scalar potentials (See Section 2 in Ref.31)

Upon the interaction of the incident beam with the cylinder, outgoing cylindrically diverging scattered waves are produced in the surrounding medium, which can be represented by

Psc(r, θ, z)= P0

+∞



n =−∞

bnCnHn(1)(krr) ei(kz z +nθ), (16)

where Hn(1)(·) is the cylindrical Hankel function of the first kind of order n, and Cn are the scat-tering coefficients that are determined by applying the boundary conditions of continuity of radial stresses and displacements, and the nullity of the tangential stresses since the surrounding fluid is nonviscous

Denoting the differential operators ∂/∂ (r,θ, z) by ∂r,θ, z, and using the expressions for the potentials given by Eqs (12)-(15), the components of the displacements (ur, uθ, uz) and stresses (σr r, σr θ, σr z) are written, respectively, as,

ur= ∂rΦc+1

uθ=1

uz= ∂zΦc+1

σr θ= µc



∂ruθ+1

r (∂θur− uθ)



TABLE I The expression for the parameters κ L , κ S , Ω n , δ and γ appearing in the matrix elements of the scattering coe fficients C n according to the longitudinal and shear waves coupling angles θ L, S = sin −1

(c/c L, S ).

κ L



k 2

L − k 2

z ,



k 2

z − k 2

L ,



k 2

z − k 2

L ,

κ S



k 2

S − k 2

z ,



k 2

S − k 2

z ,



k 2

z − k 2

S ,

σr z= µc(∂zur+ ∂ruz) , (22) where ∆ in (20) is given by,

∆= ∂rur+1

At the interface fluid-cylinder for r= a, the following boundary conditions are applied,

• continuity of the radial displacement,

ur|r =a= ρω1 ∂r(Pinc+ Psc)

r =a

• continuity of the radial stress,

σr r|r =a= −(Pinc+ Psc)|r =a, (24b)

• nullity of the tangential and shear stresses,

These boundary conditions lead to four linear equations with four unknowns coefficients An,

Bn, Cnand Dn The general solution for the scattering coefficients Cnfor the scattered pressure field

in the fluid medium, which appear in Eq (16) is given by

Cn=

det













M1 m12 m13 m14

M2 m22 m23 m24

M3 m32 m33 m34

M4 m42 m43 m44













det













m11 m12 m13 m14

m21 m22 m23 m24

m31 m32 m33 m34

m41 m42 m43 m44













where Mi and mj k are the dimensionless elements of the determinants given explicitly in Appendix A of Ref 32 These coefficients are found to equal those obtained previously from the study of the acoustic resonance scattering of plane progressive waves by an elastic infinite cylinder

at oblique incidence.32 – 35Note also that the interior (resonance) compressional field of the cylinder which depends on An, as well as the interior (resonance) SH and SV fields which depend on

Bn and Dn, respectively, can be determined in a straightforward manner by solving the matrix equation mj k {Cn, An, Bn, Dn}= {M1, M2, M3, M4} by Cramer’s rule The corresponding interior displacement potential fields for the compressional and shear waves inside the elastic cylinder were evaluated in the context of the standard RST for cylinders.36The plots for the resonating potential fields inside the elastic scattering cylinder displayed “pure” resonance peaks with no interference with any specular reflection background

A Generalized Total Form Function in the Far-Field

In the region far from the cylinder, a dimensionless representation of the acoustic scattered pressure [given by (16)] in the surrounding fluid medium is defined by a generalized (steady-state) total form function as,

f∞(ka, θ) = 

(2r/a) [Psc(r, θ, z) /P0] e−i(kz z +k r r ) (26)

Using the asymptotic limit for the cylindrical Hankel function of the first kind, Hn(1)(krr)

≈ 2/(πkrr) ei (k r r −nπ/2−π/4 ), (26) is expressed as,

f∞(ka, θ)=√ 2

iπkra

+∞



n =−∞

B Generalized Resonance Form Function in the Far-Field

The generalized total form function (27) is obtained as a PWSE, based on the linear theory

of elastodynamics.23 When the form function is plotted as a function of the dimensionless size parameter ka, variations in the form of resonance peaks or dips appear The standard RST formalism for plane waves1 showed that resonance peaks appear because of a constructive interference of inner elastic (resonance) waves with inelastic (non-resonant) outer surface waves circumnavigating the cylinder’s surface in the surrounding fluid (also known as Franz’ waves7), which cause the specular reflection echo in experimental signals.37To isolate the pure resonances, the standard RST suggested that appropriate backgrounds be subtracted from the total form function.38In particular,

it has been found that for dense cylinder materials, the background is closely approximated by the modal amplitudes of rigid-body scattering, whereas for very soft materials, it approaches the soft-body scattering amplitude However, for materials with density comparable to the external fluid, (or for example a dense but quite thin cylindrical shell material), an intermediate back-ground38 has been adequately defined and subtracted from the total form function to properly identify the resonances The main purpose of identifying and isolating resonances is to carry out remote classification of a target in underwater applications, or to characterize some of the physical and mechanical properties of the scatterer

The generalized resonance form function for a cylinder (or cylindrical shell) is defined by,

f∞res(ka, θ)=√ 2

iπkra

+∞



n =−∞

i−nbn



Cn− Cnr, s, i ei nθ, (28)

where Cnr, s, i are the scattering coefficients for a perfectly rigid, a perfectly soft or an intermediate cylinder material The scattering coefficients for the rigid and soft cylinder are given, respectively,

by,7Cr

n= −J′

n(kra) /Hn(1)′(kra), where the prime indicates a derivative with respect to the argument, and Cs

n= −Jn(kra) /Hn(1)(kra) For near-field scattering calculations, the process of isolation of the resonances can be readily applied to (16), in a procedure similar to what is performed in Eq (28)

III ACOUSTIC RADIATION FORCE COMPONENTS FOR A CYLINDER IN A 2D

ARBITRARY-SHAPED BEAM IN A NON-VISCOUS FLUID

Now that the expressions for the incident and scattered fields are established, evaluation of the acoustic radiation force components becomes possible In a non-viscous fluid, it proves advanta-geous to consider the far-field scattering in the evaluation of the force39that is expressed as

⟨F⟩ =

k r → ∞

1

2ρk2 r

 2π 0

where,

Φi s= Φ∗

sc[(i/kr) ∂rΦinc− Φinc− Φsc] (30) The incident and scattered velocity potentials Φinc and Φscin (30) can be expressed in terms of the incident and scattered pressure fields, given by (2) and (16), respectively, such that Φ{inc, sc}

= P{inc, sc}/(iρω) The cylinder’s differential surface is dS = nLr dθ where L is the characteristic length of the cylinder The outward normal unit vector is n= cos θex+ sin θey, where ex and ey are the unit vectors in the Cartesian coordinates system, the symbol⟨·⟩ denotes time-averaging, the superscript * denotes a complex conjugate

Taking the asymptotic limits for Eq (2) and Eq (16) in the far-field scattering limit (i.e kr

→ ∞), and substituting them into Eq (29) using Eq (30) as well as the property of the following angular integral,

 2π 0

ei(n′−n)θ





cos θ sin θ









dθ = π







(δn, n +1+ δn, n−1)

i(δn, n +1−δn, n−1)









where δij is the Kronecker delta function, and manipulating the result, the on-axis and off-axis components of the acoustic radiation force are expressed, respectively, as,









Fx

Fy









= ⟨F⟩ ·







ex

ey









= 







Yx

Yy









where Sc= 2aL is the cross-sectional surface of the cylinder, E0=1

2ρk2

|φ0|2 is a characteristic energy density, and the functions Y{x, y} are the on- and off-axis radiation force functions, that normalize the radiation force components Their expressions are given after some algebraic manipu-lation, respectively, by,

Yx= 1

kraℑ









+∞



n =−∞

bn(1+ Cn)b∗

n +1Cn∗+1− b∗n−1C∗n−1





Yy= − 1

kraℜ









+∞



n =−∞

bn(1+ Cn)b∗

n +1Cn∗+1+ b∗

n−1Cn−1∗ 





where the symbols ℜ{.} and ℑ{.} denote the real and imaginary parts of a complex number, respectively

IV NUMERICAL RESULTS AND DISCUSSIONS

A Numerical Validation

The analysis is started by validating a numerical code that is developed to compute a normal-ized form function,

fnorm,∞(ka, θ)=1

2

+∞



n =−∞

based on the aforementioned generalized acoustic scattering theory of an arbitrary-shaped beam incident upon a cylinder This normalized function (35) is equivalent to a simplified expression given previously for the case of plane waves (p 411 in Ref.6), which allows direct comparison and validation of the results with those given in Ref.6 The truncation order Nmaxfor the index n in the series is chosen such thatCn +N max/C0 ∼ 10−8, for n= 1,2, ., which yields a negligible truncation error

As an initial test of the arbitrary incident beam theory and the computer program, the case of infinite plane wave incidence40is assumed, such that bp= ip The computed BSCs for the infinite plane wave case using (7) are used to evaluate the magnitude of the normalized form function (35), and compare it with the result obtained using the standard RST. fnorm,∞(ka, θ) is evaluated for an aluminum elastic cylinder ( ρc= 2700 kg/m3, cL= 6200 m/s, cS= 3170 m/s) immersed in water ( ρ= 1000 kg/m3, c= 1470 m/s) for ka = 5, and the results are displayed in Fig 2 As noticed, perfect agreement is observed An additional test of the validity of the arbitrary incident beam theory and the computer program considered an incident plane wave such that the cylinder is shifted

off-axially (along the x-direction or the y-direction) The computed scattering directivity diagram

FIG 2 The plot for the modulus of the normalized form function given by Eq ( 35 ) for an aluminum cylinder immersed in water at ka = 5 in infinite plane waves with k z = 0 The plot is computed using the standard elastic scattering theory (See Fig 12 in Ref 6 ), while the dotted circles correspond to the computations obtained using the translational addition theorem Perfect agreement is observed The arrow on the left-hand side denotes the direction of the incident waves.

showed also perfect agreement with Fig 2, such that the shift did not affect the scattering, as expected for the plane wave case

B Off-Axial (Resonance) Scattering of a Cylindrical Quasi-Gaussian Beam by an Elastic Aluminum Cylinder

The present theory is now illustrated by considering numerical computations for the scattering directivity patterns of a zeroth-order quasi-Gaussian beam with kz= 0 and a dimensionless waist

kw0= 7, chosen as an example, and corresponding to a focused beam (See Fig 2 in Ref.41) The total and resonance form-functions, given respectively by Eq (28) and Eq (29) are numerically evaluated, such that the on-axis BSCs40are expressed as bp= ipIp(krxR)/I0(krxR)

Fig.3(a) displays the polar scattering directivity patterns for an aluminum cylinder centered

on the beam’s axis for k a= 5 The directivity pattern is affected by the choice of kw0, such that the scattering lobes around θ = 30◦and 330◦observed in Fig.2are reduced in Fig.3(a) The effect

of shifting the cylinder off-axially along the y−direction in arbitrary units is shown in panel (b) for yo ff= 0.3, while panel (c) corresponds to yo ff= 0.6 The resulting asymmetry in the directivity patterns is obvious when the cylinder is shifted off the beam’s axis Moreover, the patterns in (b) and (c) are asymmetric with respect to one another as expected Panels (d)-(f) show the corresponding resonancedirectivity patterns, which clearly show a quadrupole resonance vibrational mode for the aluminum cylinder at the selected ka Note that the radiating pattern is affected by the translational shift of the beam from the center of the cylinder

The analysis is further extended to calculate the total backscattering (θ = 180◦) form function

as given by Eq (27) as well as its resonance counterpart, given by Eq (28), for the aluminum elastic cylinder placed in the field of a zeroth-order quasi-Gaussian beam Magnitude plots are evaluated in the non-dimensional frequency and transverse shift ranges defined, respectively, as

0 < ka ≤ 10, and −5 ≤ yo ff ≤ 5 Panels (a) and (b) of Fig.4display the results for| f∞(ka, θ= π)| and  fres

∞ (ka, θ= π), respectively, for kw0= 7 In panels (a), (b), the maxima in the plots are observed at the center of the beam (i.e., yo ff= 0) Moreover, panel (b) clearly shows the effects of background subtraction such that the pure resonances appear clearly in the plot