Shock and Vibration 10 (2003) 339–363 339IOS Press

Dynamic viscoelastic effects on sound wavediffraction by spherical and cylindrical shellssubmerged in and filled with viscouscompressible fluids

Seyyed M. Hasheminejad and Naemeh SafariDepartment of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, IranTel.: +9821 73912936; 9821 2557166; Fax: +9821 7451143; E-mail: hashemi@iust.ac.ir

Received 13 November 2002

Revised 8 June 2003

Abstract. An analysis for sound scattering by fluid-filled spherical and cylindrical viscoelastic shells immersed in viscousfluids is outlined. The dynamic viscoelastic properties of the scatterer and the viscosity of the surrounding and core fluids arerigorously taken into account in the solution of the acoustic scattering problem. The novel features of Havriliak-Negami model forviscoelastic material dynamic behaviour description along with the appropriatewave-harmonic field expansions and the pertinentboundary conditions are employed to develop a closed-form solution in form of infinite series. Subsequently, the associatedacoustic field quantities such as the scattered far-field pressure directivity pattern, form function amplitude, transmitted intensityratio, and acoustic force magnitude are evaluated for given sets of medium physical properties. Numerical results clearly indicatethat in addition to the traditional fluid viscosity-related mechanisms, the dynamic viscoelastic properties of the shell material aswell as its thickness can be of major significance in sound scattering. Limiting cases are examined and fair agreements withwell-known solutions are established.

1. Introduction

Historically, sound wave scattering by cylindrical and spherical objects has been investigated quite extensivelysince works of Rayleigh [1] and Lamb [2]. For example, the scattering of acoustic waves has been broadly studiedfor a rigid, fixed, solid sphere and circular cylinder [2–4]; for elastic solid sphere and a circular cylinder [5–10];for elastic spherical and cylindrical shells [11–17], and for coated shells [18–20] in inviscid fluids. On the otherhand, investigations of sound scattering by various objects with allowance for various dissipation mechanisms, suchas viscous and thermal losses and complicated boundary or scatterer models, have been reported in great manypapers [21–25]. The sustained interest in these problems is due to the importance of scattering and attenuation inmany areas of research such as cloud physics, rocket propulsion, dispersion ultrasonics and underwater acoustics.The inclusion of viscosity in the fluid model was first made by Sewell [26], who treated sound absorption byrigid, fixed spheres and circular cylinders in a viscous gas. Later, Lamb [2] simplified the Sewell’s treatmentand studied sound scattering by rigid, fixed or movable spheres in viscous fluid. The most well known acoustictheory for heterogeneous systems was developed by Epstein and Carhart [27], Allegra and Hawley [28] (i.e., the

ISSN 1070-9622/03/$8.00 2003 – IOS Press. All rights reserved

340 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

so called ECAH theory). Lin and Raptis [29,30] presented analytical solutions as well as numerical results for theboundary value problem concerning the interaction of a plane sound wave with (thermo)elastic solid cylinders andspheres immersed in (thermo)viscous fluid. They studied the effects of the fluid viscosity (thermal conductivity),and scatterer’s (thermo)elasticity on acoustic-wave scattering patterns and acoustic-radiation force. Later, the sameauthors presented a general analysis for scattering of a plane sound wave obliquely incident upon a thin, elasticcircular rod immersed in an unbounded viscous fluid [31]. The scattered fields of the incident sound wave, theinteraction of the sound waves and the rod, and sound radiation by acoustoelastic vibrations of the rod were studiedin great details. Likewise, acoustic scattering from a spherical shell including viscous and thermal effects in thesurrounding fluid and also the viscoelastic losses in both core and shell material was analyzed in [32]. Hasegawa andWatanabe [33] modified the standard harmonic series theory to study the effect of hysteresis type of absorption onacoustic field of an absorbing sphere immersed in an ideal fluid. Their analysis, however, is applicable to materialswith frequency-independent sound absorption per wavelength and may not be considered to be satisfactory from thestandpoint of generality and completeness. Just recently, Hasheminejad and Harsini [34] employed the novel featuresof Havriliak-Negami (HN) model to investigate effects of dynamic viscoelastic properties on acoustic scattering by asolid sphere submerged in a viscous fluid. The prime objective of present work is to utilize HN model to investigateeffects of dynamic viscoelastic properties on acoustic interaction of plane sound waves with fluid-filled sphericaland cylindrical shells submerged in viscous fluids.

2. Viscoelastic model

Accurate mathematical modelling of viscoelastic materials is difficult mainly because their measured dynamicproperties are frequency and temperature dependent, and can also depend on the type of deformation and amplitude.Consequently, mathematical models describing the behaviour of viscoelastic materials cannot be clearly linked tothe physical principles involved and thus empirical approaches are used. The most popular approach, called thestructural damping model, uses complex constants as the material moduli. Strictly speaking, for viscoelastic andisotropic materials, two independent complex moduli are necessary for mechanical characterization; for examplethe Young’s modulusE∗(ω) and the shear modulusG∗(ω). Both moduli, in principle, are frequency dependent.The main difficulty is the simultaneous presence of the Young’s and shear complex moduli as well as the Poissonratio. Practically, however, for viscoelastic isotropic materials the hypothesis of a constant (frequency-independent)and real Poisson ratio is often adopted [35,36]. This assumption has been validated with some experiments [37]revealing that the imaginary part of the Poisson ratio of elastomers is less than 1% of the real part. For elastomers,the Poisson ratio is often supposed to have the constant value 0.5 (i.e., incompressible material), which implies that

G∗(ω) = G′(ω) + iG′′(ω) =E∗(ω)

2(1 + v)≈ 1

3E∗(ω) (1)

Frequency dependence ofG ′(ω) andG′′(ω) in the viscoelastic transition region has been the object of manyexperimental and theoretical studies [38]. The most successful description for the frequency dependence of thecomplex modulus of polymers in the glass transition region is perhaps given by Havriliak-Negami model [39].According to HN model, the real and imaginary parts of the complex modulus are given by [40]

G′(ω) = G∞ +(G0 −G∞) cos(βθ)

[1 + 2ωατα cos γ + ω2ατ2α]β/2

(2)

G′′(ω) =(G∞ −G0) sin(βθ)

[1 + 2ωατα cos γ + ω2ατ2α]β/2

whereγ = απ/2, and the loss factor is specified by

η(ω) =G′′(ω)G′(ω)

=(1 − χ) sin(βθ)

[1 + 2ωατα cos γ + ω2ατ2α]β/2 − (1 − χ) cos(βθ)(3)

in whichχ = G0/G∞, and

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 341

θ(ω) = tan−1 ωατα sin γ1 + ωατα cos γ

(4)

Here we note thatη(ω) depends only on the ratioχ = G0/G∞, not on their individual values. Furthermore,G0 (relaxed modulus) andG∞ (unrelaxed modulus) are the limiting values of the shear modulus at low and highfrequencies, respectively,τ(= 1/ω0) is the relaxation time associated with the polymer glass transition centerfrequency (loss factor peak),α is a dimensionless parameter(0 < α < 1) that governs the width of the relaxation,andβ is another dimensionless parameter(0 < β < 1) that governs the asymmetry of the relaxation.

3. Governing field equations

Following the standard methods of continuum mechanics, the linearized equations of continuity and Navier-Stokesfor a viscous non-heat-conducting compressible fluid are respectively [41]

∂ρ

∂t+ ρ∇ · u = 0

(5)

ρ∂u

∂t+ ∇p = µ∇2u+

(13µ+ µb

)∇(∇ · u)

Here,ρ is the mass density, andp represents the deviation of pressure from its mean value,u is the fluid velocityvector, andµ andµb are the shear and the expansive (bulk) coefficient of viscosity, respectively. For a barotropicfluid, the linearized equation of state isp = c2ρ, where, as usual, c is the ideal speed of sound evaluated at ambientconditions. Equation (5) can readily be combined to yield a single equation for the velocity vectoru:

∂2u

∂t2− c2∇(∇ · u) =

µ

ρ∇2 ∂u

∂t+

1ρ

(13µ+ µb

)∇

(∇ · ∂u

∂t

)(6)

Helmholtz decomposition theorem allows us to resolve the velocity fields as superposition of longitudinal andtransverse vector components

u = −∇ϕ+ ∇×ψ (7)

Introducing the above decomposition into Eq. (6), making use of problem symmetry,ψ = (0., 0., ψ), and thecalibration condition,∇ ·ψ = 0, a set of two equations is deduced

∂2ϕ

∂t2=

[c2 +

1ρ

(43µ+ µb

)∂

∂t

]∇2ϕ,

∂ψ

∂t=

µ

ρ∇2ψ (8)

Now, since the incident wave is assumed to be monochromatic, with frequencyω, we expect solutions of the form

ϕ(r, θ, t) = Re[ϕ(r, θ, ω)e−iωt], ψ(r, θ, t) = Re[ψ(r, θ, ω)e−iωt] (9)

whereRe indicates the real part of a complex number, and quantitiesϕ(r, θ, ω) andψ(r, θ, ω) may be complex.Incorporation of above assumptions in Eq. (8), after some manipulations, yields

(∇2 + k2c )ϕ = 0, (∇2 + k2

s)ψ = 0 (10)

where the complex compressional and shear wave numbers in the viscous fluid are given by [41]

kc =ω

c

(1 + i

ωµ

2ρc2

(43

+µb

µ

)), ks = (1 + i)

√ωρ

2µ(11)

The viscoelastic material under consideration is assumed to be linear, macroscopically homogeneous, and isotropicfor which the constitutive equation, for harmonic time functions, may be written as [42]

σij = λ∗(ω)δijεkk + 2µ∗s(ω)εij (12)

whereδij is Kronecker delta symbol,λ∗(ω) andµ∗s(ω) are complex, frequency dependent Lame functions which

are determined according to the standard relations

342 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

a)

b)

Fig. 1. Problem geometry; a) Spherical coordinate system, b) Cylindrical coordinate system.

λ∗(ω) =2v

1 − 2vG∗(ω), µ∗

s(ω) = G∗(ω) (13)

in which the real and imaginary parts of the complex shear modulus,G ∗(ω), are specified in Eq. (2). The wavemotion in the viscoelastic shell is governed by the classical Navier’s equation [43]

ρs∂2U

∂t2= µ∗

s∇2U + (λ∗ + µ∗s)∇(∇ · U) (14)

subject to the appropriate boundaryconditions. Here,ρ s is the solid material density, andU is the vector displacementthat can advantageously be expressed as sum of the gradient of a scalar potential and the curl of a vector potential:

U = ∇Φ + ∇× Ψ (15)

with the condition∇ · Ψ = 0, andΨ = (0., 0.,Ψ). The above decomposition enables us to separate the dynamicequation of motion into the classical Helmholtz equations:

(∇2 +K∗2

c )Φ = 0, (∇2 +K∗2

s )Ψ = 0 (16)

whereK∗c andK∗

s are complex wave numbers, known as [42]

K∗c =

ω√λ∗(ω)+2µ∗

s(ω)ρs

, K∗s =

ω√µ∗

s(ω)ρs

(17)

4. Sound wave diffraction by a spherical viscoelastic shell

In this section we consider the general problem of acoustic scattering from a (viscous) fluid-filled sphericalviscoelastic shell suspended in an unbounded viscous fluid. The geometry and the coordinate system used aredepicted in Fig. 1a. Mathematically, the dynamics of the problem may be expressed in terms of appropriate scalar

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 343

potentials. The expansion of the incident plane wave in spherical coordinate has the form [44]

ϕinc.(r.θ) = P0

∞∑n=0

(2n+ 1)injn(kc1r)Pn(cos θ) (18)

wherejn are spherical Bessel functions [45],Pn are Legendre polynomials, andP0 is the amplitude of the incidentwave. Likewise, keeping in mind the radiation condition, the solutions of the Helmholtz equations for the potentialsin the surrounding fluid medium I and the encapsulated fluid medium II can be expressed as a linear combination ofoutgoing spherical waves as follows

ϕ1(r, θ) =∞∑

n=0

(2n+ 1)inAnh(1)n (kc1r)Pn(cos θ), ϕ2(r, θ) =

∞∑n=0

(2n+ 1)inGnjn(kc2r)Pn(cos θ),

(19)

ψ1(r, θ) =∞∑

n=1

(2n+ 1)inBnh(1)n (ks1r)P 1

n(cos θ), ψ2(r, θ) =∞∑

n=1

(2n+ 1)inQnjn(ks2r)P 1n(cos θ),

whereh(1)n are spherical Hankel functions of first kind [45],P 1

n = −(d/dθ)Pn are associated Legendre functions,An, Bn, Gn andQn are unknown scattering coefficients, and the subscripts 1 and 2 refer to the surrounding fluidmedium I and the inner fluid medium II, respectively. Furthermore, the transmitted longitudinal and transversewaves in the viscoelastic shell are represented by

Φ(r, θ) =∞∑

n=0

(2n+ 1)in[Cnh(1)n (K∗

c r) +Dnh(2)n (K∗

c r)]Pn(cos θ)

(20)

Ψ(r, θ) =∞∑

n=1

(2n+ 1)in[Enh(1)n (K∗

s r) + Fnh(2)n (K∗

s r)]P1n (cos θ)

whereh(2)n are spherical Hankel functions of second kind [45] and the superscript∗ indicates that complex, frequency-

dependent viscoelastic properties are involved.Now considering the basic field equations in spherical coordinates, the velocity components of the waves inr-

andθ-directions in terms of potentials in the viscous fluids are [41]

u1r = −∂ϕ

∂r+

1r sin θ

∂(ψ1 sin θ)∂θ

u2r = −∂ϕ2

∂r+

1r sin θ

∂(ψ2 sin θ)∂θ (21)

u1θ = −1

r

∂ϕ

∂θ− 1r

∂(rψ1)∂r

u2θ = −1

r

∂ϕ2

∂θ− 1r

∂(rψ2)∂r

whereϕ = ϕinc. + ϕ1. Similarly, the displacement components in the viscoelastic shell are [43]

Ur =∂Φ∂r

+1

r sin θ∂(Ψ sin θ)

∂θUθ =

1r

∂Φ∂θ

− 1r

∂(rΨ)∂r

(22)

The stress components in the viscous fluids are [41]

σ1rr = −p1 +

(µb1 − 2

3µ1

)∆1 + 2µ1(∂u1

r/∂r) σ2rr = −p2 +

(µb2 − 2

3µ2

)∆2 + 2µ2(∂u2

r/∂r)(23)

σ1rθ = µ1

(1r

∂u1r

∂θ+∂u1

θ

∂r− u1

θ

r

)σ2

rθ = µ2

(1r

∂u2r

∂θ+∂u2

θ

∂r− u2

θ

r

)

where

p1 = −iωρ1ϕ+(µb1 +

43µ1

)∆1 p2 = −iωρ2ϕ2 +

(µb2 +

43µ2

)∆2 (24)

in which ∆1 = ∇ · u1 = −∇2ϕ = k2c1ϕ, ∆2 = ∇ · u2 = −∇2ϕ2 = k2

c2ϕ2. Likewise, the stresses in theviscoelastic scatterer are [42]

344 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

σsrr = 2µ∗

s

∂Ur

∂r+ λ∗ε σs

rθ = µ∗s

(1r

∂Ur

∂θ+∂Uθ

∂r− Uθ

r

)(25)

whereε = ∇ · U = ∇2Φ = 0 −K∗2

C Φ.The appropriate boundary conditions that must hold at the interfaces of the inner and outer fluids with the shell

are continuity of velocity and stress components, i.e.,

u1r(a, θ, ω) = −iωUr(a, θ, ω) − iωUr(b, θ, ω) = u2

r(b, θ, ω)

u1θ(a, θ, ω) = −iωUθ(a, θ, ω) − iωUθ(b, θ, ω) = u2

θ(b, θ, ω)(26)

σ1rr(a, θ, ω) = σs

rr(a, θ, ω) σsrr(b, θ, ω) = σ2

rr(b, θ, ω)

σ1rθ(a, θ, ω) = σs

rθ(a, θ, ω) σsrθ(b, θ, ω) = σ2

rθ(b, θ, ω)

The unknown scattering coefficients shall be determined by imposing the stated boundary conditions. Employingexpansions Eqs (18) through (20) in the field Eqs (21) through (25), and substituting the obtained results into theboundary conditions Eq. (26), we obtain, for then-th mode

−kc1Anh(1)′n (kc1a) +

n(n+ 1)a

Bnh(1)n (ks1a) + iωK∗

c

[Cnh

(1)′n (K∗

c a) +Dnh(2)′n (K∗

c a)]

(27a)

+iωn(n+ 1)

a

[Enh

(1)n (K∗

sa) + Fnh(2)n (K∗

sa)]

= kc1P0j′n(kc1a)

Anh(1)n (kc1a) −

[h(1)

n (ks1a) + ks1ah(1)′n (ks1a)

]Bn − iω

[Cnh

(1)n (K∗

c a) +Dnh(2)n (K∗

c a)]

(27b)−iω

{[h(1)

n (K∗sa) + k∗sah

(1)′n (K∗

sa)]En +

[h(2)

n (K∗sa) +K∗

sah(2)′n (K∗

sa)]Fn

}= −P0jn(kc1a)

[(iωρ1 − 2µ1k

2c1)h

(1)n (kc1a) − 2µ1k

2c1h

(1)′′n (kc1a)

]An

−2µ1n(n + 1)a2

[h(1)

n (ks1a) − ks1ah(1)′n (ks1a)

]Bn

−K∗2

c

{[2µ∗

sh(1)′′n (K∗

c a) − λ∗h(1)n (K∗

c a)]Cn +

[2µ∗

sh(2)′′n (K∗

c a) − λ∗h(2)n (K∗

c a)]Dn

}(27c)

+2µ∗

sn(n + 1)a2

{[h(1)

n (K∗sa) −K∗

sah(1)′n (K∗

sa)]En +

[h(2)

n (K∗sa) −K∗

sah(2)′n (K∗

sa)]Fn

}

= − [(iωρ1 − 2µ1k

2c1)jn(kc1a) − 2µ1k

2c1j

′′n(kc1a)

]P0

−2µ1

{h(1)

n (kc1a) − kc1ah(1)′n (kc1a)

]An + µ1

{[2 − n(n+ 1)]h(1)

n (ks1a) − k2s1a

2h(1)′′n (ks1a)

}Bn

+2µ∗s

{{K∗

c ah(1)′n (K∗

c a) − h(1)n (K∗

c a)]Cn +

{K∗

c ah(2)′n (K∗

c a) − h(2)n (K∗

c a)]Dn

}

−µ∗s

{[2 − n(n+ 1)]h(1)

n (K∗sa) −K∗2

s a2h(1)′′n (K∗

sa)}En (27d)

−µ∗s

{[2 − n(n+ 1)]h(2)

n (K∗sa) −K∗2

s a2h(2)′′n (K∗

sa)}Fn

= 2µ1 [jn(kc1a) − kc1aj′n(kc1a)]P0

iωK∗c

[Cnh

(1)′n (K∗

c b) +Dnh(2)′n (K∗

c b)]

+ iωn(n+ 1)

b

[Enh

(1)n (K∗

s b) + Fnh(2)n (K∗

s b)]

(27e)

−kc2Gnj′n(kc2b) +

n(n + 1)b

Qnjn(kc2b) = 0

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 345

−iω[Cnh

(1)n (K∗

c b) +Dnh(2)n (K∗

c b)]− iω

[h(1)

n (K∗s b) +K∗

s bh(1)′n (K∗

s b)]En

(27f)−iω

[h(2)

n (K∗s b) +K∗

s bh(2)′n (K∗

s b)]Fn +Gnjn(kc2b) − [jn(ks2b) + ks2bj

′n(ks2b)]Qn = 0

K∗2

c

{[λ∗h(1)

n (K∗c b) − 2µ∗

sh(1)′′n (K∗

c b)]Cn +

[λ∗h(2)

n (K∗c b) − 2µ∗

sh(2)′′n (K∗

c b)]Dn

}

−2n(n+ 1)b2

µ∗s

{[−h(1)

n (K∗s b) +K∗

s bh(1)′n (K∗

s b)]En +

[−h(2)

n (K∗s b) +K∗

s bh(2)′n (K∗

s b)]Fn

}(27g)

+[(iωρ2 − 2µ2k

2c2)jn(kc2b) − 2µ2k

2c2j

′′n(kc2b)

]Gn − 2µ2n(n + 1)

b2[jn(ks2b) − ks2bj

′n(ks2b)]Qn = 0

2µ∗s

[K∗

c bh(1)′n (K∗

c b) − h(1)n (K∗

c b)]Cn + 2µ∗

s

[K∗

c bh(2)′n (K∗

c b) − h(2)n (K∗

c b)]Dn

−µ∗s

{[2 − n(n+ 1)]h(1)

n (K∗s b) −K∗2

s b2h(1)′′n (K∗

s b)}En

(27h)−µ∗

s

{[2 − n(n+ 1)]h(2)

n (K∗s b) −K∗2

s b2h(2)′′n (K∗

s b)}Fn

−2µ2 [jn(kc2b) − kc2bj′n(kc2b)]Gn + µ2

{[2 − n(n + 1)]jn(ks2b) − µ2k

2s2b

2j′′n(ks2b)}Qn = 0

wheren = 0, 1, 2, . . . , except for Eqs (27b), (27d), (27f), (27h) wheren = 1, 2, . . .The most relevant acoustic field quantities are scattered pressure and transmitted intensity. Substitution of first of

Eq. (19) into the first of Eq. (24) yields the acoustic pressure for harmonic scattered waves. The radial component ofthe acoustic power flux vector (acoustic intensity) transmitted into the encapsulated fluid medium II may be obtainedfrom [21]

Itrans.(r, θ, ω) =12Re

{σ2

rr × conj(u2r) + σ2

rθ × conj(u2θ)

}(28)

whereconj stands for the complex conjugate operator. Correspondingly, the quotient|I trans./Iinc.| represents thepower transmitted per unit solid angle and per unit incident intensity(I inc. = P0ρω

2/2c) into the shell.The force acting on the spherical shell due to the acoustic field present in the surrounding viscous fluid medium

can be found by determining the fluid stresses on the outer surface of the shell. This force will have only onecomponent that is given by [41]

F = 2πa2

∫ π

0

(σ1rr cos θ − σ1

rθ sin θ) sin θdθ (29)

The fluid stress tensor components as obtained from Eq. (23) may be expressed in the form

σ1rr =

∞∑n=0

σ1rrnPn(cos θ) σ1

rθ =∞∑

n=1

σ1rθnP

1n(cos θ) (30)

Substituting the above expansions into the force Eq. (29), using the appropriate orthogonality relation for Legendrefunctions [45] and performing the integration, we find

F = 2πa2

(23σ1

rr1 +43σ1

rθ1

)(31)

5. Sound wave diffraction by a cylindrical viscoelastic shell

In this section we consider a compressional planar monochromatic sound wave of angular frequencyω obliquelyincident upon a (viscous) fluid-filled cylindrical viscoelastic shell immersed in an unbounded viscous fluid at rest(Fig. 1b). When the sound wave meets the shell, both the wave and the obstacle are affected by the interaction.

346 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

Consequently, flexural and compressional oscillations in the shell are induced which in turn radiate sound wavesinto the neighbouring fluid medium and affect the scattered waves. The dynamics of the problem can be expressedin terms of appropriate scalar potentials. The expansion of the incident plane wave in cylindrical coordinate has theform [44]

ϕinc. = P0

∞∑n=0

εninJn(γc1r) cosnθeikzz (32)

whereε0 = 1 andεn = 2 for n > 0, P0 is the amplitude of the incident wave,Jn are cylindrical Bessel functionsof first kind [45], andγc1 =

√k2

c1 − k2z in whichkz = Re(kc1 sinα). Next, noting that the outer fluid medium is

unbounded and keeping in mind the radiation condition, the solutions of the Helmholtz equations for the potentialsin the surrounding fluid medium I can be expressed as a linear combination of outgoing cylindrical waves as follows

ϕ1 =∞∑

n=0

AnH(1)n (γc1r) cosnθeikzz ψ1

θ = −∞∑

n=0

BnH(1)n+1(γs1r) cosnθeikzz

(33)

ψ1r =

∞∑n=1

BnH(1)n+1(γs1r) sinnθeikzz ψ1

z =∞∑

n=1

CnH(1)n (γs1r) sinnθeikzz

whereH (1)n are cylindrical Hankel functions of first kind [45],An, Bn andCn are unknown scattering coefficients,

andγs1 =√k2

s1 − k2z . Furthermore, the velocity potentials in the encapsulated fluid medium II can be expressed as

ϕ2 =∞∑

n=0

MnJn(γc2r) cosnθeikzz ψ2θ = −

∞∑n=0

NnJn+1(γs2r) cosnθeikzz

(34)

ψ2r =

∞∑n=1

NnJn+1(γs2r) sinnθeikzz ψ2z =

∞∑n=1

QnJn(γs2r) sinnθeikzz

whereMn, Nn andQn are unknown refraction coefficients,γc2 =√k2

c2 − k2z andγs2 =

√k2

s2 − k2z . Similarly,

the transmitted longitudinal and transverse waves in the viscoelastic shell medium are represented by

Φ =∞∑

n=0

[DnH

(1)n (Γcr) + EnH

(2)n (Γcr)

]cosnθeikzz

Ψr =∞∑

n=1

[FnH

(1)n+1(Γsr) +GnH

(2)n+1(Γsr)

]sinnθeikzz

(35)

Ψθ = −∞∑

n=0

[FnH

(1)n+1(Γsr) +GnH

(2)n+1(Γsr)

]cosnθeikzz

Ψz =∞∑

n=1

[KnH

(1)n (Γsr) + LnH

(2)n (Γsr)

]sinnθeikzz

whereH (2)n are cylindrical Hankel functions of second kind [45],Dn throughLn are unknown transmission

coefficients andΓc =√K∗2

c − k2z , Γs =

√K∗2

s − k2z .

Now considering the basic field equations in cylindrical coordinates, the velocity components of the waves inr-andθ-directions in terms of potentials in the viscous fluids are [29]

u1r = −∂ϕ

∂r+

1r

∂ψ1z

∂θ− ∂ψ1

θ

∂zu2

r = −∂ϕ2

∂r+

1r

∂ψ2z

∂θ− ∂ψ2

θ

∂z

u1θ = −1

r

∂ϕ

∂θ+∂ψ1

r

∂z− ∂ψ1

z

∂ru2

θ = −1r

∂ϕ2

∂θ+∂ψ2

r

∂z− ∂ψ2

z

∂r(36)

u1z = −∂ϕ

∂z+

1r

∂(rψ1θ)

∂r− 1r

∂ψ1r

∂θu2

z = −∂ϕ2

∂z+

1r

∂(rψ2θ)

∂r− 1r

∂ψ2r

∂θ

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 347

whereϕ = ϕinc. + ϕ1. Similarly, the relevant displacements in the viscoelastic shell are [43]

Ur =∂Φ∂r

+1r

∂Ψz

∂θ− ∂Ψθ

∂z

Uθ =1r

∂Φ∂θ

+∂Ψr

∂z− ∂Ψz

∂r(37)

Uz =∂Φ∂z

+1r

∂(rΨθ)∂r

− 1r

∂Ψr

∂θThe stress components in the viscous fluids are [29,31]

σ1rr = −p1 + (µb1 − 2µ1/3)∆1 + 2µ1(∂u1

r/∂r) σ1rr = −p2 + (µb2 − 2µ2/3)∆2 + 2µ2(∂u2

r/∂r)

σ1rθ = µ1

(1r

∂u1r

∂θ+∂u1

θ

∂r− u1

θ

r

)σ2

rθ = µ2

(1r

∂u2r

∂θ+∂u2

θ

∂r− u2

θ

r

)(38)

σ1rz = µ1

(∂u1

r

∂z+∂u1

z

∂r

)σ2

rz = µ2

(∂u2

r

∂z+∂u2

z

∂r

)

and the corresponding stresses in the viscoelastic shell are [43]

σsrr = 2µ∗

s

∂Ur

∂r+ λ∗sε

σsrθ = µ∗

s

(1r

∂Ur

∂θ+∂Uθ

∂r− Uθ

r

)(39)

σsrz = µ∗

s

(∂Ur

∂z+∂Uz

∂r

)

whereε = ∇ · U = ∇2Φ = −K∗2

c Φ.The appropriate boundary conditions that must hold at the fluid/solid interfaces are simply continuity of velocities

and stresses that are written as

u1r(a, θ, ω) = −iωUr(a, θ, ω) − iωUr(b, θ, ω) = u2

r(b, θ, ω)

u1θ(a, θ, ω) = −iωUθ(a, θ, ω) − iωUθ(b, θ, ω) = u2

θ(b, θ, ω)

u1z(a, θ, ω) = −iωUz(a, θ, ω) − iωUz(b, θ, ω) = u2

z(b, θ, ω)(40)

σ1rr(a, θ, ω) = σs

rr(a, θ, ω) σsrr(b, θ, ω) = σ2

rr(b, θ, ω)

σ1rθ(a, θ, ω) = σs

rθ(a, θ, ω) σsrθ(b, θ, ω) = σ2

rθ(b, θ, ω)

σ1rz(a, θ, ω) = σs

rz(a, θ, ω) σsrz(b, θ, ω) = σ2

rz(b, θ, ω)At this point the unknown scattering coefficients shall be determined by imposing the stated boundary conditions.

Employing expansions (32) through (35) in the field Eqs (36) through (39), and substituting obtained results into theboundary conditions Eq. (40), we obtain, for then-th mode

−γc1AnH(1)′n (γc1a) + ikzBnH

(1)n+1(γs1a) +

n

aCnH

(1)n (γs1a)

+iωΓc

[DnH

(1)′n (Γca) + EnH

(2)′n (Γca)

]− ωkz

[FnH

(1)n+1(Γsa) +GnH

(2)n+1(Γsa)

](41a)

+iωn

a

[KnH

(1)n (Γsa) + LnH

(2)n (Γsa)

]= P0εni

nγc1J′n(γc1a)

n

aAnH

(1)n (γc1a) + ikzBnH

(1)n+1(γs1a) − γs1CnH

(1)′n (γs1a)

−iωna

[DnH

(1)n (Γca) + EnH

(2)n (Γca)

]− ωkz

[FnH

(1)n+1(Γsa) +GnH

(2)n+1(Γsa)

](41b)

−iωΓs

[KnH

(1)′n (Γsa) + LnH

(2)′n (Γsa)

]= −P0

n

aεni

nJn(γc1a)

348 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

−ikzaAnH(1)n (γc1a) −

[(n+ 1)H(1)

n+1(γs1a) + γs1aH(1)′n+1(γs1a)

]Bn

−ωkza[DnH

(1)n (Γca) + EnH

(2)n (Γca)

]− iω

{[(n+ 1)H(1)

n+1(Γsa) + ΓsaH(1)′n+1(Γsa)

]Fn (41c)

+[(n+ 1)H(2)

n+1(Γsa) + ΓsaH(2)′

n+1(Γsa)]Gn

}= P0εni

n+1kzaJn(γc1a)

[(iωρ1 − 2µ1k

2c1)H

(1)n (γc1a) − 2µ1γ

2c1H

(1)′′n (γc1a)

]An

+2iµ1kzγs1H(1)′n+1(γs1a)Bn − 2

µ1n

a2

[H(1)

n (γs1a) − γs1aH(1)′n (γs1a)

]Cn

−[2µ∗

sΓ2c1H

(1)′′n (Γca) − λ∗K∗2

c H(1)n (Γca)

]Dn −

[2µ∗

sΓ2cH

(2)′′n (Γca) − λ∗K∗2

c H(2)n (Γca)

]En

(41d)−2iµ∗

skzΓs

[FnH

(1)′n+1(Γsa) +GnH

(2)′n+1(Γsa)

]

+2µ∗

sn

a2

{[H(1)

n (Γsa) − ΓsaH(1)′n (Γsa)

]Kn +

[H(2)

n (Γsa) − ΓsaH(2)′n (Γsa)

]Ln

}

= −P0εnin

[(iωρ1 − 2µ1k

2c1)Jn(γc1a) − 2µ1γ

2c1J

′′n(γc1a)

]

−2µ1n[H(1)

n (γc1a) − aγc1H(1)′n (γc1a)

]An − ikzµ1

[(n+ 1)aH(1)

n+1(γs1a) − γs1a2H

(1)′n+1(γs1a)

]Bn

−µ1

[n2H(1)

n (γs1a) + γ2s1a

2H(1)′′n (γs1a) − aγs1H

(1)′n (γs1a)

]Cn

−2µ∗sn

{[H(1)

n (Γca) − aΓcH(1)′n (Γca)

]Dn +

[H(2)

n (Γca) − aΓcH(2)′n (Γca)

]En

}

+ikzµ∗s

{[(n + 1)aH(1)

n+1(Γsa) − Γsa2H

(1)′n+1(Γsa)

]Fn

(41e)+

[(n+ 1)aH(2)

n+1(Γsa) − Γsa2H

(2)′n+1(Γsa)

]Gn

}

+µ∗s

[n2H(1)

n (Γsa) + Γ2sa

2H(1)′′n (Γsa) − aΓsH

(1)′n (Γsa)

]Kn

+µ∗s

[n2H(2)

n (Γsa) + Γ2sa

2H(2)′′n (Γsa) − aΓsH

(2)′n (Γsa)

]Ln

= 2µ1nP0εnin [Jn(γc1a) − aJ ′

n(γc1a)]

−2ikzµ1a2γc1AnH

(1)′n (γc1a)

+µ1

{[n+ 1 − a2k2

z

]H

(1)n+1(γs1a) − γs1a(n+ 1)H(1)′

n+1(γs1a) − γ2s1a

2H(1)′′n+1 (γs1a)

}Bn

+ikzµ1anCnH(1)n (γs1a) − 2ikzµ

∗s1a

2Γc

[DnH

(1)′n (Γca) + EnH

(2)′n (Γca)

]

−µ∗s

{[n + 1 − a2k2

z

]H

(1)n+1(Γsa) − Γsa(n + 1)H(1)′

n+1(Γsa) − Γ2sa

2H(1)′′n+1 (Γsa)

}Fn (41f)

−µ∗s

{[n + 1 − a2k2

z

]H

(2)n+1(Γsa) − Γsa(n + 1)H(2)′

n+1(Γsa) − Γ2sa

2H(2)′′n+1 (Γsa)

}Gn

−ikzµ∗san

[KnH

(1)n (Γsa) + LnH

(2)n (Γsa)

]= 2P0µ1εni

n+1kzγc1a2J ′

n(γc1a)

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 349

Table 1Parameter values used in calculations

Parameter Gylcerine Olive oil

µ (g/cm.s) 9.50 0.933µb (g/cm.s) 9.50 0.933c (cm/s) 1.91× 105 1.44× 105

ρ (g/cm3) 1.25 0.9

iωΓc

[DnH

(1)′n (Γcb) + EnH

(2)′n (Γcb)

]− ωkz

[FnH

(1)n+1(Γsb) +GnH

(2)n+1(Γsb)

]

+iωn

b

[KnH

(1)n (Γsb) + LnH

(2)n (Γsb)

]− γc2MnJ

′n(γc2b) + ikzNnJn+1(γs2b) (41g)

+n

bQnJn(γs2b) = 0

−iωnb

[DnH

(1)n (Γcb) + EnH

(2)n (Γcb)

]− ωkz

[FnH

(1)n+1(Γsb) +GnH

(2)n+1(Γsb)

]

−iωΓs

[KnH

(1)′n (Γsb) + LnH

(2)′n (Γsb)

]+n

bMnJn(γc2b) + ikzNnJn+1(γs2b) (41h)

−γs2QnJ′n(γs2b) = 0

−kzωb[DnH

(1)n (Γcb) + EnH

(2)n (Γcb)

]

−iω[(n+ 1)H(1)

n+1(Γsb) + ΓsbH(1)′n+1(Γsb)

]Fn − iω

[(n+ 1)H(2)

n+1(Γsb) + ΓsbH(2)′n+1(Γsb)

]Gn (41i)

−ikzbMnJn(γc2b) −[(n + 1)Jn+1(γs2b) + γs2bJ

′n+1(γs2b)

]Nn = 0

[−λ∗K∗2

c H(1)n (Γcb) + 2µ∗

sΓ2cH

(1)′′n (Γcb)

]Dn +

[−λ∗K∗2

c H(2)n (Γcb) + 2µ∗

sΓ∗2

c H(2)′′n (Γcb)

]En

+2ikzµ∗sΓs

[FnH

(1)′n+1(Γsb) +GnH

(2)′n+1(Γsb)

]

−2µ∗sn

b2

{[H(1)

n (Γsb) − bΓsH(1)′n (Γsb)

]Kn +

[H(2)

n (Γsb) − bΓsH(2)′n (Γsb)

]Ln

}(41j)

− [(iωρ2 − 2µ2k

2c2)Jn(γc2b) − 2µ2γ

2c2J

′′n(γc2b)

]Mn

−2iµ2kzγs2J′n+1(γs2b)Nn + 2µ2

n

b2[Jn(γs2b) − γs2bJ

′n(γs2b)]Qn = 0

2µ∗sn

{[H(1)

n (Γcb) − bΓcH(1)′n (Γcb)

]Dn +

[H(2)

n (Γcb) − bΓcH(2)′n (Γcb)

]En

}

−ikzµ∗s

{[(n + 1)bH(1)

n+1(Γsb) − Γsb2H

(1)′n+1(Γsb)

]Fn +

[(n+ 1)bH(2)

n+1(Γsb) − Γsb2H

(2)′n+1(Γsb)

]Gn

}

−µ∗s

[n2H(1)

n (Γsb) + Γ2sb

2H(1)′′n (Γsb) − bΓsH

(1)′n (Γsb)

]Kn

(41k)−µ∗

s

[n2H(2)

n (Γsb) + Γ2sb

2H(2)′′n (Γsb) − bΓsH

(2)′n (Γsb)

]Ln

+2µ2n [Jn(γc2b) − bγc2J′n(γc2b)]Mn + +ikzµ2

[(n + 1)bJn+1(γs2b) − γs2b

2J ′n+1(Γs2b)

]Nn

+µ2

[n2Jn(γs2b) + γ2

s2b2J ′′

n (γs2b) − bγs2J′n(γs2b)

]Qn = 0

350 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

Table 2Havriliak-Negami fitting parameters [40]

Polymer 18 Polymer 19

G0 (dynes/cm2) 3.372 e7 5.019 e7G∞ (dynes/cm2) 1.453 e10 0.8089 e10τ (sec) 3.139 e-9 1.702 e-1α 0.4822 0.4941β 0.4116 0.1356ρs (g/cm3) 1.101 1.096ν 0.5 0.5

Fig. 2. Angular distribution of the scattered far-field pressure for a unit amplitude planewaveincident upon an olive oil-filled spherical viscoelasticshell immersed in glycerine for selected dimensionlesswavenumbers and shell thicknesses.

2ikzµ∗sb

2Γc

[DnH

(1)′n (Γcb) + EnH

(2)′n (Γcb)

]

+µ∗s

{[n + 1 − b2k2

z

]H

(1)n+1(Γsb) − Γsb(n+ 1)H(1)′

n+1(Γsb) − Γ2sb

2H(1)′′n+1 (Γsb)

}Fn

+µ∗s

{[n + 1 − b2k2

z

]H

(2)n+1(Γsb) − Γsb(n+ 1)H(2)′

n+1(Γsb) − Γ2sb

2H(2)′′n+1 (Γsb)

}Gn

+ikzµ∗sbn

[KnH

(1)n (Γsb) + LnH

(2)n (Γsb)

](41l)

+2ikzµ2b2γc2MnJ

′n(γc2b)

−µ2

{[n+ 1 − b2k2

z

]Jn+1(γs2b) − γs2b(n + 1)J ′

n+1(γs2b) − γ2s2b

2J ′′n+1(γs2b)

}Nn

−ikzµ2bnQnJn(γs2b) = 0

wheren = 0, 1, 2, . . . , except for Eqs (41b), (41e), (41h), (41k) wheren = 1, 2, . . .Substitution of first of Eq. (33) into first of Eq. (24) yields the acoustic pressure for harmonic scattered waves.

The radial component of the acoustic power flux vector (acoustic intensity) transmitted into the encapsulated fluid

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 351

Fig. 3. Variation of the form function for the fluid-filled spherical viscoelastic shell with changes in shell thickness.

medium II per unit shell length may be obtained from [21]

Itrans.(r, θ, z, ω) =12Re

{σ2

rr × conj(u2r) + σ2

rθ × conj(u2θ) + σ2

rz × conj(u2z)

}(42)

whereconj stands for the complex conjugate operator. Furthermore, since the incident wave is being propagated inan oblique direction with respect to the shell, the resultant force (per unit length) acting on the shell will thereforehave two components that are given by [29,31]

Fx = a

∫ 2π

0

(σ1rr cos θ − σ1

rθ sin θ)dθ, Fz = a

∫ 2π

0

σ1rzdθ (43)

in which the fluid stress tensor components as obtained from Eq. (38) may be expressed in the form

σ1rr =

∞∑n=0

σ1rrn cosnθeikzz

σ1rθ =

∞∑n=1

σ1rθn sinnθeikzz (44)

352 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

Fig. 4. Variation of Transmitted acoustic intensity per unit incidentwaveintensity at the center of the fluid-filled spherical viscoelastic shell withthe shell thickness.

σ1rz =

∞∑n=0

σ1rzn cosnθeikzz

Substituting the above expressions for the stress components into the force Eq. (43) and performing the integration,we find

F =√F 2

x + F 2z =

√[πα(σ1

rr1 − σ1rθ1)]

2 + [2πaσ1rz0]

2 (45)

where the appropriate orthogonality relations for the transcendental functions are employed.

6. Numerical results and discussion

In order to illustrate the nature and general behaviour of the solution, we consider numerical examples in thissection. Realizing the large number of parameters involved here, no attempt is made to exhaustively evaluate the

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 353

Fig. 5. Variation of the normalized acoustic force magnitude on the fluid-filled spherical viscoelastic shell with the shell thickness.

effect of varying each of them. The intent of the collection of data presented here is merely to illustrate the kinds ofresults to be expected from some representative and physically realistic choices of values for these parameters. Fromthese data some trends are noted and general conclusions made about the relative importance of certain parameters.Noting the crowd of parameters that enter into the final expressions and keeping in view the availability of numericaldata, we shall confine our attention to a particular model. The surrounding fluid is taken to be glycerine and theinner fluid is assumed to be olive oil with their assumed properties displayed in Table 1 [46]. As there are noreliable data found for the bulk viscosity of olive oil and glycerine, their numerical values are presumed to beequal to their shear viscosities. The viscoelastic shell material is supposed to be elastomeric with a fixed outerradius ofa = 0.05 cm. Hartmann et al. [40], for the first time, reported all the input parameters necessary for acomplete description of viscoelastic material properties for a set of (polyurethane) polymers within the context ofHavriliak-Negami theory. The HN fitting parameters for two selected polymers with distinctively different dynamicviscoelastic properties in the frequency range of interest are compiled in Table 2. The corresponding fit of HNequations for shear modulus,G ′(ω), and loss factor,η(ω), (e.g., Eqs (2) and (3)) for the selected polymers in a

354 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

Fig. 6. Form function amplitude versus nondimensional frequency for a hollow duraluminium sphere immersed in water.

wide frequency range, are displayed in Figs 2a and 2b of [34], respectively. Polymer 18 is found to have the highestdamping (loss factor), and polymer 19 is found to have the lowest damping in the frequency range of our interest (i.e.,in 0.1 < ka < 10). Accurate computation of Bessel functions of complex argument is achieved using MATLABspecialized math functions “besselh” and “besselj”. The derivatives of Bessel functions were calculated by utilizing(9.1.27), (10.1.19) and (10.1.22) in [45]. A MATLAB code was constructed for treating boundary conditions and tocalculate the unknown scattering coefficients and relevant acoustic field quantities as functions of nondimensionalfrequencyka wherek = Re(kc1) = ωa/c1.

Figure 2 displays the angular distribution of the scattered far-field pressure for a unit amplitude plane wave

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 355

Fig. 7. Angular distribution of the scattered far-field pressure due to normal incidence of a unit amplitude planewave on an oliveoil-filledcylindrical viscoelastic shell immersed in glycerine for selected dimensionlesswavenumbers and shell thicknesses.

(P0 = 1.) incident upon an olive oil-filled spherical viscoelastic shell immersed in glycerine at selected dimensionlesswave numbers and shell thicknesses. The far-field value of the radial coordinate in each case was simply chosenby making several computer runs while seeking for the convergence of the scattered pressure directivity patterns.The choice ofr∞ = 10a was found to be adequate for all cases. Furthermore, a maximum number of forty modeswere included in all summations in order to assure convergence in the high frequencies. It is very interesting tonotice the change in directionality of the scattered waves as the frequency is varied. As seen from the figure, at lowand intermediate frequencies (i.e., atka = 0.1, 1) the pressure patterns are very uniform and they show a very lowdirectionality, especially for the polymeric shell 18 that has the highest loss factor. Furthermore, increasing shellthickness has no important effect on the general far-field pressure distribution at these frequencies. However, atka = 10 pressure directionality for both cases dramatically amplifies, especially for polymeric coating 19 that hasthe lowest loss factor. Furthermore, adding to the thickness of the polymeric shell 18 leads to a noticeable reductionin pressure pattern directionality at this frequency.

Figure 3 displays the variation of the forward-scattered far-field pressure magnitude (i.e.,|p scat.(r∞, π)/pinc.|,or the form function amplitude) with nondimensional frequency and shell thickness. Here the main observationfor polymeric shell 18 is that as its thickness increases, resonances in the pressure curves begin to shift to lowerfrequencies. Forh/a > 0.2 there are almost no resonances in the intermediate and high frequencies and only a smallpeak is observed atka ≈ 0.5. For the polymeric shell 19, on the other hand, there are no resonances observed forh/a < 0.4. As the shell thickness increases beyond this critical value, relatively small amplitude resonances appearin the entire frequency range.

Figure 4 presents variations of the transmitted acoustic intensity (e.g., Eq. (28)) per unit incident wave intensity(Iinc. = P 2

0 ρω2/2c) at the center of the spherical shell with frequency and shell thickness. A similar trend as in

the previous figure is evident. In particular, relatively high peaks for polymeric shell 18 (polymeric shell 19) ath/a = 0.1, 0.2 (h/a � 0.7) are observed. The change in normalized acoustic force magnitude (i.e.,|F |/πρωa 2P0)

356 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

Fig. 8. Variation of the form function for the fluid-filled cylindrical viscoelastic shell with the shell thickness at normalwaveincidence (α = 0).

acting on the viscoelastic spherical shell with nondimensional frequency of the incident wave and shell thickness arepresented in Fig. 5. The results are plotted in dimensionless form; namely, they are the acoustic radiation force perunit area and per unit pressure of the incident wave for the spherical shell. Once again, similar behaviour as in theprevious two figures is observed. Here, the dimensionless force coefficients are almost linearly proportional to thedimensionless wave numberka when the frequency of the incident wave is low (i.e.,ka < 1). The linearity breaksdown when the incident wave frequency increases. Also, recurring peaks in the force amplitudes of the polymericshell 18 are observed forh/a � 0.4. Adding to the thickness of polymeric shell 18 (i.e.,h/a > 0.5) has in generalno considerable effect on the force magnitudes as they become almost identical and non-oscillatory (i.e., there isonly a single peak observed nearka = 1). A more regular pattern is noted for polymeric shell 19, especially at lowshell thicknesses in the entire frequency range. This behaviour could be due to its lower loss factor and higher shear

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 357

Fig. 9. Variation of the form function for the fluid-filled cylindrical viscoelastic shell with the shell thickness at obliquewaveincidence (α = π/4).

modulus in the frequency range of interest. Furthermore, a distinctively large peak in the force magnitude appearsat ka = 1.7 andh/a = 0.1 (ka = 2.6 andh/a > 0.9) for polymeric shell 18 (polymeric shell 19). Moreover,as the incident wave frequency is increased, the force coefficient amplitudes gradually decrease for both polymericshells. Lastly, in order to check overall validity of the first part of the work we computed the form function versusnondimensional frequency for a hollow duraluminium sphere immersed in water. The numerical results are shownin Fig. 6 that closely follow the curves displayed in Fig. 3 of [47]. The relatively small differences are mainly dueto the inclusion of fluid viscosity in the current formulation.

Figure 7 displays the angular distribution of the scattered far-field pressure due to normal incidence (α = 0) ofa unit amplitude plane wave on an olive oil-filled cylindrical viscoelastic shell immersed in glycerine at selecteddimensionless wave numbers and shell thicknesses. Here, observations analogous to the previous case can be made.Just as in the spherical shell situation, the patterns are very uniform in the low and intermediate frequencies for both

358 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

Fig. 10. Variation of the form function for the fluid-filled cylindrical viscoelastic shell with the shell thickness at obliquewave incidence(α = π/3).

polymeric shells. However, at the high frequency ofka = 10, the pressure curves are much more directional with arelatively high forward scattering magnitude, especially for the polymeric shell 18.

Figures 8 through 10 display the variations of the forward-scattered far-field pressure magnitude (i.e.,|pscat.(r∞, π)/pinc.|, or the form function amplitude) with nondimensional frequency and shell thickness for se-lected angles of wave incidence, i.e.,α = 0, π/4, π/3, respectively. Similar to the spherical shell case, the mainobservation for polymeric shell 18 is that as the shell thickness increases, resonances in the pressure curves beginto shift to lower frequencies. For this polymer, there are almost no resonances forh/a > 0.2 at the intermediateand high frequencies while a perceptible peak is observed atka ≈ 0.2. For the polymeric shell 19, on the other

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 359

Fig. 11. Transmitted acoustic intensity at the center of the fluid-filled viscoelastic cylindrical shell per unit incidentwave intensity versusnondimensional frequency and angle ofwaveincidence for selected shell thicknesses.

hand, relatively small magnitude resonances are observed for small shell thicknesses (i.e.,h/a < 0.4). As theshell thickness is increased, the magnitude and number of these resonances increase in the entire frequency range ofinterest for all angles of wave incidence.

Figure 11 presents variations of the transmitted acoustic intensity (e.g., Eq. (42)) per unit incident wave intensity(Iinc. = P0ρω

2/2c) at the center of the cylindrical shell with nondimensional frequency and angle of wave incidenceat selected shell thicknesses (i.e.,h/a = 0.01, 0.1, 0.5). Here, we first observe that forh/a = 0.01, 0.1 the intensityratio magnitudes display very regular and nearly harmonic surface patterns for both polymeric shells. In case ofthe very thick shell (i.e.,h/a = 0.5), pattern oscillations decrease, especially for polymeric shell 18 that displaysa relatively large peak at the low frequency ofka ≈ 0.8 for α < π/3. Furthermore, the surface patterns for thepolymeric shell 19 become very irregular with the peaks occurring predominantly in the high frequency and largeangle of incidence range.

Figure 12 displays the variations of normalized force magnitude (i.e.,|F |/2πρωaP 0) acting on the cylindricalshells with respect to the nondimensional frequency and the angle of wave incidence for selected shell thicknesses.The results are plotted in dimensionless form; namely, they are the acoustic radiation force per unit area (per unit shelllength). Here, we first note that the force magnitudes display regular, nearly harmonic and slightly damped surfacepatterns for both polymeric shells ath/a = 0.01, 0.1. In case of the thick polymeric shell 18 (i.e.,h/a = 0.5),the overall force magnitude and the pattern oscillation level drastically decrease with nondimensional frequency.For the polymeric shell 19, the surface patterns turn out to be very irregular with relatively large peaks occurringpredominantly in the high frequency and high angle of incidence range. Finally, in order to check overall validity ofthe second part of the work we computed the form function amplitude versus nondimensional frequency for a solidaluminium cylinder (h/a ≈ 1) immersed in water at various angles of incidence. The numerical results are shown

360 S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction

Fig. 12. Normalized force magnitude on the fluid-filled viscoelastic cylindrical shell versus nondimensional frequency and angle ofwaveincidence for selected shell thicknesses.

in Fig. 13 that closely follow the curves displayed in Fig. 2 of [48]. The small differences are mainly due to theinclusion of fluid viscosity in the current formulation.

7. Conclusion

This work presents analytical solutions as well as numerical results for two fundamental boundary value problemsconcerning the interaction of a plane sound wave with (viscous) fluid-loaded viscoelastic spherical and cylindricalshells filled with viscous fluids. The prime objective is to investigate the effects of dynamic viscoelastic shellmaterial properties on acoustic scattering and its associated field quantities. Numerical results reveal the importantconsequencesarising due to inclusion of Havriliak-Negamidynamic model for description of the viscoelastic materialproperties. The proposed model can lead to a better understanding of dynamic response of fluid-filled viscoelasticshells in an acoustic field. It may equally be useful in complementing the inverse scattering techniques that havebeen developed to determine the physical properties of viscoelastic materials from laboratory measurements of thescattered acoustic field [49].

Acknowledgements

The authors wish to sincerely thank professors Daniel Levesque, Roderic Lakes, Yves Berthelot, S. Temkin,and Andrei Dukhin for valuable and productive consultations on dynamic theory of viscoelasticity and acoustics of(thermo)viscous media.

S.M. Hasheminejad and N. Safari / Dynamic viscoelastic effects on sound wave diffraction 361

Fig. 13. Form function amplitude versus nondimensional frequency for a solid aluminium cylinder (h/a ≈ 1) immersed in water for selectedangles ofwaveincidence.

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