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The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering
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The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering夽 N.V. Larin, L.A. Tolokonnikov Tula, Russia
a r t i c l e
i n f o
Article history: Received 22 May 2014
a b s t r a c t An analytical solution of the problem of the scattering of a plane sound wave by a homogeneous elastic cylinder with a discrete-layered covering is obtained. The results of calculations of the radiation patterns of the scattered field for a cylinder with continuously-homogeneous and discrete-layered coverings are presented. It is shown that a radially inhomogeneous coating can be modelled by a system of homogeneous elastic layers. © 2015 Elsevier Ltd. All rights reserved.
There are various approaches to the change in the sound-reflecting characteristics of bodies in certain directions. A change in the sound scattering characteristics of elastic bodies can be obtained using coverings in the form of a continuously inhomogeneous elastic layer. This representation of the covering turns out to be more convenient in a number of cases, for example, when solving inverse problems. Continuously inhomogeneous coverings can be constructed using a multilayer coating – a system of homogeneous elastic layers with different values of the mechanical parameters (the density and elastic constants). A large number of publications, for example, Ref. 1, have been devoted to investigating reflections of plane sound waves from a system of plane homogeneous elastic layers. The scattering of sound waves by homogeneous isotropic elastic cylinders has been considered in a number of papers. For example, the case of the normal incidence of a plane wave2 and the case of inclined incidence3 have been investigated. The problem of the scattering of plane sound waves by an inhomogeneous elastic cylinder has been solved.4,5 The scattering of sound waves by a transversely isotropic inhomogeneous cylindrical layer has been investigated in Ref. 6. The diffraction of a plane sound wave by an inhomogeneous anisotropic hollow cylinder has been considered in the general case of anisotropy.7 A solution of the problem of the scattering of a plane sound wave by an inhomogeneous elastic hollow cylinder in a viscous fluid has been obtained.8 The diffraction of a plane sound wave by an inhomogeneous thermoelastic cylindrical layer, bounded by non-viscous heat-conducting fluids has been investigated.9 The scattering of sound waves by an absolutely rigid cylinder with an inhomogeneous elastic coating has been considered.10 The problem of the scattering of an inclined incident plane sound wave by an elastic cylinder with a radially inhomogeneous covering has been solved.11 Below we solve the problem of the scattering of a plane monochromatic sound wave, incident obliquely on an elastic circular cylinder with a discrete-layered covering. The possibility of modelling the discrete-homogeneous covering of a system of homogeneous elastic layers is discussed. 1. Statement of the problem Consider an infinite homogeneous isotropic elastic cylinder of radius r0 , the material of which has a density 0 and elastic constants 0 and 0 . The cylinder has a covering in the form of a system of N thin coaxial cylindrical layers of radii rj (j = 1, 2,..., N). Each j-th homogeneous isotropic elastic layer has a density j and elasticity moduli j and j . The fluid surrounding the body is ideal, its equilibrium density is and the velocity of sound is c. A cylindrical system of coordinates r, , z is chosen so that the z coordinate axis is the axis of rotation of the cylinder. Suppose a plane sound wave, the velocity potential of which is
夽 Prikl. Mat. Mekh., Vol. 79, No. 2, pp. 242–250, 2015. E-mail address: [email protected] (L.A. Tolokonnikov). http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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is incident from the external space onto a cylinder of arbitrary form, where A is the wave amplitude, 0 and 0 are the polar and azimuthal angles of incidence of the wave, k = /c is the wave number in the external region and is the angular frequency. The time factor e−it will henceforth be omitted. 2. Determination of the wave fields The propagation of small perturbations in an ideal fluid, in the case of steady oscillations, is described by the Helmholtz equation12
where N+1 = 0 + s is the velocity potential of the total acoustic field in the external region, and s is the velocity potential of the scattered field. The velocity of the particles and the acoustic pressure p in the fluid are given by the formulae
We will represent the velocity potential of an obliquely incident plane wave in the form13
where Jn (x) is a cylindrical Bessel function of order n. Here and everywhere henceforth, unless otherwise stated, summation is carried out over n from −∞ to ∞. Taking the radiation conditions at infinity into account, we will seek the function s in the form (2.1) where Hn (x) is the cylindrical Hankel function of the first kind of order n. We will now consider the equations describing the propagation of small perturbations in an elastic cylinder with a discrete-layered coating. We will represent the displacement vector u(j) of the particles of the j-th elastic isotropic homogeneous layer in the form
where (j) and (j) are the scalar and vector potentials of the displacement in the j-th layers (j = 0 corresponds to a cylinder of radius r0 ). In the case of harmonic motion, the displacement potentials are the solutions of the Helmholtz wave equations12
(j)
(j)
(j)
(j)
(j)
where kl = /cl , k = /c and cl =
(j)
j + 2j /j , c =
(2.2) j /j are the wave numbers and the velocities of longitudinal and
transverse elastic waves in the j-th layer, respectively. We will represent the vector (j) in the form14
where L(j) and M(j) are scalar functions of the spatial coordinates r, , z, and e¯ z is the unit vector of the z axis. Vector equation (2.2) is then replaced by two Helmholtz scalar equations in the functions L(j) and M(j)
We will seek the functions (j) , L(j) and M(j) (j = 0, 1,..., N) in the form
(2.3) where
where Nn (x) is the cylindrical Neumann function of order n. Taking into account the boundedness condition of the functions (0) , L(0) and M(0) we will have
The coefficients of expansions (2.1) and (2.3) are to be determined from the boundary conditions. Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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The boundary conditions on the outer surface of the body are the equality of the normal velocities of the particles of the elastic medium and the fluid, the equality on it of the normal stress and acoustic pressure, and no shear stresses (2.4) On passing through the interface of the j-th and (j + 1)-th elastic layers (j = 0, 1,..., N – 1) the components of the particle displacement vector and also the normal and shear stresses must be continuous:
(2.5) The relations between the components of the stress tensor layer are written in the form15
(j) (j) (j) rr , r , rz
and the components of the displacement vector
u(j)
in the j-th
The components of the displacement vector u(j) , written in terms of the functions (j) , L(j) and M(j) in a cylindrical system of coordinates, have the form
(2.6) Using the formulae derived above, we can express the components of the stress tensor L(j) and M(j) :
(j) (j) rr , r
and
(j) rz
in terms of the functions (j) ,
(2.7) A direct way of solving the problem is to use all the boundary conditions to construct 6N + 4 linear algebraic equations in the unknown expansion coefficients (2.1) and (2.3), and then solving this system. However, it is preferable to solve the problem by a matrix method.16,17 (j) The components of the displacement vector u(j) and the components of the stress tensor kl are periodic functions of the coordinate with period 2. Hence, the functions
will be sought in the form
(2.8) Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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Here, for each j-th layer (j = 1, 2,..., N) and a cylinder of radius r0 (j = 0), we introduce into consideration the displacement–stress vector (j) Sn (r). We also introduce the vector of the coefficients
These vectors are connected by the formula
(j)
(j)
(j)
(2.9)
where Tn (r) is a sixth-order matrix with elements tkl = tkl (r) rj−1 ≤ r ≤ rj , r−1 = 0 . Taking expressions (2.6) – (2.8) into account, we obtain
where
(j)
Here Zn (x) = Jn (x) when q = 1, and Zn (x) = Nn (x) when q = 2. The elements tkl must be given the subscript n, which, for simplicity, we will omit. Consider the arbitrary j-th layer (j = 1, 2,..., N). On its boundaries, for r = rj – 1 and r = rj we obtain from (2.9) the relations
(j)
Using the constancy of the vector Kn inside the j-th layer, we obtain (2.10) Substituting (2.10) into boundary conditions (2.5) with r = rj
– 1
we have
Relation (2.10) can then be rewritten in the form (2.11) Successive application of formula (2.11) enables us to relate the value of the displacement–stress vector on the boundary of the media r = rN to the value of the displacement–stress vector on the boundary r = r0 : (2.12) Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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From relation (2.12) we obtain
(2.13) where Pms (m, s = 1, 2, 3, the subscript n is omitted) are the elements of the matrix Pn . We will not use the equality of the second, and also of the third, elements of the vectors on the left and right sides of equality (2.12), since, on the boundary of the elastic body and the fluid, the continuity of the tangential components of the displacement vector breaks down. Substituting expansions (2.8) into boundary conditions (2.4) with r = rN , we obtain
(2.14) From (2.9) with j = 0 and r = r0 we have
(2.15) Substituting expressions (2.14) and (2.15) into (2.13), we arrive at a system of four linear algebraic equations with the unknowns (0) (0) (0) An , B1n , C1n and D1n . We find An from this system. As a result we obtain an analytical description of the scattered acoustic field in the form (2.1). 3. Results of the calculations Consider the far zone of the acoustic field. Using the asymptotic formula for a cylindrical Hankel function of the first kind for large values of the argument18 (br » 1)
we obtain from (2.1)
where
The expression for F() in the case of a radially inhomogeneous covering was presented previously in Ref. 11. To estimate the possibility of modelling a radially-inhomogeneous covering by a system of homogeneous elastic layers we calculated the scattering amplitudes |F()| both for a continuously inhomogeneous covering, and for a discrete-layered covering with a different number of homogeneous layers. The replacement of a radially inhomogeneous covering by a multilayer system is equivalent to the approximation of continuous functions, characterizing the variable parameters of the inhomogeneous layer, by piecewise-constant functions, describing the mechanical parameters of the homogeneous layers. The scattering amplitudes |F()| were calculated for an aluminium cylinder of radius r0 = 0.1 m (0 = 2.7 × 103 kg/m3 , 0 = 5.3 × 1010 N/m2 and 0 = 2.6 × 1010 N/m2 , in water (0 = 103 kg/m3 and c = 1485 m/s). We considered the normal incidence on a cylinder (0 = 90◦ ) of a plane sound wave of unit amplitude in the direction 0 = 0◦ . We assumed that the wave dimension of the body krN = 12, while the ratio of the outer radius rN to the inner radius r0 of the covering is 1.2. We considered a radially inhomogeneous covering, the mechanical characteristics of which varied over the thickness of the cylindrical layer as follows:
Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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Fig. 1.
where
The factor a was chosen so that the mean value of the function f(r) over the thickness of the covering was equal to unity (a = 1.56). We ¯ = 3.9 × 109 N/m2 and ¯ = 9.8 × 108 N/m2 then considered a covering with a basic density ¯ = 1.07 × 103 kg/m3 and Lamé elastic constants (polyvinylbuteral). The radiation patterns of the scattered field for a discrete-layered covering was calculated for homogeneous layers of the same thickness hN = (rN – r0 )/N for a different number of them (N = 3, 6, 12). We assumed that when rj−1 < r ≤ rj
where
We show in Figure 1 the scattering amplitudes |F()| as a function of the polar angle . The continuous curve corresponds to a continuously inhomogeneous covering, and the dotted, dashed and dot-dash curves correspond to the discrete-layered covering with N = 3, N = 6 and N = 12 layers, respectively. The arrow indicates the direction of propagation of the incident plane wave. The calculations show that, as the number of homogeneous layers in the covering increases, the radiation pattern for continuously inhomogeneous and layered inhomogeneous coverings become less and less different. This indicates the possibility of modelling a radially inhomogeneous coating by a coating consisting of a system of homogeneous layers. Acknowledgements This research was supported by the Russian Foundation for Basic Research (13-01-97514) and by the Ministry of Education and Science (State Assignment 1.1333.2014K). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Brekhovskikh LM. Waves in Layered Media. New York: Academic; 1980. Faran JJ. Sound scattering by solid cylinders and spheres. Acoust Soc Amer 1951;23(4):405–20. Flax L, Varadan VK, Varadan VV. Scattering of an obliquely incident acoustic wave by an infinite cylinder. J Acoust Soc Amer 1980;68(6):1832–5. Bezrukov AV, Prikhod’ko VYu, Tyutekin VV. The scattering of sound waves by elastic radially layered cylindrical bodies. Akust Zh 1986;32(6):762–6. Kovalenko GP. The diffraction of an acoustic wave by an inhomogeneous rigid body. Akust Zh 1987;33(6):1060–3. Skobel’tsyn SA, Tolokonnikov LA. The scattering of sound waves by a transversely isotropic inhomogeneous cylindrical layer. Akust Zh 1995;41(1):134–8. Tolokonnikov LA. Diffraction of sound waves by an inhomogeneous anisotropic hollow cylinder. Oboron Tekh 1998;4–5:11–4. Romanov AG, Tolokonnikov LA. Scattering of a plane sound wave by an inhomogeneous hollow elastic cylinder in a viscous fluid. Izv Tul GU Yestestv Nauki 2009;1:62–70. Larin NV, Tolokonnikov LA. Diffraction of a plane acoustic wave by an inhomogeneous thermoelastic cylindrical layer, bounded by inviscid heat-conducting fluids. JAMM 2009;73(3):336–43. Romanov AG, Tolokonnikov LA. The scattering of acoustic waves by a cylinder with a non-uniform elastic coating. JAMM 2011;75(5):595–600. Tolokonnikov LA. The scattering of an obliquely incident plane sound wave by an elastic cylinder with an inhomogeneous covering. Izv Tul GU Yestestv Nauki 2013;2(2):265–74. Shenderov YeL. Wave Problems of Hydroacoustics. Leningrad: Sudostroyeniye; 1972. Ivanov YeA. Diffraction of Electromagnetic Waves by Two Bodies. Minsk: Nauka i Tekhnika; 1968. Morse P, Feshbach H. Methods of Theoretical Physics, Vol. 2. New York: McGraw-Hill; 1953. Nowacki W. Theory of Elasticity. Warsaw: PWN; 1973. Molotkov LA. The Matrix Method in the Theory of the Propagation of Waves in Stratified Elastic and Fluid Media. Leningrad: Nauka; 1984. Brekhovskikh LM, Godin OA. Acoustics of Stratified Media, Vols 1 and 2. Berlin: Springer; 1990. Lebedev NN. Special Functions and Their Applications. New York: Dovet; 1972.
Translated by R.C.G.
Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
ARTICLE IN PRESS
JAMM-2271; No. of Pages 6
Journal of Applied Mathematics and Mechanics xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering夽 N.V. Larin, L.A. Tolokonnikov Tula, Russia
a r t i c l e
i n f o
Article history: Received 22 May 2014
a b s t r a c t An analytical solution of the problem of the scattering of a plane sound wave by a homogeneous elastic cylinder with a discrete-layered covering is obtained. The results of calculations of the radiation patterns of the scattered field for a cylinder with continuously-homogeneous and discrete-layered coverings are presented. It is shown that a radially inhomogeneous coating can be modelled by a system of homogeneous elastic layers. © 2015 Elsevier Ltd. All rights reserved.
There are various approaches to the change in the sound-reflecting characteristics of bodies in certain directions. A change in the sound scattering characteristics of elastic bodies can be obtained using coverings in the form of a continuously inhomogeneous elastic layer. This representation of the covering turns out to be more convenient in a number of cases, for example, when solving inverse problems. Continuously inhomogeneous coverings can be constructed using a multilayer coating – a system of homogeneous elastic layers with different values of the mechanical parameters (the density and elastic constants). A large number of publications, for example, Ref. 1, have been devoted to investigating reflections of plane sound waves from a system of plane homogeneous elastic layers. The scattering of sound waves by homogeneous isotropic elastic cylinders has been considered in a number of papers. For example, the case of the normal incidence of a plane wave2 and the case of inclined incidence3 have been investigated. The problem of the scattering of plane sound waves by an inhomogeneous elastic cylinder has been solved.4,5 The scattering of sound waves by a transversely isotropic inhomogeneous cylindrical layer has been investigated in Ref. 6. The diffraction of a plane sound wave by an inhomogeneous anisotropic hollow cylinder has been considered in the general case of anisotropy.7 A solution of the problem of the scattering of a plane sound wave by an inhomogeneous elastic hollow cylinder in a viscous fluid has been obtained.8 The diffraction of a plane sound wave by an inhomogeneous thermoelastic cylindrical layer, bounded by non-viscous heat-conducting fluids has been investigated.9 The scattering of sound waves by an absolutely rigid cylinder with an inhomogeneous elastic coating has been considered.10 The problem of the scattering of an inclined incident plane sound wave by an elastic cylinder with a radially inhomogeneous covering has been solved.11 Below we solve the problem of the scattering of a plane monochromatic sound wave, incident obliquely on an elastic circular cylinder with a discrete-layered covering. The possibility of modelling the discrete-homogeneous covering of a system of homogeneous elastic layers is discussed. 1. Statement of the problem Consider an infinite homogeneous isotropic elastic cylinder of radius r0 , the material of which has a density 0 and elastic constants 0 and 0 . The cylinder has a covering in the form of a system of N thin coaxial cylindrical layers of radii rj (j = 1, 2,..., N). Each j-th homogeneous isotropic elastic layer has a density j and elasticity moduli j and j . The fluid surrounding the body is ideal, its equilibrium density is and the velocity of sound is c. A cylindrical system of coordinates r, , z is chosen so that the z coordinate axis is the axis of rotation of the cylinder. Suppose a plane sound wave, the velocity potential of which is
夽 Prikl. Mat. Mekh., Vol. 79, No. 2, pp. 242–250, 2015. E-mail address: [email protected] (L.A. Tolokonnikov). http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007 0021-8928/© 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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is incident from the external space onto a cylinder of arbitrary form, where A is the wave amplitude, 0 and 0 are the polar and azimuthal angles of incidence of the wave, k = /c is the wave number in the external region and is the angular frequency. The time factor e−it will henceforth be omitted. 2. Determination of the wave fields The propagation of small perturbations in an ideal fluid, in the case of steady oscillations, is described by the Helmholtz equation12
where N+1 = 0 + s is the velocity potential of the total acoustic field in the external region, and s is the velocity potential of the scattered field. The velocity of the particles and the acoustic pressure p in the fluid are given by the formulae
We will represent the velocity potential of an obliquely incident plane wave in the form13
where Jn (x) is a cylindrical Bessel function of order n. Here and everywhere henceforth, unless otherwise stated, summation is carried out over n from −∞ to ∞. Taking the radiation conditions at infinity into account, we will seek the function s in the form (2.1) where Hn (x) is the cylindrical Hankel function of the first kind of order n. We will now consider the equations describing the propagation of small perturbations in an elastic cylinder with a discrete-layered coating. We will represent the displacement vector u(j) of the particles of the j-th elastic isotropic homogeneous layer in the form
where (j) and (j) are the scalar and vector potentials of the displacement in the j-th layers (j = 0 corresponds to a cylinder of radius r0 ). In the case of harmonic motion, the displacement potentials are the solutions of the Helmholtz wave equations12
(j)
(j)
(j)
(j)
(j)
where kl = /cl , k = /c and cl =
(j)
j + 2j /j , c =
(2.2) j /j are the wave numbers and the velocities of longitudinal and
transverse elastic waves in the j-th layer, respectively. We will represent the vector (j) in the form14
where L(j) and M(j) are scalar functions of the spatial coordinates r, , z, and e¯ z is the unit vector of the z axis. Vector equation (2.2) is then replaced by two Helmholtz scalar equations in the functions L(j) and M(j)
We will seek the functions (j) , L(j) and M(j) (j = 0, 1,..., N) in the form
(2.3) where
where Nn (x) is the cylindrical Neumann function of order n. Taking into account the boundedness condition of the functions (0) , L(0) and M(0) we will have
The coefficients of expansions (2.1) and (2.3) are to be determined from the boundary conditions. Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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The boundary conditions on the outer surface of the body are the equality of the normal velocities of the particles of the elastic medium and the fluid, the equality on it of the normal stress and acoustic pressure, and no shear stresses (2.4) On passing through the interface of the j-th and (j + 1)-th elastic layers (j = 0, 1,..., N – 1) the components of the particle displacement vector and also the normal and shear stresses must be continuous:
(2.5) The relations between the components of the stress tensor layer are written in the form15
(j) (j) (j) rr , r , rz
and the components of the displacement vector
u(j)
in the j-th
The components of the displacement vector u(j) , written in terms of the functions (j) , L(j) and M(j) in a cylindrical system of coordinates, have the form
(2.6) Using the formulae derived above, we can express the components of the stress tensor L(j) and M(j) :
(j) (j) rr , r
and
(j) rz
in terms of the functions (j) ,
(2.7) A direct way of solving the problem is to use all the boundary conditions to construct 6N + 4 linear algebraic equations in the unknown expansion coefficients (2.1) and (2.3), and then solving this system. However, it is preferable to solve the problem by a matrix method.16,17 (j) The components of the displacement vector u(j) and the components of the stress tensor kl are periodic functions of the coordinate with period 2. Hence, the functions
will be sought in the form
(2.8) Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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Here, for each j-th layer (j = 1, 2,..., N) and a cylinder of radius r0 (j = 0), we introduce into consideration the displacement–stress vector (j) Sn (r). We also introduce the vector of the coefficients
These vectors are connected by the formula
(j)
(j)
(j)
(2.9)
where Tn (r) is a sixth-order matrix with elements tkl = tkl (r) rj−1 ≤ r ≤ rj , r−1 = 0 . Taking expressions (2.6) – (2.8) into account, we obtain
where
(j)
Here Zn (x) = Jn (x) when q = 1, and Zn (x) = Nn (x) when q = 2. The elements tkl must be given the subscript n, which, for simplicity, we will omit. Consider the arbitrary j-th layer (j = 1, 2,..., N). On its boundaries, for r = rj – 1 and r = rj we obtain from (2.9) the relations
(j)
Using the constancy of the vector Kn inside the j-th layer, we obtain (2.10) Substituting (2.10) into boundary conditions (2.5) with r = rj
– 1
we have
Relation (2.10) can then be rewritten in the form (2.11) Successive application of formula (2.11) enables us to relate the value of the displacement–stress vector on the boundary of the media r = rN to the value of the displacement–stress vector on the boundary r = r0 : (2.12) Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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From relation (2.12) we obtain
(2.13) where Pms (m, s = 1, 2, 3, the subscript n is omitted) are the elements of the matrix Pn . We will not use the equality of the second, and also of the third, elements of the vectors on the left and right sides of equality (2.12), since, on the boundary of the elastic body and the fluid, the continuity of the tangential components of the displacement vector breaks down. Substituting expansions (2.8) into boundary conditions (2.4) with r = rN , we obtain
(2.14) From (2.9) with j = 0 and r = r0 we have
(2.15) Substituting expressions (2.14) and (2.15) into (2.13), we arrive at a system of four linear algebraic equations with the unknowns (0) (0) (0) An , B1n , C1n and D1n . We find An from this system. As a result we obtain an analytical description of the scattered acoustic field in the form (2.1). 3. Results of the calculations Consider the far zone of the acoustic field. Using the asymptotic formula for a cylindrical Hankel function of the first kind for large values of the argument18 (br » 1)
we obtain from (2.1)
where
The expression for F() in the case of a radially inhomogeneous covering was presented previously in Ref. 11. To estimate the possibility of modelling a radially-inhomogeneous covering by a system of homogeneous elastic layers we calculated the scattering amplitudes |F()| both for a continuously inhomogeneous covering, and for a discrete-layered covering with a different number of homogeneous layers. The replacement of a radially inhomogeneous covering by a multilayer system is equivalent to the approximation of continuous functions, characterizing the variable parameters of the inhomogeneous layer, by piecewise-constant functions, describing the mechanical parameters of the homogeneous layers. The scattering amplitudes |F()| were calculated for an aluminium cylinder of radius r0 = 0.1 m (0 = 2.7 × 103 kg/m3 , 0 = 5.3 × 1010 N/m2 and 0 = 2.6 × 1010 N/m2 , in water (0 = 103 kg/m3 and c = 1485 m/s). We considered the normal incidence on a cylinder (0 = 90◦ ) of a plane sound wave of unit amplitude in the direction 0 = 0◦ . We assumed that the wave dimension of the body krN = 12, while the ratio of the outer radius rN to the inner radius r0 of the covering is 1.2. We considered a radially inhomogeneous covering, the mechanical characteristics of which varied over the thickness of the cylindrical layer as follows:
Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007
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ARTICLE IN PRESS N.V. Larin, L.A. Tolokonnikov / Journal of Applied Mathematics and Mechanics xxx (2015) xxx–xxx
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Fig. 1.
where
The factor a was chosen so that the mean value of the function f(r) over the thickness of the covering was equal to unity (a = 1.56). We ¯ = 3.9 × 109 N/m2 and ¯ = 9.8 × 108 N/m2 then considered a covering with a basic density ¯ = 1.07 × 103 kg/m3 and Lamé elastic constants (polyvinylbuteral). The radiation patterns of the scattered field for a discrete-layered covering was calculated for homogeneous layers of the same thickness hN = (rN – r0 )/N for a different number of them (N = 3, 6, 12). We assumed that when rj−1 < r ≤ rj
where
We show in Figure 1 the scattering amplitudes |F()| as a function of the polar angle . The continuous curve corresponds to a continuously inhomogeneous covering, and the dotted, dashed and dot-dash curves correspond to the discrete-layered covering with N = 3, N = 6 and N = 12 layers, respectively. The arrow indicates the direction of propagation of the incident plane wave. The calculations show that, as the number of homogeneous layers in the covering increases, the radiation pattern for continuously inhomogeneous and layered inhomogeneous coverings become less and less different. This indicates the possibility of modelling a radially inhomogeneous coating by a coating consisting of a system of homogeneous layers. Acknowledgements This research was supported by the Russian Foundation for Basic Research (13-01-97514) and by the Ministry of Education and Science (State Assignment 1.1333.2014K). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
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Translated by R.C.G.
Please cite this article in press as: Larin NV, Tolokonnikov LA. The scattering of a plane sound wave by an elastic cylinder with a discrete-layered covering. J Appl Math Mech (2015), http://dx.doi.org/10.1016/j.jappmathmech.2015.07.007