Acta Mech 224, 1009–1018 (2013) DOI 10.1007/s00707-012-0801-2
Qiang Zhao · P. J. Wei
The reflection and transmission of elastic waves through a plane of spheres in periodic arrangement
Received: 18 October 2012 / Published online: 8 January 2013 © Springer-Verlag Wien 2013
Abstract The work presented in this paper focuses on the reflection and transmission coefficients of an incident plane wave which impinges obliquely a plane of identical spheres arranged periodically in a homogeneous host with infinite extension. The Bloch theorem of periodic structure and the addition theorem of spherical wave functions are used to obtain the total scattering wave from all spherical scatterers periodically arranged in a plane. The total scattering wave in series form of spherical wave functions is then transformed into plane wave form in order to derive the reflection and transmission coefficients. Some numerical examples are given for different size, material constants and array patterns of spherical scatterers, and their influences on the reflection and transmission coefficients of a plane of spheres are discussed based on the numerical results. This study implies that a plane of spheres can be elaborately designed to serve as a sound barrier at a certain frequency range.
1 Introduction The elastic wave scattering problem has attracted considerable attention over the past decades due to its application in quantitative nondestructive evaluation (NDE) of mechanical engineering. Ying and Truell [1] first studied the scattering problem when an incident longitudinal plane wave impinges a spherical scatterer embedded in an isotropic solid with infinite extension. Then, Einspruch et al. [2] studied the wave scattering by a sphere when the incident wave is a transverse plane wave. In their study, the incident plane wave is expanded into series of spherical wave functions in order to treat the boundary conditions on the surface of a spherical scatterer. The method is the well-known plane wave expansion method and thereafter was used widely in the scattering problem of various scatterers with regular shapes like sphere or cylinder. Pao and Mao [3] presented a detailed discussion on the application of this method for elastic wave scattering by a cylinder (with a circular or an elliptical cross section) or a sphere embedded in a homogeneous host. Lauchle [4] and Oien and Pao [5] further studied the scattering of a spheroidal obstacle. However, for the scatterer with irregular shapes, the weakness of the plane wave extension method comes out. In this case, the integral equation method [6–8] and the T matrix method [9–11] can be used instead. Recently, the elastic wave scattering by multiple scatterers is becoming interesting. Gaunaurd et al. [12] studied the scattering of acoustic waves by a pair of spheres. Fang et al. [13] extended the scattering of acoustic waves by a pair of spheres to the elastic wave case. Zhang et al. [14] further studied the scattering of elastic waves by an array of cylindrical holes. In their studies, the addition theorem of the spherical or cylindrical wave function plays an important role in order to transfer the scattering wave from one local coordinate system to another. The multiple scattering problem by a large number of scatterers, namely the effective wave in the composite with randomly dispersive scatterers [15–17] and the Bloch wave in the composite with periodically arranged scatterers [18–20] have also been studied. Q. Zhao · P. J. Wei (B) Department of Applied Mechanics, University of Science and Technology Beijing, Beijing 100083, China E-mail:
[email protected]
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In this paper, the main concern is focused on the transmission behavior of incident elastic waves through a plane of spheres arranged periodically in two-dimensional lattices. The reflection and transmission coefficients are first formulated based on the addition theorem of spherical wave functions and the Bloch theorem of elastic waves in periodic structures. Then, some numerical examples are demonstrated and the numerical results are depicted graphically to show the influence of the size, material constants and array patterns of spheres as scatterers. 2 The reflection and transmission through a plane of spheres The displacement equation of motion in a homogeneous isotropic elastic medium can be expressed as (λ + 2μ) ∇ (∇ · u) − μ∇ × (∇ × u) + ρω2 u = 0,
(1)
where u(r) is the displacement vector. λ and μ are Lamé coefficients, and ρ is the mass density of the elastic medium. ω is the angular frequency of an elastic wave. The general solutions of Eq. (1) are as follows: u (r) = (2) [almσ J lmσ (r ) + blmσ H lmσ (r)] , lmσ
where 1 ∇ jl (αr )Ylm (θ, ϕ) , α 1 J lm2 (r) = √ ∇ × r jl (βr ) Ylm (θ, ϕ) , l (l + 1) 1 J lm3 (r) = √ ∇ × ∇ × r jl (βr )Ylm (θ, ϕ) , β l (l + 1) J lm1 (r) =
1 ∇ h l (αr )Ylm (θ, ϕ) , α 1 H lm2 (r) = √ ∇ × rh l (βr ) Ylm (θ, ϕ) , l (l + 1) 1 H lm3 (r) = √ ∇ × ∇ × rh l (βr ) Ylm (θ, ϕ) . β l (l + 1)
H lm1 (r) =
jl (r ) and h l (r ) are the spherical Bessel and Hankel functions, respectively. Ylm (θ, ϕ) is the spherical harmonic function. σ = 1 denotes the longitudinal mode and σ = 2, 3 denotes two transverse modes. m(= −l, −l + 1, . . . , 0, 1, . . . , l) and l(= 0, 1, 2, . . .) denote the orders of the spherical harmonic function. almσ and blmσ are combination coefficients. α = cωl and β = cωt are the wave numbers of the longitudinal √ √ and transverse waves, respectively. c = (λ + 2μ)/ρ and ct = μ/ρ are wave speeds. ∇ is the gradient operator. Physically, the first term in Eq. (2), that is, almσ J lmσ (r), represents the incoming wave toward the center of a sphere and the second term, that is, blmσ H lmσ (r), represents the outgoing wave away from a sphere. When an incident plane wave impinges a single sphere embedded in a homogeneous isotropic medium, the incident wave and the scattering wave can be expressed as (in) uin (r) = almσ J lmσ (r) , (3) lmσ
u (r) = sc
(sc) blmσ H lmσ (r) ,
(4)
lmσ
where r is measured from the center of a sphere. By using the continuity of the displacement and of the surface traction at the spherical interface, u+ = u− , t+ = t− , where ∂u r trr = λ∇ · u + 2μ , ∂r ∂u θ 1 ∂u r uθ tr θ = μ + − , ∂r r ∂θ r ∂u φ uφ 1 ∂u r , + − tr φ = μ r sin θ ∂φ ∂r r (sc) (in) can be related with the coefficient almσ by the coefficient blmσ
(5) (6)
The reflection and transmission of elastic waves through a plane of spheres (sc)
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(in)
blmσ = Tlmσ l m σ al m σ ,
(7)
and then, the scattering wave can be determined by the incident wave given. When an incident plane wave impinges a plane of spheres periodically embedded in a homogeneous isotropic medium, the incident wave on a given sphere consists of two parts. One is the externally incident wave in(1) i i ui (8) almσ J lmσ (ri ) = (ri ) , lmσ
and another is the scattered wave from other spheres, j
j in(2) ui blmσ H lmσ r j . (ri ) =
(9)
j =i lmσ
It is noted that ri and r j in Eq. (9) indicate the same spatial point but are measured from different origins oi and o j (centers of the sphere i and of the sphere j). In order to sum the scattered waves from different spheres, the addition theorem of spherical wave function
H lmσ r j = (10) G lmσ l m σ oi − o j J l m σ (ri ) lmσ
is used, and Eq. (9) can be rewritten as j
in(2) ui blmσ G lmσ l m σ oi − o j J l m σ (ri ) . (ri ) =
(11)
j =i lmσ l m σ
A plane of spheres is depicted in Fig. 1. All spheres are assumed identical and arranged periodically in the xy-plane. The sites of the two-dimensional lattice are indicated by the direct lattice vector Rn = n 1 a1 + n 2 a2 ,
(12)
where a1 and a2 are primitive vectors of the two-dimensional direct lattice, and n 1 and n 2 are arbitrary integers. The incident wave can come from the left or the right of the plane of spheres. Due to the periodicity of arrangement of spheres, the wave field satisfies the Bloch theorem u (r + Rn ) = u (r) eik ·Rn ,
(13)
which means the displacement vector of the Bloch wave with the wave number k at different sites differs from each other only by a phase factor eik ·Rn . By use of the Bloch theorem, Eq. (11) can be further expressed as in(2) (14) blmσ eik ·R G lmσ l m σ (R) J l m σ (ri ) = al m σ J l m σ (ri ) , ui (ri ) = l m σ lmσ
lmσ
R =0
where al m σ =
lmσ
blmσ
eik ·R G lmσ l m σ (R) .
R =0
For an arbitrary sphere in the plane, the scattered wave can be expressed as usc (r) = blmσ H lmσ (ri ) , lmσ
where
blmσ = Tlmσ l m σ al m σ + al m σ = Z lmσ l m σ al m σ ,
(15)
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Fig. 1 A plane of spheres parallel to the xy-plane with spheres arranged on sites of the two-dimensional lattice. a the spherical scatterer periodically arranged in 2D lattice; b the reflected and transmitted wave for the normal incident wave; c the polar and azimuthal angles (θ and ϕ) of an oblique incidence wave
which means the multiple scattering effects among spheres in this plane are taken into consideration. The total scattered waves contain the contribution from all spheres in the plane and can be expressed as usc (r) =
i
blmσ H lmσ (ri ) =
lmσ
blmσ
eik ·R H lmσ (r − R) .
(16)
R
lmσ
Considering that the incident wave is in the plane wave form, the total scattered wave can be further transformed into the plane wave form by using the summation of lattices
eik ·R H lm1 (r − R) = eik ·R H lm2 (r − R) =
±
g±
ik± βg ·r
(17.1)
,
(17.2)
g± ik± βg ·r
(17.3)
Blm2 e
g
R
R
g±
Blm1 eikαg ·r ,
g
R
eik ·R H lm3 (r − R) =
g
Blm3 e
,
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where k± αg
2 = k + g ± α 2 − k + g ez .
g±
The explicit expression of Blmσ is referred to [21]. The superscript ± indicates the propagation direction of the plane waves. Then, the scattered wave on the left and the right sides of the plane of spheres can be expressed in the form of plane waves as sc± usc (r) = usc± α (r) + uβ (r) =
±
ikαg ·r U sc± + U sc± αg e βg e
ik± βg ·r
,
(18)
g
where U sc± αg =
g±
blm1 Blm1 ,
lm
U sc± βg
g± g± blm2 Blm2 + blm3 Blm3 . = lm
Let the incident plane wave be expressed as in± uin± (r) = uin± α (r) + uβ (r) =
±
ikαg ·r U in± + U in± αg e βg e
ik± βg ·r
.
(19)
g
Then, the expansion coefficients of the scattered wave and the incident wave can be related by ⎡ ⎣ ⎡ ⎣
U sc+ α U sc+ β U sc− α U sc− β
⎤
⎡
⎦=⎣ ⎤
⎡
⎦=⎣
M ++ αα
M ++ αβ
M ++ βα
M ++ ββ
M −+ αα
M −+ αβ
M −+ βα
M −+ ββ
⎤⎡ ⎦⎣ ⎤⎡ ⎦⎣
U in+ α U in+ β U in+ α U in+ β
⎤
⎡
⎦+⎣ ⎤
⎡
⎦+⎣
M +− αα
M +− αβ
M +− βα
M +− ββ
M −− αα
M −− αβ
M −− βα
M −− ββ
⎤⎡ ⎦⎣ ⎤⎡ ⎦⎣
U in− α U in− β U in− α U in− β
⎤ ⎦,
(20.1)
⎤ ⎦.
(20.2)
The first terms in the right sides of Eqs. (20.1) and (20.2) represent the transmission amplitude and the reflection amplitude when the incident wave comes from the left side of the plane of spheres; the second ones represent the reflection amplitude and the transmission amplitude when the incident wave comes from the right side of the plane of spheres. It is noted that all plane wave expansions, including the incident waves’ and the scattered waves’, are referred to the central scatterer in the plane. Therefore, the total displacement amplitudes at the plane z = 0− (side 1) and at the plane z = 0+ (side 2) are, respectively, ⎡ ⎣ ⎡ ⎣ ⎡ ⎣ ⎡ ⎣
U+ α (1) U+ β (1) U− α (1) U− β (1) U− α (2) U− β (2) U+ α (2) U+ β (2)
⎤
⎡
⎦=⎣ ⎤
⎡
⎦=⎣ ⎤
⎡
⎦=⎣ ⎤
⎡
⎦=⎣
U in+ α U in+ β U sc− α U sc− β U in− α U in− β U sc+ α U sc+ β
⎤ ⎦, ⎤
(21) ⎡
⎦+⎣ ⎤
U in− α U in− β
⎤ ⎦,
⎦, ⎤
(22)
(23) ⎡
⎦+⎣
U in+ α U in+ β
⎤ ⎦.
(24)
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Substituting Eq. (??) into Eqs. (22) and (24) leads to ⎤⎡ + ⎤ ⎡ ⎤⎡ − ⎤ ⎡ − ⎤ ⎡ −+ U α (1) U α (2) U α (1) M αα M −+ I + M −− M −− αα αβ αβ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦, ⎣ − ⎦ = ⎣ −+ + U β (1) U+ U− M βα M −+ M −− I + M −− β (1) β (2) ββ βα ββ ⎡ + ⎤ ⎡ ⎤ ⎡ + ⎤ ⎡ +− ⎤⎡ − ⎤ U α (2) U α (1) U α (2) M ++ I + M ++ M αα M +− αα αβ αβ ⎣ + ⎦ = ⎣ ++ ⎦ ⎣ ⎦ ⎣ ⎦ ⎦. ⎣ + U β (2) U+ U− M βα I + M ++ M +− M +− β (1) β (2) ββ βα ββ
(25)
(26)
If the side 1 and side 2 are used to indicate the planes at z = −a and z = +a, respectively, Eqs. (25) and (26) should be replaced by ⎤ ⎡ + ⎤ ⎡ −− ⎤⎡ − ⎤ ⎡ − ⎤ ⎡ −+ U α (1) U α (2) U α (1) Qαα Q−+ Qαα Q−− αβ αβ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦, ⎣ − ⎦ = ⎣ −+ + (27) U β (1) U+ U− Qβα Q−+ Q−− Q−− β (1) β (2) ββ βα ββ ⎡ + ⎤ ⎡ ++ ⎤ ⎡ + ⎤ ⎡ +− ⎤⎡ − ⎤ U α (2) U α (1) U α (2) Qαα Q++ Qαα Q+− αβ αβ ⎣ + ⎦ = ⎣ ++ ⎦ ⎣ + ⎦ + ⎣ +− ⎦⎣ − ⎦, (28) ++ +− U β (2) U β (1) U β (2) Qβα Qββ Qβα Qββ where
s s s ss s Qss κκ = κ κ δκκ δss + κ Mκκ κ ,
s s s a , ex p isk a , · · · ex p isk a . sκ = diag ex p iskκg κg2 z κg N z 1z
Considering that the energy flux of the longitudinal wave and the transverse wave can be estimated by, respectively, 2 ql = ρcl2 ω U αg kαg , (29.1) 2 2 qt = ρct ω U βg kβg , (29.2) the reflection and transmission coefficients can be estimated by 2 2 2 tr(ref) ± 2 tr(ref) ± cl U αg kagz + ct U βg kβgz T (R) =
g
2 2 2 in in+ 2 in in+ cl U αg kagz + ct U βg kβgz
.
(30)
g
3 Numerical example and discussion In the numerical examples, different sizes, material constants and array patterns of spheres as scatterers are separately investigated, the reflection and transmission coefficients are calculated and the influences of the size, material constants and array patterns of scatterers are discussed. Polyester (ρ = 1,220 kg/m3 , cl = 2,490 m/s, ct = 1,180 m/s) is chosen as the host material. Steel spheres, aluminum spheres, copper spheres and glass spheres are chosen as the scatterer, respectively. Square lattice, rectangular lattice and hexagonal lattice are considered. The first numerical example aims at showing the influence of the radius of the spherical scatterers. The reflection and transmission coefficients of five different radi of steel spheres are numerically calculated and shown graphically, as in Figs. 2 and 3. The radius of the spheres considered is r = 0.1a0 , 0.2a0 , 0.25a0 , 0.4a0 and 0.5a0 , respectively. The scatterers in the plane are arrayed in hexagonal lattice, and the lattice constant is a0 . By comparing the five curves, transmission coefficients versus normalized frequency, it is found that each curve has a deep dip at a certain frequency. When the radius of the spheres gets larger, the dip gets deeper and wider. Meanwhile, the frequency corresponding to the dip shifts gradually toward the low frequency. The phenomena can be explained by the interaction of a Bloch wave in the plane of sphere with the incident wave in
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Fig. 2 The influences of the radius of the spherical scatterers on the transmission coefficients. (c0 is the speed of the longitudinal wave in the host medium)
Fig. 3 The influences of the radius of the spherical scatterers on the reflection coefficients
the host. The interaction makes the scattering wave interfering destructively with the incident wave at a certain frequency and leads to a dramatical decrease of the transmission wave. As the radius of spheres increases, the destructive interfering becomes stronger and the dips in the transmission curve become deeper and wider. It is motivated that the plane of spheres can be used as a sound barrier to insulate the sound within certain frequency range by an elaborate design. The reflection coefficient is complementary with the transmission coefficient, see Figs. 2 and 3. That is because the energy of the incident wave is divided into two parts without lost. One is carried by the transmission wave and another is carried by the reflection wave. The second numerical example aims at showing the influence of different materials of scatterers. The transmission coefficients of steel spheres (ρ = 7,800 kg/m3 , cl = 5,940 m/s, ct = 3,200 m/s), aluminum spheres (ρ = 2,700 kg/m3 , cl = 6,260 m/s, ct = 3,080 m/s), copper spheres (ρ = 8,900 kg/m3 , cl = 4,700 m/s, ct = 2,260 m/s) and glass spheres (ρ = 2,500 kg/m3 , cl = 5,700 m/s, ct = 3,400 m/s) are separately calculated and depicted graphically in Fig. 4. The radius of spheres is fixed at 0.25a0 , and the array pattern is the hexagonal lattice in all cases. It is found that case 1 (scatterers are steel spheres) and case 2 (scatterers are copper spheres) are similar, while case 3 (scatterers are aluminum spheres) and case 4 (scatterers are glass spheres) are similar. Consider that the densities of steel and copper are much larger than those of aluminum and glass, but speeds of the longitudinal and the transverse waves are only slightly different. It can be concluded that the density of the scatterer plays a more important role than the elastic constants of it. The larger the density of spheres, the deeper and wider the dip of the transmission curve, and more evidently the frequency corresponding with the dip shifts toward the low frequency.
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Fig. 4 The influences of material constants of the spherical scatterers on the transmission coefficients
Fig. 5 The influences of array patterns of the spherical scatterers on the transmission coefficients. (steel sphere r = 0.25a0 , and a2 = 1.5a1 for the rectangle lattice)
The third numerical example shows the influence of array patterns of the spherical scatterers. The transmission coefficients in the case of steel spheres arrayed in a square lattice, rectangular lattice and hexagonal lattice are separately calculated, and the numerical results are depicted graphically in Fig. 5. The radius of the spheres is fixed at 0.25a0 in three cases. By comparing these curves, it is found that there is only one dip on the transmission curves in the case of a hexagonal lattice and two dips on the transmission curve in the case of a square patterns, a deep one and a shallow one. Further, there are even three dips on the transmission curves in the case of a rectangular lattice. This means that the locally resonant phenomenon of scattering happens more than once when the scatterers are arrayed in square and rectangular lattice. Obviously, the array pattern of spheres also has influence on the transmission behavior, although the influence of array pattern is not so evident as the size and the density of the spheres. The last numerical example shows the influence of the incident angle. In Figs. 6 and 7, the transmission curves corresponding to four different incident angles are shown. It is found that the frequency of the deep dip does almost not change. But the depth and the width of the dip have somewhat change, namely the dip becomes shallow and wide with the increase in the incident angle θ . The transmission curves at different azimuthal angle ϕ for given polar angle θ have only slight changes. The changes can be noticed only at high frequency range. The azimuthal angles ϕ = 0◦ ∼ 30◦ are only considered due to the rotation symmetry of the hexagonal
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Fig. 6 The influences of the polar angle (θ) of the incidence wave on the transmission coefficients. (r = 0.25a0 , steel sphere, hexagonal lattice)
Fig. 7 The influences of the azimuthal angle (ϕ) of the incidence wave on the transmission coefficients. (r = 0.25a0 , steel sphere, hexagonal lattice)
lattice. This observation means that the plane of spheres as sound barrier can insulate the sound from different directions within a certain frequency range. The frequency range is only determined by the plane of spheres, and it can be elaborately designed by taking different array pattern, size and material constants of spheres.
4 Conclusions The reflection and transmission coefficients of elastic waves through a plane of spheres embedded in a homogeneous isotropic host medium with infinite extension are studied. Main concerns are placed on the influences of the size, material constants and array patterns of spheres in the plane. It is found that at least one deep dip appears at the curves of transmission coefficient versus normalized frequency. The larger the radius and the density of the sphere, the deeper and wider the dip gets. Meanwhile, the frequency corresponding to the dip shifts toward the low frequency more evidently. The array pattern can also influence the transmission behavior of elastic waves through the plane of spheres to some extent but not as evidently as the radius and the density of the spheres. This study implies that a plane of spheres can be elaborately designed to serve as a sound barrier. It can insulate the sound from different directions within a certain frequency range. The frequency range is only determined by the plane of spheres, and it can be designed by taking different array pattern, size and material constants of the spheres.
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Acknowledgments This work is supported by the National Natural Science Foundation of China (No. 10972029).
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