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Surface waves guided by
topography in an anisotropic
elastic half-space
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Y. B. Fu1,2 , G. A. Rogerson1 and W. F. Wang1
1 Department of Mathematics, Keele University, Staffordshire
Research
Cite this article: Fu YB, Rogerson GA, Wang
WF. 2013 Surface waves guided by topography
in an anisotropic elastic half-space. Proc R
Soc A 469: 20120371.
http://dx.doi.org/10.1098/rspa.2012.0371
Received: 21 June 2012
Accepted: 25 September 2012
Subject Areas:
applied mathematics
Keywords:
trapped modes, guided waves, Stroh
formalism, surface waves, anisotropic
Author for correspondence:
Y. B. Fu
e-mail: [email protected]
ST5 5BG, UK;
2 Department of Mechanics, Tianjin University, Tianjin 300072,
P.R. China
We consider the propagation of free surface waves
on an elastic half-space that has a localized geometric
inhomogeneity perpendicular to the direction of wave
propagation (such waves are known as topographyguided surface waves). Our aim is to investigate how
such a weak inhomogeneity modifies the surfacewave speed slightly. We first recover previously
known results for isotropic materials and then present
additional results for a generally anisotropic elastic
half-space assuming only one plane of material
symmetry. It is shown that a topography-guided
surface wave in the present context may or may
not propagate depending on a number of factors.
In particular, they cannot propagate if the original
two-dimensional surface wave on a flat half-space is
supersonic with respect to the speed of anti-plane
shear waves. For the case when a topographyguided surface wave may exist, the existence and
computation of wave speed correction is reduced to
the solution of a simple eigenvalue problem whose
properties are previously well understood. As a byproduct of our analysis, we deduce that there exists
at least one topography-guided surface wave on an
isotropic elastic half-space, and that it is unique when
the geometric inhomogeneity has sufficiently small
amplitude.
1. Introduction
This paper is concerned with the effects of a localized
hump or depression on the propagation of an elastic
surface wave along the otherwise flat surface of an elastic
half-space; see figure 1. A (classical) surface wave is a
wave propagating along a free flat surface, for which the
associated displacement and stress decay away from the
c 2012 The Author(s) Published by the Royal Society. All rights reserved.
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x3
surface. They appear in seismology, signal processing, non-destructive testing and within many
other physical scenarios. Lord Rayleigh [1] was the first to show that the plane traction-free
surface of an elastic isotropic half-space could support such a surface wave. The intuitive
approach initially adopted for an isotropic half-space is not capable of resolving the existence
and uniqueness question for surface waves on a generally anisotropic elastic half-space. For the
latter, the Stroh formalism [2] is the necessary tool and it was with this tool that Barnett et al. [3],
Barnett & Lothe [4] and Chadwick & Smith [5] gave the first existence and uniqueness proof. We
refer to Barnett & Lothe [6] for an updated account of the existence and uniqueness theory and
to the book by Ting [7] for a more detailed description of the theory and for a comprehensive
collection of relevant references.
A surface-wave solution may also be referred to as a trapped mode in the sense that
propagation energy is confined to within a few wavelengths of the free surface. At the same
time as extensions of Rayleigh’s classical surface-wave solution were made to anisotropic elastic
and/or viscoelastic materials, much interest was also shown in trapped modes around thin
rectangular plates, wedges, or similar structures attached to a half-space. There are two important
limits corresponding to this set-up. When the rectangular plates are thin and long, the trapped
modes are expected to localize near the free edge and the associated modes are also known as
edge waves; e.g. Konenkov [8], Norris et al. [9], Fu & Brookes [10], Lawrie & Kaplunov [11],
and references therein. When the extrusions are almost flat, we expect the trapped modes to be
the classical surface-wave mode slightly modified by the geometrical inhomogeneity. It is the
latter that we are interested in the present paper. For the intermediate case, existence results have
been obtained by Bonnet-Ben Dhia et al. [12] using the min–max principle, and numerical results
have been given by Burridge & Sabina [13]; see also Biryukov et al. [14] for a review of some
approximate methods.
Our present work is partly motivated by the recent series of studies by Kaplunov et al. [15],
Gridin et al. [16,17], Adams et al. [18] and Postnova & Craster [19,20], in which a multiple-scale
approach was fruitfully used to study trapped modes in a variety of problems, but attention
was invariably focused on isotropic materials. In this paper, we demonstrate that the same
methodology, coupled with the Stroh formalism, can be used to deal with anisotropic materials.
It turns out that our general formulation also enables us to settle the existence and uniqueness
question left open by Adams et al. [18] concerning topography-guided surface waves on an
isotropic elastic half-space.
The paper is organized as follows. After formulating the problem in §2, we present
and solve the leading-order and second-order problems in the subsequent two sections,
respectively. In §5, we write down the third-order problem and obtain the amplitude equation
by imposing a solvability condition. It is shown that this amplitude equation can be reduced
to the time-independent Schrödinger equation whose spectral properties are well known.
After demonstrating in §6 that our formulation can recover the isotropic results of Adams
et al. [18], we solve our amplitude equation in §7 numerically. Illustrative examples are
given that demonstrate a rich variety of behaviour associated with anisotropy: we may have
zero, a single or multiple topography-guided solutions, and the speed may be higher or
lower than the speed corresponding to a flat surface. The paper is concluded in §8 with
additional discussions.
..................................................
Figure 1. A half-space whose surface is not flat but suffers a localized and slowly varying perturbation. Surface waves propagate
along the x1 -direction and decay as x2 → ∞ and x3 → ±∞.
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x2
2
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2. Problem formulation
3
h(±∞) → 0,
(2.1)
where xi are Cartesian coordinates, h is a smooth localized function and ε is a small positive
parameter. We shall use the wavelength L as the reference length scale and assume that h/L
is of order unity. The specified function h is thus slowly varying and is localized in the
x3 -direction; see figure 1. The surface-wave problem is governed by the equation of motion and
constitutive relation
σij,j = ρ üi
and σij = cijkl uk,l ,
(2.2)
together with the traction-free boundary condition and decay condition
σij nj = 0 on x2 = −h(εx3 )
and ui → 0 as x2 → ∞,
(2.3)
where ui and σij (i, j = 1, 2, 3) are the displacement and stress components, ρ is the density, cijkl
are the elastic moduli and (ni ) is the unit normal to the free surface. Here and hereafter, we use
the summation convention on repeated suffices and use a comma to signify partial differentiation
with respect to spatial coordinates, and a superimposed dot to denote differentiation with respect
to time.
We assume that the moduli obey the symmetry relations cijkl = cklij = cjikl , and the strong
convexity condition
cpqrs Spq Srs > 0 ∀ non-zero real symmetric tensor S.
(2.4)
For numerical illustrations, we shall consider two representative anisotropic elastic materials. The
first is a transversely isotropic elastic material with moduli given by
cijkl = λδij δkl + μt (δik δjl + δil δjk ) + α(δij mk ml + mi mj δkl ) + βmi mj mk ml
+ (μl − μt )(mi mk δjl + mi ml δjk + mj mk δil + mj ml δik ),
(2.5)
where λ, α, β, μt and μl are material constants and mi is the preferred direction; see Spencer [21].
These constants are related to Young’s moduli El and Et and Poisson’s ratio νlt as follows:
El = β̂ −
(α + λ)2
,
λ + μt
Et =
4μt (β̂(λ + μt ) − (α + λ)2 )
β̂(λ + 2μt ) − (α + λ)2
and νlt =
α+λ
,
2(λ + μt )
where β̂ = λ + 2α + 4μl − 2μt + β. We shall take ρ = 1852 kg m−3 , νlt = 0.324 and
(El , Et , μl , μt ) = (42.7, 11.6, 4.69, 6.07) × 109 N m−2 ,
(2.6)
given by Rikards et al. [22] for a typical glass–epoxy composite. The second material is the cubic
material silicon with material constants taken from Farnell [23],
(c1111 , c1122 , c2323 ) = (165.7, 63.9, 79.913) × 109 N m−2
and ρ = 2340 kg m−3 .
(2.7)
For a surface wave propagating in the x1 -direction with speed v, the dependence on x1 and t
is through x1 − vt, and so equations (2.2) and (2.3)1 may be written as
cijkl uk,lj = ρv 2 ui,11 ,
−h(εx3 ) < x2 < ∞
(2.8)
..................................................
x2 = −h(εx3 ),
rspa.royalsocietypublishing.org Proc R Soc A 469: 20120371
We consider surface-wave propagation on a half-space whose surface is defined by
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and
4
cijkl uk,l nj = 0, on x2 = −h(εx3 ).
(2.9)
(2.10)
We now introduce the variable transformation
x1 = x1 ,
x2 = x2 + h(εx3 )
and x3 = εx3 ,
so that in terms of the new coordinates, the free surface is given by x2 = 0. We have
∂
∂
=
,
∂x1 ∂x1
∂
∂
=
∂x2 ∂x2
and
∂
∂
∂
= ε + εh (x3 ) ,
∂x3
∂x3
∂x2
and in terms of the new coordinates, (2.8) and (2.10) become
ciαkβ uk,αβ + ε(ciαk3 + ci3kα )(uk,3α + h uk,2α ) + ε2 ci3k3 (uk,33 + h uk,2
+ 2 h uk,23 + h2 uk,22 ) = ρv 2 ui,11 ,
0 < x2 < ∞,
(2.11)
and
ci2kα uk,α + εci3kα uk,α h (εx3 ) + εci2k3 (uk,3 + h uk,2 )
+ ε2 ci3k3 (h uk,3 + h2 uk,2 ) = 0,
on x2 = 0,
(2.12)
where here and hereafter, Greek subscripts range from 1 to 2, and a comma now signifies partial
differentiation with respect to the primed coordinates. To simplify notation, we shall from now
on drop the primes on the coordinates.
We now look for a perturbation solution of the form
ρv 2 = X0 + ε2 X1 + · · ·
and u = u(0) + εu(1) + ε2 u(2) + · · · ,
(2.13)
where X0 and X1 are constants and u(k) (k = 0, 1, 2, . . .) are functions to be determined. Obviously,
the solution (X0 , u(0) ) describes a surface wave when the surface is flat. Our aim is to find the
leading-order correction ε2 X1 when the surface is specified by (2.1).
On substituting (2.13) into (2.11) and (2.12) and then equating coefficients of like powers of ε,
we obtain, from the first three orders, respectively, that
(0)
(0)
for 0 < x2 < ∞,
(2.14)
(0)
on x2 = 0,
(2.15)
ciαkβ uk,αβ − X0 ui,11 = 0,
ci2kα uk,α = 0,
(1)
(1)
(0)
(0)
ciαkβ uk,αβ − X0 ui,11 = −(ciαk3 + ci3kα )(uk,3α + h uk,2α ),
(1)
(0)
(0)
for 0 < x2 < ∞,
(0)
ci2kα uk,α = −h (ci3kα uk,α + ci2k3 uk,2 ) − ci2k3 uk,3 , on x2 = 0,
(2)
(2)
(0)
(1)
(2.16)
(2.17)
(1)
ciαkβ uk,αβ − X0 ui,11 = X1 ui,11 − (ciαk3 + ci3kα )(uk,3α + h uk,2α )
(0)
(0)
(0)
(0)
− ci3k3 (uk,33 + h uk,2 + 2 h uk,23 + h2 uk,22 ),
and
(2)
(1)
(1)
for 0 < x2 < ∞,
(2.18)
(1)
ci2kα uk,α = −h (ci3kα uk,α + ci2k3 uk,2 ) − ci2k3 uk,3
(0)
(0)
− ci3k3 (h uk,3 + h 2 uk,2 ), on x2 = 0.
(2.19)
..................................................
ci2kl uk,l + εci3kl uk,l h (εx3 ) = 0, on x2 = −h(εx3 ).
rspa.royalsocietypublishing.org Proc R Soc A 469: 20120371
The free surface is defined by x2 + h(εx3 ) = 0, and so its unit normal is in the direction
(0, 1, εh (εx3 )), where the prime indicates differentiation with respect to the argument εx3 . Thus,
the traction-free boundary condition (2.9) may be written as
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3. The leading-order problem
5
z(kx2 ) = e−kx2 E z(0),
(3.2)
where E is a 3 × 3 matrix whose eigenvalues determine the decay rate; see Fu & Mielke [24]. We
shall initially set k = 1, and then show at a later stage (see the discussion leading to (5.16)) how to
obtain the corresponding results when k is not unity.
On substituting (3.1) with k = 1 into (2.14) and (2.15), we find that the equations of motion
reduce to
(3.3)
Tz (x2 ) + i(R + RT )z (x2 ) − Q(v) z(x2 ) = 0, 0 < x2 < ∞,
and the boundary conditions to t(0) = 0, where the reduced traction vector t is given by
− t(x2 ) = Tz (x2 ) + iRT z(x2 ),
(3.4)
and the matrices T, R and Q(v) are defined by their components
Tkl = ck2 l2 ,
Rkl = ck1 l2
(v)
and Qkl = ck1 l1 − X0 δkl .
(3.5)
On substituting a trial solution of the form z = a eipx2 into (3.3), we obtain
(p2 T + p(R + RT ) + Q(v) )a = 0,
(3.6)
so that the values of p are determined by setting the determinant of coefficient of a to zero. For v
in the subsonic interval [5], none of the values of p can be purely real. Furthermore, when the six
values of p are distinct (as three pairs of complex conjugates), the surface-wave solution can be
written as
z(x2 ) =
3
−1
ck a(k) eipk x2 = A eipx2 c = A eipx2 A−1 z(0) = eiApA
x2
z(0),
(3.7)
k=1
where p1 , p2 and p3 are the eigenvalues of (3.6) with positive imaginary part, a(k) (k = 1, 2, 3) are
the associated eigenvectors, c = (c1 , c2 , c3 )T is a constant vector,
eipx2 = diag{eip1 x2 , eip2 x2 , eip3 x2 },
p = diag{p1 , p2 , p3 }
and A = [a(1) , a(2) , a(3) ].
The last expression in (3.7) then shows that in this case, the matrix E in (3.2) is equal to −iApA−1 .
When the six values of p are not distinct, the matrix E can be determined as follows. First, with
the use of (3.3) and (3.4), the first-order derivatives z (x2 ) and t (x2 ) can easily be written as a
linear combination of z(x2 ) and t(x2 ), thus leading to
z(x2 )
N1 N2
z (x2 )
,
(3.8)
= iN
, N=
N3 N1T
it (x2 )
it(x2 )
the Stroh formulation, with N1 , N2 and N3 given by
N1 = −T−1 RT ,
N2 = T−1
and N3 = RT−1 RT − Q(v) .
(3.9)
Next, we define the surface-impedance matrix M through
− t(0) = Mz(0).
(3.10)
Since the half-space is homogeneous, the above relation implies that
− t(x2 ) = Mz(x2 )
∀x2 > 0.
(3.11)
..................................................
where c.c. denotes the complex conjugate of the preceding term, k is the wavenumber and f (kx3 )
is a function to be determined. The depth variation z(kx2 ) may be represented by
rspa.royalsocietypublishing.org Proc R Soc A 469: 20120371
The leading-order problem is the classical surface-wave problem, the solution of which may be
written in the form
(3.1)
u(0) = f (kx3 )z(kx2 ) eik(x1 −vt) + c.c.,
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On substituting (3.11) and (3.2) into (3.8), we find that the matrix E in (3.2) and the surface
impedance matrix are related by
and that M satisfies the Riccati matrix equation
(M − iR)T−1 (M + iRT ) − Q(v) = 0.
(3.13)
It follows from the definition (3.10) of the impedance matrix that the traction-free boundary
condition t(0) = 0 can be satisfied only if
det M = 0,
(3.14)
which is the secular equation determining X0 . When a root of this equation is found, a solution
of z(0) can be found by solving Mz(0) = 0, and the corresponding surface-wave solution is then
given by (3.1) and (3.2).
Finally, we note that although (3.13) has multiple solutions, it has a unique solution for M that
is positive definite for v less than the unique surface-wave speed, and this unique solution is what
should be selected in calculating the surface-wave speed according to (3.14) and E according to
(3.12) [24].
For an orthotropic elastic half-space whose axes of symmetry coincide with the coordinate
axes, the three matrices T, R and Q(v) take the simple form
⎞
⎛
⎞
⎛
⎞
⎛
T1 0
0
0
0
0 R1 0
Q1
⎟
⎜
⎟
⎜
⎟
⎜
T = ⎝ 0 T2 0 ⎠, R = ⎝R2 0 0⎠ and Q(v) = ⎝ 0 Q2
(3.15)
0 ⎠.
0
0 T3
0
0 0
0
0 Q3
It can be shown that the impedance matrix M then assumes the form
⎛
⎞
M1
M3 + iM4
0
⎜
⎟
M = ⎝M3 − iM4
M2
0 ⎠, Mi ∈ R.
0
0
M5
(3.16)
The Riccati matrix equation (3.14) can be solved exactly to find the unique solution
mentioned above,
⎫
⎪
⎪
T1 R 1 + R 2 2
T2
⎪
⎪
, M2 = γ M1 ,
M1 = T1 Q1 −
⎪
⎬
T2
1+γ
T1
(3.17)
⎪
⎪
T1 Q2 ⎪
γ R1 − R2
⎪
and M5 = T3 Q3 , γ =
.⎪
M3 = 0, M4 =
⎭
1+γ
T2 Q1
The secular equation (3.14) then takes the form
T1 T2 Q1 Q2 −
γ R21 + R22
= 0.
1+γ
(3.18)
For an isotropic elastic half-space, we have
T1 = T3 = μ,
T2 = λ + 2μ,
Q2 = Q3 = μ − X0 ,
Q1 = λ + 2μ − X0 ,
R1 = λ
and R2 = μ,
and then the secular equation reduces to the familiar form
X03 − 8μX02 +
λ+μ
8μ2 (3λ + 4μ)
X0 − 16μ3
= 0.
λ + 2μ
λ + 2μ
(3.19)
..................................................
(3.12)
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E = T−1 (M + iRT ),
6
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4. The second-order problem
7
(1)
(1)
(1)
u3 = (f w1 + fh w2 ) ei(x1 −vt) + c.c.,
(4.1)
where w1 and w2 , both functions of x2 only, satisfy the following equations derived from (2.16)
and (2.17):
c3232 w1 + 2ic3132 w1 − (c3131 − X0 )w1 = −ig(1) · z − g(2) · z ,
ic3231 w1 + c3232 w1 = −g(4) · z, on x2 = 0,
c3232 w2
+ 2ic3132 w2
(1)
− (c3131 − X0 )w2 = −ig
·z −g
(2)
·z ,
ic3231 w2 + c3232 w2 = −(ig(3) · z + g(2) · z ), on x2 = 0.
and
(4.2)
(4.3)
(4.4)
(4.5)
The four vectors g(1) , g(2) , g(3) and g(4) in the above equations are defined by their components
g(1)
α = c31α3 + c33α1 ,
g(2)
α = c32α3 + c33α2 ,
g(3)
α = c33α1
and g(4)
α = c32α3 .
(4.6)
To solve these two problems explicitly, we rewrite (3.7) in the form
zα = bαβ eipβ x2 ,
(4.7)
where the constants bαβ can in principle be obtained by comparing (4.7) with (3.2) and (3.7). For
instance, when the half-space is isotropic, these constants are given, to within a multiplicative
constant, by
b12 = p21 − 1, b21 = 2p2 , b22 = p2 (p21 − 1),
X0
X0
and p2 = i 1 −
.
p1 = p3 = i 1 −
μ
(λ + 2μ)
b11 = −2p1 p2 ,
To solve the problem for w1 , we first note that a particular integral of (4.2) is given by
w1 = s1 eip1 x2 + s2 eip2 x2 = sβ eipβ x2 ,
(4.8)
where s1 and s2 are constants to be determined.
On substituting this into (4.2), making use of (4.7), and then comparing the coefficients of eipβ x2 ,
we obtain
sβ =
(1)
(2)
igα bαβ + igα bαβ pβ
c3232 p2β + 2c3132 pβ + c3131 − X0
,
no summation on β.
(4.9)
When the material is isotropic, the denominator of (4.9) becomes zero when β = 1 owing to the
fact that p1 = p3 . In this case, (4.8) is modified to
w1 = s1 x2 eip1 x2 + s2 eip2 x2 ;
(4.10)
however, for brevity, the corresponding expression for s1 is not written out here. Numerically, the
case of isotropy can be dealt with by first considering a nearly isotropic material with e.g. moduli
given by
cijkl = λδij δkl + μ(δik δjl + δil δjk ) + γ δjl δi2 δk2 ,
and then taking the limit γ → 0.
(4.11)
..................................................
(1)
choosing u1 = u2 = 0 and seeking a solution for u3 in the form
rspa.royalsocietypublishing.org Proc R Soc A 469: 20120371
We now assume that the x3 = 0 plane is a plane of material symmetry, which is the case for
(0)
monoclinic materials. In this case, we may set u3 = 0. Also, there is no loss of generality in
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vt
1.80
8
1.65
vR
1.60
1.55
20
40
60
80
q
Figure 2. Variation of the leading-order surface-wave speed vR and anti-plane shear wave speed vt against θ , showing that
the two-dimensional surface wave is subsonic only for 0◦ ≤ θ < 24.37◦ , or 54.67◦ < θ ≤ 90◦ . The unit of speed is km s−1 .
6.0
vt
5.5
5.0
vR
4.5
0
20
40
q
60
80
Figure 3. Variation of the leading-order surface-wave speed vR and anti-plane shear wave speed vt against θ for a silicon
material, showing that the two-dimensional surface wave is subsonic for all values of θ .
A general solution of (4.2) is then given by
w1 = s3 eip3 x2 + sβ eipβ x2 ,
(4.12)
where p3 is the root of
c3232 p2 + 2c3132 p + c3131 − X0 = 0
with positive imaginary part, and s3 is a disposable constant. We observe that such a p3 exists
only if c23132 − c3232 (c3131 − X0 ) < 0, that is if X0 < ρvt2 , where vt is the anti-plane shear body wave
speed. In other words, the type of topography-guided surface-wave solution under consideration
only exists if the original surface wave on the associated flat half-space is subsonic. For the
composites defined by (2.5) and (2.6) with m = (cos θ, sin θ, 0)T , figure 2 shows the variation of
√
vR (= X0 /ρ) and vt with respect to the angle θ . It is seen that the subsonic condition is satisfied
only for 0◦ ≤ θ < 24.37◦ or 54.67◦ < θ ≤ 90◦ .
√
In figure 3, we have shown the variation of vR (= X0 /ρ) and vt with respect to the angle
θ between the [100] direction and that of wave propagation, assuming that the surface waves
propagate on the (001) plane of the silicon material described by (2.7). In this case, the surface
wave is subsonic for all the values of θ considered [23].
..................................................
1.70
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1.75
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On substituting (4.12) into the boundary condition (4.3), we obtain
(4.13)
Similarly, the problem for w2 can be solved to yield
w2 = q3 eip3 x2 + qβ eipβ x2 ,
(4.14)
where
(1)
qβ = −
(2)
gα bαβ pβ + gα bαβ p2β
no summation on β,
(4.15)
gα (bα1 + bα2 ) + gα bαβ pβ + c3231 (q1 + q2 ) + c3232 pβ qβ
.
c3231 + c3232 p3
(4.16)
c3232 p2β + 2c3132 pβ + c3131 − X0
,
and
q3 = −
(3)
(2)
Thus, under the assumption that x3 = 0 is a plane of material symmetry, the second-order problem
can be solved without having to impose any solvability conditions.
5. The third-order problem
If we write u(2) as
u(2) = y(x2 , x3 )ei(x1 −vt) + c.c.,
(5.1)
then the unknown vector function y(x2 , x3 ) satisfies the following equations derived from (2.18)
and (2.19):
Ty + i(R + RT )y − Q(v) y = h(1)
(5.2)
Ty + iRT y = h(2) , on x2 = 0,
(5.3)
and
where a prime denotes partial differentiation with respect to x2 ,
h(1) = −fX1 z − ig(1) {f w1 + f h (w2 + w1 ) + f (h w2 + h2 w2 )}
− g(2) {f w1 + f h (w2 + w1 ) + f (h w2 + h2 w2 )}
− f Sz − 2f h Sz − f (h Sz + h2 Sz ),
(5.4)
h(2) = −i(g(1) − g(3) )(f h w1 + fh2 w2 ) − (g(2) − g(4) )[f w1 + (f h + fh )w2 ]
− g(2) (f h w1 + fh2 w2 ) − f h Sz − fh2 Sz
(5.5)
and S is the matrix with components cα3β3 .
It can easily be shown, by integrating by parts followed by the use of (3.3), that
∞
z̄ · (Ty + i(R + RT )y − Q(v) y) dx2 = z̄ · (Ty + iRT y)|∞
0 .
0
Thus, the solvability condition for the inhomogeneous problem (5.2) and (5.3) is given by
∞
z̄ · h(1) dx2 = −z̄ · h(2) |x2 =0 .
(5.6)
0
We note that a similar solvability condition can be written down for the second-order problem,
but it can be shown that it is automatically satisfied.
..................................................
+ bα2 ) − c3231 (s1 + s2 ) − c3232 pβ sβ
.
c3231 + c3232 p3
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s3 =
(4)
igα (bα1
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On substituting (5.4) and (5.5) into (5.6), we obtain
10
where
c0 =
c1 =
c2 =
∞
0
∞
0
∞
0
|z|2 dx2 ,
(5.8)
{iw2 z̄ · g(1) − w2 z̄ · g(2) + z̄ · Sz } dx2 − w2 (0)z̄(0) · g(4) ,
(5.9)
{w2 (−iz̄ · g(1) + z̄ · g(2) ) − z̄ · Sz } dx2
+ w2 (0)g(2) · z̄ (0) − iw2 (0)g(3) · z̄(0),
∞
{i(w2 + w1 )z̄ · g(1) + (w1 + w2 )z̄ · g(2) + 2z̄ · Sz } dx2
c3 =
(5.10)
0
+ iw1 (0)z̄(0) · (g(1) − g(3) ) + w2 (0)z̄(0) · (g(2) − g(4) )
+ w1 (0)z̄(0) · g(2) + z̄(0) · Sz(0)
∞
c4 =
{w1 (iz̄ · g(1) − z̄ · g(2) ) + z̄ · Sz} dx2 − w1 (0)z̄(0) · g(4) .
and
(5.11)
(5.12)
0
The coefficient c0 is obviously real and positive. In appendix A, we show that c2 and c4 are real,
but c3 is purely imaginary and is related to c1 by c3 = 2i Im(c1 ). Thus, on writing f in the polar form
f (x3 ) = r(x3 ) eiθ(x3 ) ,
(5.13)
(i)
it is then straightforward to deduce that θ (x3 ) = −c1 h(x3 )/c4 and that the variation of r(x3 ) is
governed by the equation
r + (d2 h2 + d1 h + d0 X1 )r = 0,
(5.14)
where
d0 =
(r)
c0
,
c4
(r)
d1 =
c1
c4
and d2 =
c2
+
c4
(i)
c1
c4
2
,
(i)
and c1 and c1 denote the real and imaginary parts of c1 , respectively.
Thus, the problem of finding the speed correction X1 is reduced to solving the eigenvalue
problem of (5.14) subject to the decay conditions r(x3 ) → 0 as x3 → ±∞. We recall, however, that
the above eigenvalue problem has been derived under the assumption that the wavenumber k in
(3.1) is unity. For the case when k is not unity, we may observe that under the substitutions kx1 →
x1 , kx2 → x2 and kx3 → x3 , equations (2.14)–(2.19) remain the same, except that h (x3 ) and h (x3 )
should be replaced by h (k−1 x3 ) and k−1 h (k−1 x3 ), respectively. Equation (5.14) is then replaced by
x d
x 3
1
3
+ h
+ d0 X1 r(x3 ) = 0,
r (x3 ) + d2 h2
k
k
k
(5.15)
or equivalently, with another substitution x3 /k → x3 followed by r(kx3 ) → r(x3 ),
r (x3 ) + (d2 k2 h2 (x3 ) + d1 kh (x3 ) + d0 k2 X1 )r(x3 ) = 0,
which shows that the speed correction is dependent on the wavenumber.
(5.16)
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(5.7)
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The amplitude equation (5.16) is recognized as a special case of the so-called time-independent
Schrödinger equation
associated with the potential well V(x3 ) and energy level E (a constant, which should not be
confused with the matrix E in (3.2)). Under the assumption that
∞
(1 + |x|)|V(x)| dx < ∞,
−∞
Simon [25] and Klaus [26] have established the following general results concerning its negative
eigenvalues.
(i) If there exists an eigenvalue when the potential well V(x3 ) is of sufficiently small
amplitude, then the eigenvalue is unique and is given by
√
1
1 ∞
−E = I −
V(x){|x| ∗ V(x)} dx + O(δ 3 ),
(5.18)
2
4 −∞
where the star denotes convolution, I is defined by
∞
I=
V(x) dx
−∞
and δ is a small positive parameter characterizing the amplitude of V(x3 ).
(ii) A necessary and sufficient condition for the existence of the above-mentioned single
eigenvalue is that the right-hand side of (5.18) is positive. Thus, under the assumption
that the second term on the right-hand side of (5.18) is one order of magnitude smaller than the
first term, this condition is I ≥ 0 (note that if I = 0, the second term on the right-hand side
of (5.18) is automatically positive; see Simon [25, p. 284]).
(iii) Since increasing the amplitude of a potential can only increase the number of eigenvalues
[25, p. 284], the condition I ≥ 0 is also sufficient for the existence of at least one eigenvalue,
even when the amplitude of V is not small.
For the special form of V in (5.16), we have
I = d2 k2
∞
−∞
h2 dx3 ,
and so I ≥ 0 if and only if d2 ≥ 0. However, when h is of small amplitude, the two terms in the
asymptotic expansion (5.18) are of the same order of magnitude, and they combine to give
∞
1
−d0 X1 = (d2 + d21 )k
h2 dx3 + O(δ 3 ).
(5.19)
2
−∞
Thus, the existence condition is in fact given by
d2 + d21 > 0.
(5.20)
This special case of V serves to demonstrate the fact that eigenvalues may still exist even if
I < 0 (whether V has small amplitude or not). We highlight this fact since some authors have
previously claimed that I ≥ 0 is also necessary for the existence of an eigenvalue. This claim is only
valid if the condition in italics in item (ii) above is satisfied and if the potential is of sufficiently
small amplitude.
For the case corresponding to figure 2, figures 4–6 show the variations of the three coefficients
for values of the angle θ for which the two-dimensional surface wave is subsonic. The blowup behaviour corresponds to the fact that c4 vanishes at θ = 14.8◦ or 55.7◦ approximately. It is
seen that d2 is positive in both subsonic ranges (except at the two isolated values of θ where it
blows up), and so it follows immediately from the above general results that (5.14) has at least
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(5.17)
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r (x3 ) + (V(x3 ) + E)r(x3 ) = 0,
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1.5
12
d0
20
40
–0.5
60
80
q
–1.0
–1.5
Figure 4. Variation of d0 with respect to θ for the case considered in figure 2.
2
1
d1
20
40
60
80
q
–1
–2
Figure 5. Variation of d1 with respect to θ for the case considered in figure 2.
2.0
1.5
d2 1.0
0.5
0
20
40
q
60
Figure 6. Variation of d2 with respect to θ for the case considered in figure 2.
80
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0.5
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0.05
13
d0
30
40
q
d 21 + d 2
–0.05
d2
–0.10
–0.15
Figure 7. Variation of d0 , d12 + d2 and d2 with respect to θ for the case considered in figure 2.
one eigenvalue. This is confirmed in our numerical calculations later. In contrast, for the silicon
material defined by equation (2.7), figure 7 shows that d2 + d21 is negative for all values of θ, and
so the existence of eigenvalues cannot be established using the general results above.
6. Isotropic materials
Before solving (5.16) subject to the decay conditions r(±∞) → 0 numerically, we first consider the
special case when the material is isotropic. This case has previously been studied by Adams et al.
[18] using a procedure that is particularly developed for isotropic materials. We now show that
our formulae recover their results in this special case.
First, to facilitate comparison, we write X0 for X0 /μ throughout this section and introduce κ
√
through κ = (λ + 2μ)/μ. Equation (3.19) can be rewritten as
X03 − 8X02 +
8(3κ 2 − 2)
κ2 − 1
X0 − 16
= 0.
2
κ
κ2
(6.1)
The second-order problem now reduces to
w1 − (1 − X0 )w1 = −ig(1) · z − g(2) · z ,
(6.2)
w1 = −g(4) · z, on x2 = 0,
(6.3)
w2 − (1 − X0 )w2 = −ig(1) · z − g(2) · z
w2 = −ig(3) · z − g(2) · z , on x2 = 0,
and
where
(6.4)
2
g(1)
α = (κ − 1)δ1α ,
2
g(2)
α = (κ − 1)δ2α ,
g(3)
α = λδ1α
(6.5)
and g(4)
α = μδ2α .
It is then easy to show that the solutions are given by
w1 = s1 x2 eip1 x2 + s2 eip2 x2 + s3 eip3 x2
with
s1 = 0,
q1 = 0,
and w2 = q1 x2 eip1 x2 + q2 eip2 x2 + q3 eip3 x2 ,
X0
s3 = −2i 1 − X0 1 − 2 ,
κ
X0
X0
q2 = i(X0 − 2) 1 − 2 and q3 = 2i 1 − 2 .
κ
κ
s2 = i(2 − X0 ),
As a result, the expressions (5.8)–(5.12) can be evaluated explicitly to give c3 = 0, c2 = 0,
c0 =
2ĉ0
,
2 − X0
c1 =
ĉ1
κ4
and c4 =
32(1 − κ 2 )ĉ4
,
(X0 − 2)2 ( 1 − X0 + 1 − X0 /κ 2 )κ 6
√
(6.6)
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where
X0
κ2
14
X0
X0
+ (1 − X0 )3/2 2 − 2 + 7 1 − X0 −1 + 2 ,
κ
κ
ĉ1 = X02 (14κ 2 − 4κ 4 ) + 4X0 (8 − 14κ 2 + 3κ 4 ) − 8(4 − 5κ 2 + κ 4 )
and
ĉ4 = 52 − 64κ 2 + 12κ 4 + X0 (−44 + 81κ 2 − 17κ 4 ) + X02 (−8 − 13κ 2 + 4κ 4 ).
By cross-multiplication followed by repeated use of (6.1) to eliminate powers of X0 higher than
3, we have verified using MATHEMATICA that our c4 /c1 and c4 /c0 are equal to those of Adams
et al. [18], βA/B and A/C, respectively. Additionally, as a check on our derivations, we have used
the elastic moduli given by (4.11) to compute the coefficients for increasingly small γ and have
obtained the same values as the above explicit expressions.
We observe that with d2 = 0, the sufficient condition (5.20) is satisfied automatically, and so
we may conclude that topography-guided surface waves always exist in an isotropic material.
This settles the existence question left open in Adams et al. [18]. We may further conclude that
when the geometric inhomogeneity is of sufficiently small amplitude, there can only exist a single
guided surface
√ wave. Finally, it can easily be verified numerically that c0 /c4 (= d0 ) is positive
for all κ > 2/ 3. We thus conclude that all topography-guided surface waves on an isotropic
elastic half-space travel at a slower speed than the classical Rayleigh wave (see (7.2) and also
Bonnet-Ben et al. [12, Theorem 1]).
7. Numerical results
We shall now explain our numerical procedure by focusing on the case k = 1. Thus, we return to
the amplitude equation (5.14) and solve it subject to the decay conditions r(±∞) → 0. In the limit
x3 → ±∞, equation (5.14) can be approximated by
r (x3 ) + d0 X1 r(x3 ) = 0.
(7.1)
It is clear that r(x3 ) will have the required decay behaviour as x3 → ±∞ only if
d0 X1 < 0,
(7.2)
and when this is satisfied, we have
√
r(x3 ) ∼ e∓
−d0 X1 x3
,
as x3 → ±∞.
(7.3)
It then follows that if d0 < 0, then the topography-guided surface waves travel at a lower speed
than their two-dimensional counterpart, whereas if d0 > 0, then the topography-guided surface
waves are faster than their two-dimensional counterpart. When h(x3 ) is an even function of x3 ,
r(−x3 ) is a solution of (5.7) whenever r(x3 ) is a solution. Thus, the eigensolutions of (5.14) are
either even or odd. For the even (symmetric) modes, we may impose, without loss of generality,
the conditions
r(0) = 1
r (0) = 0,
(7.4)
−d0 X1 r(L) = 0,
(7.5)
and
and the decay behaviour through
e(X1 ) ≡ r (L) +
where L is a sufficiently large positive constant and the first equation in (7.5) defines the error
function e(X1 ). For each fixed X1 , the e(X1 ) can be evaluated after integrating (5.14) subjected to
the initial conditions (7.4). We first plot e(X1 ) against X1 to show the approximate locations of any
zeros and then use the Newton–Raphson method to find the exact values of the zeros.
..................................................
1 − X0 + 4 1 −
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ĉ0 =
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r
15
1.0
0.4
0.2
–20
10
–10
20
x3
Figure 8. Lateral variation of the only topography-guided surface wave possible when θ = 2.187◦ , whose speed is lower than
its two-dimensional counterpart (X1 = −0.146).
For the odd (anti-symmetric) modes, the initial conditions (7.4) are replaced by
r(0) = 0
and
r (0) = 1,
(7.6)
and we integrate (5.14) subject to the initial conditions (7.6) and iterate on X1 in order to satisfy
the decay condition (7.5).
When h(x3 ) is not an even function of x3 , the eigenmodes do not have any symmetry
properties. We may, for instance, integrate (5.14) subject to (7.5) and
r (−L) − −d0 X1 r(−L) = 0,
(7.7)
respectively, and iterate on X1 so that the two solutions have the same gradient at a matching
point, say x3 = 0.
The above numerical scheme is first tested on the eigenvalue problem
r (x3 ) + (−λ + n(n + 1) sech2 x3 )r(x3 ) = 0,
r(±∞) → 0,
n integer,
(7.8)
which is known to have n eigenvalues given by λ = i2 (i = 1, 2, . . . , n), with odd i corresponding to
odd modes and even i to even modes; see Drazin & Johnson [27]. Our scheme is able to reproduce
these exact results correctly. The numerical scheme is then applied to solve (5.14) for a variety
of h(x3 ) and different values of d1 and d2 . Our numerical results confirm the validity of the
asymptotic formula (5.19) and the existence condition (5.20). In particular, we invariably find that
for h sufficiently small, the single eigenvalue tends to zero when d2 + d21 approaches zero from
above, and there is no eigenvalue when d2 + d21 ≤ 0.
We now present illustrative calculations for the composite material defined by (2.6) and for the
2
case when the topography is described by the ‘Gaussian bump’ h(x3 ) = e−x3 . When θ = 2.187◦ ,
we have d0 = 0.36 and the corresponding X1 must necessarily be negative. It is found that there
is only a single solution that is symmetric with X1 = −0.146 (the unit of X1 is 109 N m−2 ), and
so the topography has the effect of reducing the two-dimensional surface-wave speed. The
corresponding profile of r(x3 ) is shown in figure 8.
When θ = 16.767◦ , we have d0 = −0.77, and the corresponding X1 must necessarily be positive.
It is found that there exist one odd solution and one even solution, corresponding to X1 = 0.186
and 0.469, respectively. The topography has the effect of producing two variations of the original
two-dimensional surface wave. Both waves have a higher speed than their two-dimensional
counterpart, and their lateral variations are shown in figure 9.
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r
r
0.5
1.0
–10
–5
–0.5
x3
5
10
0.5
–1.0
–10
–5
x3
5
10
Figure 9. Lateral variations of the two topography-guided surface waves possible when θ = 16.767◦ , whose speeds are higher
than their two-dimensional counterpart. The anti-symmetric and symmetric profiles correspond to the eigenvalues X1 = 0.186
and 0.469, respectively.
We have also carried out calculations for the silicon material defined by (2.7) for which d2 +
d21 < 0 for all values of θ. We have tried a large number of h profiles, but have found no eigenvalues
for (5.14).
8. Concluding remarks
We have extended the analysis of Adams et al. [18] to a generally anisotropic elastic material with
x3 = 0 a plane of material symmetry. It is shown that determination of the wave speed correction
is also reduced to the solution of a simple eigenvalue problem. However, the current anisotropic
problem has a much richer variety of behaviour and raises new questions. For instance, the
existence of a topography-guided surface wave can be eliminated if the original two-dimensional
surface wave is supersonic with respect to anti-plane shear motions, and surface inhomogeneity
may either increase or decrease the speed of the original two-dimensional surface wave. There
also exists the possibility that the coefficient of the second-order derivative term in the reduced
eigenvalue problem vanishes, in which case, a topography-guided surface wave, if it exists, may
be expected to behave very differently.
The methodology that we employed is based on the assumption that a slowly varying
perturbation to the otherwise flat surface will give rise to a small-amplitude perturbation to the
surface-wave speed. This may, however, not be the case. For instance, Farnell [23] showed that as
the direction of propagation is slightly turned in the x1 x3 -plane (in terms of the notation in the
present paper), a two-dimensional surface wave will become a three-dimensional surface wave,
and the wave speed may experience a finite change, no matter how small the turning is. Thus,
when the associated two-dimensional surface wave is supersonic with respect to anti-plane shear
motions, we cannot exclude the existence of a topography-guided surface wave whose wave
speed differs significantly from the two-dimensional surface-wave speed. Finally, when the x3 = 0
plane is not a plane of material symmetry, the present methodology does not apply either. We
leave these questions to future studies.
The contribution of W.F.W. was supported by a PhD studentship jointly funded by China Scholarship Council
and Keele University. We thank all three referees for their constructive comments and suggestions.
Appendix A. Proof that c2 and c4 are real and c3 = 2i Im (c1 )
We first define a differential operator L by
L[w1 ] = c3232 w1 + 2ic3132 w1 − (c3131 − X0 )w1 ,
(A 1)
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see (4.2). Then for any twice differentiable function v(x2 ), we have, by integrating by parts,
∞
∞
v̄L[w1 ] dx2 =
w1 L[v] dx2 − (ic3132 w1 + c3232 w1 )v̄|x2 =0
(A 2)
where an overline signifies complex conjugation. With v replaced by w1 , the above identity shows
that the expression
∞
w1 L[w1 ] dx2 + (ic3132 w1 + c3232 w1 )w1 |x2 =0
0
is real. This implies, through the further use of (4.2) and (4.3), that the expression
∞
(iw1 z̄ · g(1) − w1 z̄ · g(2) ) dx2 − g(4) · z̄(0)w1 (0)
0
is real. The reality of c4 then follows from this result and the fact that S is a symmetric matrix so
that the extra term z̄ · Sz is also real.
In a similar manner, upon replacing v and w1 in (A 2) both by w2 and making use of (4.4) and
(4.5), we find that c2 is also real.
If, on the other hand, the v in (A 2) is replaced by w2 and use is made of (4.2)–(4.5), we obtain
∞
∞
w̄1 (iz̄ · g(1) + z̄ · g(2) ) dx2
w2 (iz̄ · g(1) − z̄ · g(2) ) dx2 +
0
0
= w2 (0)g(4) · z̄(0) − w̄1 (0)[ig(3) · z̄(0) + g(2) · z̄ (0)].
(A 3)
The relation c3 − c1 = −c̄1 then follows from this result after straightforward manipulations.
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