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Proc. R. Soc. A (2005) 461, 1181–1197
doi:10.1098/rspa.2004.1431
Published online 6 April 2005
Trapped modes in curved elastic plates
B Y D MITRI G RIDIN , R ICHARD V. C RASTER
AND
A LEXANDER T. I. A DAMOU
Department of Mathematics, Imperial College London,
South Kensington Campus, London SW7 2AZ, UK
(r.craster@imperial.ac.uk)
We investigate the existence of trapped modes in elastic plates of constant thickness,
which possess bends of arbitrary curvature and flatten out at infinity; such trapped
modes consist of finite energy localized in regions of maximal curvature. We present both
an asymptotic model and numerical evidence to demonstrate the trapping. In the
asymptotic analysis we utilize a dimensionless curvature as a small parameter, whereas
the numerical model is based on spectral methods and is free of the small-curvature
limitation. The two models agree with each other well in the region where both are
applicable. Simple existence conditions depending on Poison’s ratio are offered, and
finally, the effect of energy build-up in a bend when the structure is excited at a resonant
frequency is demonstrated.
Keywords: trapped modes; elastic waveguide; bound states; curved plate;
perturbation methods
1. Introduction
The trapping of vibrational or electromagnetic energy is an interesting, but often
undesirable, phenomenon occurring in many important applications in acoustics,
electromagnetics, quantum mechanics and elasticity. It is therefore important to
be able to predict the occurrence of trapped modes and understand their physical
origin. Trapped modes are non-trivial, finite energy solutions to the homogeneous
(source-free) time-harmonic problem. They are known to exist in, for instance,
acoustic waveguides with obstacles (Evans et al. 1994 and references therein),
water-wave channels with immersed bodies (e.g. McIver et al. 2003) and in
the vicinity of diffraction gratings (e.g. Porter & Evans 1999). In some cases, the
existence of trapped modes has been proved rigorously, while in other cases only
numerical evidence has been presented.
One particularly simple and ubiquitous structure supporting scalar trapped
modes is that of a two-dimensional waveguide that is bent arbitrarily but which
straightens out eventually at infinity. For this geometry, it has been proved in
the last decade or so that the Helmholtz equation with Dirichlet boundary
conditions possesses trapped-mode solutions localized in the regions of maximal
curvature, and that there is at least one resonant frequency below the cut-off
(see Duclos & Exner 1995; Londergan et al. 1999 and references therein).
Less work has been done on trapped modes in elasticity. One well-known
example is the so-called edge resonance, when the elastic energy is localized near
the edge of a semi-infinite stress-free plate (Shaw 1956; Roitberg et al. 1998).
Received 23 June 2004
Accepted 20 October 2004
1181
q 2005 The Royal Society
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1182
D. Gridin and others
2h
0
z
x
h
s
a
Figure 1. The geometry of the problem. The plate is infinite in the direction normal to the plane
of the figure.
The existence of trapped modes in elastic plates and bars with discontinuously
non-uniform cross-sections has also been demonstrated, both numerically and
experimentally (see Johnson et al. 1996 and references therein). In this paper we
consider the case of a stress-free elastic plate which has a bend (or bends) and
flattens out at infinity. We present convincing evidence (but provide no rigorous
proof) for the existence of trapped modes localized in the regions of maximal
curvature, and offer predictions of when and why such trapping occurs. To this
end, two methods are developed, one is asymptotic, which assumes smallness of
dimensionless curvature, and the other is numerical.
This paper is organized as follows. First, we develop our asymptotic method
and provide conditions on the problem parameters for which trapped modes
may exist. Then we describe briefly our numerical scheme, which provides
numerical evidence for the existence of trapped modes. The results of the
two methods are then compared. Finally, we demonstrate numerically the effect
of energy build-up in a bend when the structure is excited at a resonant
frequency.
2. Formulation
The geometry of the problem is shown is figure 1. We consider a curved plate of
constant thickness, 2h, which is made of a homogeneous and isotropic linearly
elastic material. The density of the solid is r and its Lamé constants are l and m.
The geometry is two-dimensional and an orthogonal curvilinear coordinate
system (s, h) is adopted, where h is the signed shortest distance from the
observation point to the centreline of the waveguide, Kh%h%h, and s is
the arc-length along the centerline. The shape of the plate is characterized by the
angle a between a tangent to the centreline and a fixed line (here we choose this
to be the x-axis). Thus, the curvature of the centreline is as; here, and
throughout, the paper subscripts denote partial derivatives with respect to the
corresponding variables. We assume that the curvature vanishes at infinity,
as(GN)Z0. In fact, the curvature should
Ð N decay faster than 1/s at infinity,
so that the full angle of the bend, a0 Z K
N as ds, is finite.
We study time-harmonic motion in the plate, and the common factor exp(Kiut),
where u in the angular frequency, will be considered understood, and is henceforth
suppressed. The two-dimensional displacement vector u can be represented in
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1183
Trapped modes in elastic plates
terms of two scalar potentials, f and j, that satisfy the Helmholtz equations
Df C kL2 f Z 0;
Dj C kT2 j Z 0;
(2.1)
where kLZu/cL and kTZu/cT are the bulk p
longitudinal
pffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and transverse
wavenumbers, with respective wavespeeds cL Z ðlC 2mÞ=r and cT Z m=r.
The Laplacian operator in our curvilinear coordinates is
D Z k2 v2ss C v2hh C k3 ass hvs K kas vh ;
k Z ð1 K as hÞK1 :
(2.2)
In terms of s and h, the displacement components along the s and h directions, u
and v, respectively, are given by
u Z kfs C jh ;
v Z fh K kjs ;
(2.3)
and for the stress tensor we have
t11 ¼ mðgK2 Df K 2fhh þ 2kjhs þ 2k2 as js Þ;
t12 ¼ mð2kfhs þ 2k2 as fs K Dj þ 2jhh Þ;
t22 ¼ m½ðgK2 K 2ÞDf þ 2fhh K 2kjhs K 2k2 as js ;
9
>
>
>
=
>
>
>
;
(2.4)
where 11, 12 and 22 correspond
to the ss, sh and hh components. Here, we
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
have introduced gZ cT =cL Z ð1K 2nÞ=2ð1K nÞ, where n is Poisson’s ratio.
Traction-free boundary conditions are assumed at the plate faces, so that we have
t12 ðs;GhÞ Z t22 ðs;GhÞ Z 0:
(2.5)
Let us now introduce dimensionless variables s and h by
s Z s=h;
h Z h=h;
(2.6)
and re-write equations (2.1) as
C g2 Lf Z 0;
Df
C Lj Z 0;
Dj
(2.7)
and the boundary conditions (2.5) as
C 2jhh Z 0;
2kfhs C 2k2 asfs K Dj
h ZG1;
C 2fhh K 2kjhs K 2k2 asjs;
ðgK2 K 2ÞDf
9
=
h ZG1; ;
(2.8)
2 Z ðuh=cT Þ2 is a square of the dimensionless
respectively. In (2.7) and (2.8), LZ u
frequency, and the Laplacian becomes
Z k2 v2ss C v2hh C k3 asshvs K kasvh;
D
k Z ð1 K ashÞK1 :
(2.9)
In this paper, we consider the following problem: we seek values of L such that
there exist non-trivial functions f and j that satisfy equations (2.7) and (2.8) and
the following decay condition at infinity:
f/ 0;
Proc. R. Soc. A (2005)
j/ 0 as
s/GN:
(2.10)
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1184
D. Gridin and others
Such a value of L corresponds to a trapped mode with frequency u, which we will
call a resonant frequency. Note that if we do not impose the decay condition (2.10),
then non-trivial f and j exist for any positive L and correspond to propagating
(Lamb) modes.
3. Asymptotic method
In this section, we develop an asymptotic method for obtaining approximations
to the trapped-mode solutions. To this end we introduce a slow variable xZes,
where e is a small dimensionless parameter, and assume that the angle a is a
smooth function of x. The small parameter e may be thought of as a ratio of the
half-thickness h to a typical radius of curvature, and the limiting case eZ0
corresponds to a flat plate.
x=h and h, equations (2.7) and (2.8) become
In terms of xZ
C g2 Lf Z 0;
Df
C Lj Z 0;
Dj
(3.1)
and
C 2jhh Z 0;
2ekfhx C 2e2 k2 axfx K Dj
h ZG1;
C 2fhh K 2ekjhx K 2ek2 axjx;
ðgK2 K 2ÞDf
(3.2)
h ZG1;
respectively, where we have
Z e2 k2 v2 C v2hh C e3 k3 axxhvx K ekaxvh;
D
xx
k Z ð1 K eaxhÞK1 :
(3.3)
Note that ax, axx, etc., are all of the order one. In fact, the displacements below
with all x derivatives of order one. This is because
will also be functions of x,
xZ Oð1Þ is a natural lengthscale of the problem, with all significant changes
taking place on it.
To devise our asymptotic scheme we need to assume an ansatz suitable for the
problem at hand. In the corresponding scalar (Helmholtz) problem with Dirichlet
boundary conditions (Duclos & Exner 1995; Gridin et al. 2004), the resonant
frequencies occur near the cut-off frequencies of a straight waveguide. This has a
physical explanation: the cut-off frequencies of a circular annulus are lower than
those of a straight layer, thus there are frequencies that correspond to
propagating modes in a curved section but are cut off in straight sections.
The same argument holds here, and we will return to it later in this section.
There are two types of cut-off frequencies in a flat elastic plate (e.g. Achenbach
1984), which we distinguish by L and T superscripts, given respectively by
Lm ¼
au
mp
;
2g
T
u
m ¼
mp
2
ðm ¼ 1; 2; 3; .Þ:
(3.4)
At the L frequencies, the wavefield is purely compressional and corresponds to
the transverse resonance of longitudinal waves with wavefronts parallel to the
plate faces. The T frequencies correspond to the shear resonance. In each case,
m is a number of half-oscillations of the wavefield across the guide.
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Trapped modes in elastic plates
1185
(a) Near L cut-off frequencies
Let us start with the near L cut-off case. The wavefield is predominantly
compressional, and we assume the following asymptotic expansions:
hÞ xfð0Þ ðx;
hÞ C efð1Þ ðx;
hÞ C e2 fð2Þ ðx;
hÞ C/;
fðx;
hÞ xejð1Þ ðx;
hÞ C e2 jð2Þ ðx;
hÞ C/;
jðx;
(3.5)
and
Lm Þ2 C eLL1 C e2 LL2 C/:
LL xðu
(3.6)
Note that it is unnecessary for us to assume the zeroth order term in the
expansion (3.6), as it would naturally emerge in the analysis below, but we do it
for reasons of physical clarity.
Substituting (3.5) and (3.6) into (3.1), and equating to zero the coefficients of
individual powers of e, we obtain a hierarchy of equations for f(i), j(i) and LLi .
Similarly, the boundary conditions (3.2) produce a hierarchy of boundary
conditions. These herarchies should be resolved from the lowest order up.
(i) Equations of motion and boundary conditions at e0
The e0 equation of the hierarchy is
mp 2
ð0Þ
afhh þ
fð0Þ ¼ 0;
2
(3.7)
subject to the boundary conditions
ð0Þ
2fhh
mp
K ð1 K 2g Þ
2g
2
2
fð0Þ ¼ 0;
h ¼G1:
(3.8)
Given (3.7), equation (3.8) is re-written as
fð0Þ Z 0;
h ZG1:
(3.9)
The problems (3.7) and (3.9) have two families of solutions, one of which is
symmetric (even) with respect to the centreline hZ 0, and the other is
antisymmetric (odd). The symmetric solution is
ð2n K 1Þp
h
fð0;sÞ ¼ f ð0Þ ðxÞcos
;
2
(3.10)
with
ðL;sÞ
L2nK1 Þ2 C eL1
LðL;sÞ xðu
ðL;sÞ
C e 2 L2
C/ ðn Z 1; 2; 3; .Þ;
(3.11)
and the antisymmetric solution is
fð0;aÞ ¼ f ð0Þ ðxÞsin
np
h;
Proc. R. Soc. A (2005)
(3.12)
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D. Gridin and others
with
ðL;aÞ
L2n Þ2 C eL1
LðL;aÞ xðu
ðL;aÞ
C e2 L2
C/ ðn Z 1; 2; 3; .Þ:
(3.13)
Here, f (0) is unknown and in fact our aim is to find it, or at least an equation for
it, as f (0) governs the longitudinal behaviour of the solution.
(ii) Equations of motion and boundary conditions at e1
Let us proceed with the antisymmetric case; final formulae will be given for
both cases. The hierarchy at order ei for each i R 1 contains equations for both
f(i) and j(i), but they conveniently split into two uncoupled problems. At e1, the
equation for f(1) is
ð1Þ
hð0Þ K g2 LðL;aÞ fð0Þ ;
fhh þ ðnpÞ2 fð1Þ ¼ axf
1
subject to the boundary conditions
2
np
ð1Þ
ðL;aÞ
2
fð1Þ ¼ ð1 K 2g2 ÞL1 fð0Þ ;
2fhh K ð1 K 2g Þ
g
h ¼G1:
(3.14)
(3.15)
Substituting the general solution of (3.14), which is
1 ð0Þ fð1Þ ¼ f ð1Þ ðxÞsin
np
h þ gð1Þ ðxÞcos
np
h þ axf
ðxÞ
h sin np
h
2
þ
g2 ðL;aÞ ð0Þ f ðxÞ
h cos np
h;
L
2np 1
(3.16)
ðL;aÞ
into the boundary conditions (3.15), we find that L1
Z 0 and
2
¼ 2g axf
ð0Þ ðxÞ;
gð1Þ ðxÞ
np
(3.17)
so that f(1) becomes
2g2 ð0Þ 1 ð0Þ axf ðxÞcos np
fð1Þ ¼ f ð1Þ ðxÞsin
np
hþ
h þ axf
ðxÞ
h sin np
h;
2
np
and f ð1Þ ðxÞ
are still unknown.
where both f ð0Þ ðxÞ
(3.18)
The equation for j(1) is
ð1Þ
jhh
np
þ
g
2
jð1Þ ¼ 0;
subject to the boundary conditions
2
np
ð1Þ
ð0Þ
jð1Þ ¼ K2fx
2jhh þ
h ;
g
Proc. R. Soc. A (2005)
(3.19)
h ¼G1;
(3.20)
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1187
Trapped modes in elastic plates
for which the solution is
jð1Þ ¼
2g2 ðK1Þn
np
ð0Þ
h:
f x cos
g
np cosðnp=gÞ
(3.21)
(iii) Equations of motion and boundary conditions at e2
The equation for f(2) is
ð2Þ
ð0Þ
2
ðL;aÞ ð0Þ
hð1Þ þ ax hf
fhh þ ðnpÞ2 fð2Þ ¼ axf
hð0Þ K fxx K g2 L2
subject to the boundary conditions
2
np
ð2Þ
ð1Þ
ðL;aÞ ð0Þ
2
2
2fhh K ð1 K 2g Þ
fð2Þ ¼ 2jx
f ;
h þ ð1 K 2g ÞL2
g
f ;
h ¼G1:
(3.22)
(3.23)
Substituting the general solution of (3.22) into the boundary conditions (3.23),
we obtain the following ordinary differential equation (ODE) for the unknown
function f ð0Þ ðxÞ:
2 ð0Þ
1
ðL;aÞ
ðL;aÞ d f
Cn
C 4 K 2 a2xf ð0Þ Z L2 f ð0Þ ;
(3.24)
2
4g
dx
where we use notation
CnðL;aÞ ¼
8g tanðnp=gÞ
1
K 2:
np
g
(3.25)
The ODE (3.24) together with the decay condition
0;
f ð0Þ ðxÞ/
(0)
x/G
N
(3.26)
ðL;aÞ
L2 ,
constitutes an eigenproblem for f and
which, in general, has to be solved
numerically. The basic physics near cut-off has been distilled, via the asymptotic
procedure, into this differential eigenvalue problem. Once it is solved, the
approximation of our original eigenvalue L(L,a) is
ðL;aÞ
L2n Þ2 C e2 L2
LðL;aÞ xðu
C Oðe3 Þ;
(3.27)
and the eigenfunctions are
np
h þ Oð3Þ;
fðaÞ xf ð0Þ ðxÞsin
jðaÞ xOð3Þ:
(3.28)
The analysis for the symmetric case is similar, and the corresponding ODE is
2 ð0Þ
1
ðL;sÞ
ðL;sÞ d f
Cn
C 4 K 2 a2xf ð0Þ Z L2 f ð0Þ ;
(3.29)
2
4g
dx
where we use notation
CnðL;sÞ ¼ K
Proc. R. Soc. A (2005)
16g cot½ð2n K 1Þp=ð2gÞ
1
K 2;
ð2n K 1Þp
g
(3.30)
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1188
D. Gridin and others
and the approximations of the eigenvalue and eigenfunctions are
ðL;sÞ
L2nK1 Þ2 C e2 L2
LðL;sÞ xðu
C Oðe3 Þ
(3.31)
and
ð2n K 1Þp
h
þ Oð3Þ;
fðsÞ xf ð0Þ ðxÞcos
2
jðsÞ xOð3Þ;
(3.32)
respectively.
(b) Near T cut-off frequencies
Now the wavefield is predominantly shear, and we assume the following
asymptotic expansions:
)
hÞ x3fð1Þ ðx;
hÞ þ 32 fð2Þ ðx;
hÞ þ /;
fðx;
(3.33)
hÞ xjð0Þ ðx;
hÞ þ 3jð1Þ ðx;
hÞ þ 32 jð2Þ ðx;
hÞ þ /;
jðx;
and
2
T
2 T
T
LT xðu
m Þ C eL1 C e L2 C/
(3.34)
The analysis is similar to the longitudinal case; the following two ODEs result
for the symmetric and antisymmetric leading components, respectively:
CnðT;sÞ
d2 f ð0Þ 15 2 ð0Þ
C axf
2
4
dx
ðT;sÞ ð0Þ
f
Z L2
ðT;sÞ
2
2
T
with LðT;sÞ xðu
2n Þ C e L2
C Oðe3 Þ
(3.35)
and with eigenfunctions
fðsÞ xOð3Þ;
jðsÞ xf ð0Þ ðxÞsin
np
h þ Oð3Þ:
(3.36)
and
CnðT;aÞ
d2 f ð0Þ 15 2 ð0Þ
C axf
2
4
dx
ðT;aÞ ð0Þ
Z L2
f
ðT;aÞ
2
2
T
with LðT;aÞ xðu
2nK1 Þ C e L2
C Oðe3 Þ
(3.37)
and with eigenfunctions
fðaÞ xOð3Þ;
ðT;sÞ
The coefficients Cn
CnðT;sÞ ¼
jðaÞ xf ð0Þ ðxÞcos
8g tanðgnpÞ
K 1;
np
Proc. R. Soc. A (2005)
ðT;aÞ
and Cn
ð2n K 1Þp
h
þ Oð3Þ:
2
(3.38)
are given by
CnðT;aÞ ¼ K
16g cot½gð2n K 1Þp=2
K 1:
ð2n K 1Þp
(3.39)
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Trapped modes in elastic plates
1189
(c) Existence of trapped modes
Given equations (3.24), (3.29), (3.35) and (3.37) for f (0), we can now
investigate the possibility of trapping in more depth. We aim to find conditions
on the problem parameters that will allow for the existence of trapped modes.
Let us consider one case in detail, for instance, the symmetric, near
longitudinal cut-off case, equation (3.29), and first assume that gO1/4; this
condition holds for most naturally occurring materials. Then, the coefficient of
ðL;sÞ
the second term in the left-hand side (LHS) of (3.29) is positive. Now, if Cn
is
negative, then the operator in the LHS of (3.29) acting on f (0) is positive. This
implies that there are no negative eigenvalues and therefore no eigenfunctions
that decay at infinity (since the second term in the LHS vanishes as x/G
N
ðL;sÞ
leaving on ODE with oscillatory solutions). If, however, Cn is positive, that is,
CnðL;sÞ ¼ K
16g cot½ð2n K 1Þp=ð2gÞ
1
K 2 O 0;
ð2n K 1Þp
g
(3.40)
then a negative eigenvalue may exist, and consequently, there is a possibility
of existence of trapped modes in the original problem.
The following interesting connection exists between (3.40) and the sign of
L2nK1 . Let us consider the
group velocity near the flat-plate cut-off frequencies u
symmetric Rayleigh–Lamb equation for a flat plate,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 K k 2 u
2 K k 2 Þ
2 K k 2
tanðu
4k 2 g2 u
C
Z0
(3.41)
2
2
2
2
Kk Þ
tanðg u
2 K 2k 2 Þ
ðu
(e.g. Achenbach 1984, §6.7), where k is a dimensionless wavenumber of a
Lamb mode. If we consider the long-wavelength regime, k/1, and use
Taylor expansions in (3.41), we obtain that
ðL;sÞ
U wK
Cn
k2;
L2nK1
2u
(3.42)
u
L2nK1 is the difference between (dimensionless) frequency and the
where UZ uK
cut-off frequency. Equation (3.42) is the usual square dependence near cut-off.
Thus, the group velocity vgZvU/vk near cut-off is
ðL;sÞ
vg wK
ðL;sÞ
Cn
k:
L2nK1
u
(3.43)
If Cn is negative then the corresponding dispersion curve is ‘regular’, that is,
ðL;sÞ
it has a positive group velocity. If Cn
is positive then the mode is ‘backward’,
that is, its group velocity is negative. Thus, the condition (3.40) for the
possibility of existence of trapped modes is equivalent to requiring the negativity
L2nK1 .
of group velocity near the flat-plate cut-off frequency u
When g!1/4, which is not practically interesting, the situation reverses,
L2nK1 is
and the condition allowing the existence of trapped modes near u
ðL;sÞ
Cn ! 0, that is, it requires the group velocity to be positive. The other ODEs
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1190
D. Gridin and others
0.4
k
S1
S2
0
2.7
3.0
3.3
Figure 2. Dispersion curves for the fourth and fifth modes in a circular annulus (solid lines) and a
flat plate (dashed lines).
Table 1. Conditions allowing the existence of trapped modes.
cut-off frequency
L2nK1 ¼ ð2nK 1Þp=2g
u
L2n ¼ np=g
au
T
u
2n ¼ np
T
u
2nK1 ¼ ð2nK 1Þp=2
gO1/4
ðL;sÞ
Cn
O 0 (vg!0)
ðL;aÞ
Cn O 0
ðT;sÞ
Cn O 0
ðT;aÞ
Cn O 0
(vg!0)
(vg!0)
(vg!0)
g!1/4
ðL;sÞ
Cn
! 0 (vgO0)
ðL;aÞ
Cn ! 0
ðT;sÞ
Cn O 0
ðT;aÞ
Cn O 0
(vgO0)
(vg!0)
(vg!0)
can be analysed similarly; all the conditions are summarized in table 1, with the
coefficients given by equations (3.25), (3.30) and (3.39).
Let us now exemplify the above analysis and provide a physical explanation
for the existence of trapped modes. In figure 2, we choose the wavespeeds of steel,
cLZ5960 and cTZ3260 m sK1, so that gz0.547, and plot the fourth and fifth
modes for a flat plate (dashed line) and for a circular annulus with a centreline of
radius RZ5h (solid lines). For a flat plate these curves are solutions of the
symmetric Rayleigh–Lamb equations (3.41)—they are the so-called S1 and S2
modes, respectively. Dispersion relations for a circular annulus can be found in,
for instance, Liu & Qu (1998) or Gridin et al. (2003). The S1 cut-off frequency is
L1 Z p=ð2gÞ z2:87, and the S2 cut-off frequency is u
T
u
2 Z p. For S1, we find from
ðL;sÞ
(3.30) that C1 O 0 and indeed the group velocity is negative (see figure 2),
ðT;sÞ
while for S2, we obtain C1 ! 0 and the group velocity is positive. The cut-off
frequencies for the fourth and fifth annular modes are shifted to the right, that is
T
L1 and u
they are larger than u
2 , respectively; this also follows equations (3.29)
and (3.35) with ax Z 1 (constant curvature). We argue that a trapped mode may
occur near the S1 cut-off frequency, since there is a range of frequencies at which
there are propagating modes for the annulus but not for the flat plate, and thus
the trapping of energy in curved sections of our structure is possible. For the
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Trapped modes in elastic plates
1191
‘forward’ S2 mode there is no such range of frequencies, and no trapping is
possible.
(d) A note of the near cut-off propagation in weakly curved plates
At this point we would like to mention that the asymptotic scheme developed
in this section is also applicable for a near cut-off or a long-wavelength
propagation regime in weakly curved plates. In Gridin & Craster (2004),
asymptotic expressions have been derived for propagating quasi-modes in an
arbitrarily curved elastic plate. However, it was implicitly assumed there that
the frequency was not very close to a cut-off frequency, and the final expression
(3.37) of that article breaks down if it is. This is because the ansatz used in
Gridin & Craster (2004) is not suitable for the long-wavelength regime. Instead,
one should use the asymptotic theory developed in this section. For instance,
the ODE (3.24) becomes
2 ð0Þ
1
ðL;aÞ d f
2
K2 2
L 2
K ðu
2n Þ f ð0Þ Z 0;
Cn
C 4 K 2 ax K e ½u
(3.44)
2
4g
dx
and the decay conditions at infinity should be replaced by some suitable initial/
radiation conditions, since we are no longer solving an eigenvalue problem.
A similar description for curved plates in three dimensions, by using a slightly
different asymptotic method, is given in Kaplunov et al. (1998) and the references
therein; these authors have made much use of the long wave theory in this
context.
4. Numerical method
In this section, a general numerical scheme for solving the eigenproblem
(2.7), (2.8) and (2.10) is briefly described. The method is totally free of the smallcurvature limitation and provides a tool for the verification of the asymptotics
derived above as well as for investigating the problem for a wider range of
parameters.
We base our scheme on spectral methods which are particularly suitable for
solving eigenvalue problems in rectangular domains with smoothly varying
parameters. Readers unfamiliar with these methods can find good introductions
in several recent monographs (Fornberg 1995; Trefethen 2000; Boyd 2001).
The spectral approach reduces the partial differential eigenproblem to a matrix
eigenproblem, which is then solved using a standard algorithm. This has a
computational cost attached which we seek to minimize by using a numerical
scheme that accurately represents the differential operators with as few grid
points as possible.
For simplicity, let us assume the symmetry aðK
sÞZKað
sÞ, that is, the plate is
symmetric with respect to the z-axis (see figure 1). We now solve the problem
for a half-plate 0% s!N, with suitable boundary conditions at sZ 0, and build
the full solution from symmetry considerations. Since the plate is semi-infinite
along the s direction, it is convenient to use a Laguerre method (with M grid
points) in this direction, which automatically incorporates the decay condition at
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D. Gridin and others
infinity. Moreover, the Laguerre grid points cluster near the maximal curvature
at sZ 0, where they are required for accurate resolution of the curvature function
as. A Chebyshev method (with N grid points) is used in the h direction, and a
demonstration of incorporating the stress-free boundary conditions at hZG1
can be found in, for instance, Adamou & Craster (2004), and we use their
approach here.
There are two possible sets of symmetry conditions:
fðK
s; hÞ Z fð
s; hÞ;
jðK
s; hÞ Z Kjð
s; hÞ;
(4.1)
jðK
s; hÞ Z jð
s; hÞ;
(4.2)
and
fðK
s; hÞ Z Kfð
s; hÞ;
These lead to two respective sets of boundary conditions at sZ 0:
s Z 0; hÞ Z 0;
fsð
jð
s Z 0; hÞ Z 0;
(4.3)
fð
s Z 0; hÞ Z 0;
s Z 0; hÞ Z 0:
jsð
(4.4)
and
Solving two eigenproblems for a semi-infinite plate using either (4.3) or (4.4),
we can then build the solution for the whole plate using (4.1) and (4.2),
respectively. Note that the general case, when there is no symmetry about sZ 0,
can also be treated by using, say, a sinc or Hermite spectral method instead of
Laguerre, but that requires more grid points; we leave this tangential numerical
issue aside for the time being.
5. Numerical results and discussions
We have carried out numerical tests for various values of parameters and
curvature functions. Typical results are given below. We take the following angle
function a:
að
sÞ Z
a0
tanhðe
sÞ;
2
(5.1)
so that a0 is the full angle of the bend. The slowness parameter e can be varied,
so we can make the dimensionless curvature,
sÞ Z
asð
ea0
1
;
2 cosh2 ðe
sÞ
(5.2)
small if desired and then compare with the results from the asymptotics.
The curvature has a maximum at sZ 0 and decays exponentially at infinity.
For the numerical computations shown here, we choose gz0.547, corresponding to the wavespeeds of steel, cLZ5960 and cTZ3260 m sK1, and the right-angle
bend a0Zp/2. The slowness parameter e is chosen to be 0.25, so that the
maximal curvature is approximately 0.2. In the numerical scheme, the numbers
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Trapped modes in elastic plates
y/h
20
18
16
14
12
10
8
6
4
2
0
–20
–15
–10
–5
0
x/h
5
10
15
20
Figure 3. Elastic energy density of the lowest trapped mode. Lighter shading corresponds to
higher energy.
of grid points are MZ80 and NZ20; the accuracy of the results have been
verified by increasing these numbers.
For the parameters chosen, the lowest dimensionless resonant frequency as
found using the numerical scheme on the full partial differential equation (PDE)
0 z2:8741. It is close to u
L1 ¼ p=ð2gÞ and corresponds to the longitudinal
is u
ðL;sÞ
symmetric case, when we have C1 O 0. The elastic energy density distribution
for this resonant frequency is shown in figure 3; one can clearly see the energy
localization at the bend.
Solving the appropriate ODE (3.29) derived using the asymptotic scheme,
0 z2:8755.
we find an approximate value of the resonance frequency as u
This gives a relative error of just 0.05%, which, particularly considering the
relatively large value of e chosen, is very accurate. To solve the ODE (3.29),
which must also be undertaken numerically, we once more utilize spectral
methods, namely a Laguerre method, although only in s this time; it is
computationally efficient since we are now dealing with an ODE. In figure 4, the
displacement n at the inner side of the plate, hZ 1, is plotted and exhibits
exponential decay at infinity. The agreement between the exact and asymptotic
solutions is quite good; the asymptotic code is about 10 000 times faster than the
numerical solution of the full PDE.
To further verify and investigate the trapping of modes we consider timedependent problems. In figure 5a,b, we demonstrate effects owing to the
excitation of our structure by a source acting at a resonant and a non-resonant
frequency, respectively. The material and geometrical parameters are chosen as
above. However, since we are now solving a propagation and not an eigenvalue
problem, the structure is truncated at sZG30. The problem is time-dependent,
so that instead of (2.7) we have
g2 ftt Z Df;
jtt Z Dj;
(5.3)
where the dimensionless time t is t Z t=t0 , with t0Zh/cT. The plate face hZK1 is
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D. Gridin and others
v (arb. units)
1
0
−200 −150 −100 −50
0
s/h
50
100
150
200
Figure 4. Displacement v at hZ 1 as calculated using the exact numerics (solid line) and the
asymptotic approximation (dashed line).
assumed to be traction-free, that is,
t12 ð
s; K1Þ Z t22 ð
s; K1Þ Z 0;
(5.4)
while on the face hZ 1 a transducer with a Gaussian distribution of normal
stresses centred at sZ 10 is applied. We model the transducer by
t12 ð
s; 1Þ Z 0;
t22 ð
s; 1Þ Z Fðt Þexp½K2ð
s K 10Þ2 ;
where FðtÞ is given by
8
0;
>
>
<
t Þ;
Fðt Þ ¼ 0:5½1 K cosð10pt Þcosðu
>
>
:
t Þ;
cosðu
(5.5)
if t % 0;
if 0! t ! 0:1;
(5.6)
if t R 0:1:
The function FðtÞ is chosen so that the stresses and their derivatives are not
discontinous as t Z 0 and *+*+t Z 0:1*+; and after -, the structure is excited at
a single frequency *.
In our exact numerical scheme, spectral methods are used again to
approximate the spatial derivatives. More specifically, we use a Chebyshev
method in the h direction (with NZ30 grid points), and a Fourier method in the
s direction (with MZ200 grid points). The Fourier method is chosen because
it automatically incorporates periodicity conditions at sZG30, which allows us
to avoid problems with numerical instabilities at the corners of the computational domain that may occur if the ends were stress-free too; such edges would
themselves possibly support localized edge modes, and that too is undesirable.
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Trapped modes in elastic plates
(a)
20
y/h
15
10
5
0
20
(b)
y/h
15
10
5
0
–20
–15
–10
–5
0
x/h
5
10
15
20
Figure 5. Excitation at resonant (a) and a non-resonant (b) frequency.
Physically, the periodicity conditions mean that whatever leaves at one end of
the plate re-enters at the other, so that no energy leaves the structure and elastic
waves constantly move through the structure allowing energy to localize if such
localization is to occur; the leap-frog formula is used to approximate the timederivatives.
In figure 5a, the energy density is shown at t z439, which corresponds to
approximately 200 cycles of cosine in Fðt Þ. Since the structure is excited at the
0 , the energy build-up at the bend is clearly seen owing to
resonant frequency u
excitation of the trapped mode. This figure is very similar to figure 3 obtained for
the eigenproblem and provides alternative numerical evidence for the existence
of trapped modes. In figure 5b, the transducer operates at a non-resonant
2:5, and the energy distribution at the same time is almost
frequency uZ
uniform throughout the plate.
6. Concluding remarks
We have demonstrated the existence of trapped modes in elastic plates with
bends; these modes are shown to localize their energy in the regions of maximal
curvature and decay exponentially at infinity. We have provided no proof of their
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D. Gridin and others
existence, but provide convincing evidence by using two methods: a direct
numerical scheme and an asymptotic method. Simple existence conditions
depending on Poisson’s ratio have been offered, and these provide guidance as to
when the trapping occurs; we also describe physically why trapped modes should
be expected.
Both the asymptotic and the numerical methods developed in this paper can
be extended to deal with many other geometries, such as, for example, plates of
varying width, where trapped modes may also exist. Interestingly such trapping
of modes does appear to occur in such situations; we have become aware of the
ongoing work by Julius Kaplunov and Graham Rogerson on localized vibrations
in straight plates of non-uniform width with mixed stress-free/clamped boundary
conditions, where they develop an asymptotic method not dissimilar to ours and
trapping also occurs. This suggests that trapped modes are a common feature of
many elastic guiding problems and that practitioners should be aware of this
possibility and of the capability to predict, and thereby avoid, their occurrence.
Three-dimensional elastic structures, such as curved bars and pipes, and plates
curved in two dimensions (see Duclos et al. 2001 for a quantum-mechanical
counterpart), also present significant interest and challenges, and will be the
focus of our future work.
We are grateful to the Engineering and Physical Sciences Research Council (EPSRC), UK for their
financial support via grant no. GR/R32032/01. We also thank Peter Cawley, Mike Lowe and
Jimmy Fong from the Non-Destructive Testing Group at Imperial College London for their interest
in and encouragement with this work.
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