## Abstract

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.

## 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). 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, 2*h*, which is made of a homogeneous and isotropic linearly elastic material. The density of the solid is *ρ* and its Lamé constants are *λ* and *μ*. The geometry is two-dimensional and an orthogonal curvilinear coordinate system (*σ*, *η*) is adopted, where *η* is the signed shortest distance from the observation point to the centreline of the waveguide, −*h*≤*η*≤*h*, and *σ* is the arc-length along the centerline. The shape of the plate is characterized by the angle *α* 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 *α*_{σ}; here, and throughout, the paper subscripts denote partial derivatives with respect to the corresponding variables. We assume that the curvature vanishes at infinity, *α*_{σ}(±∞)=0. In fact, the curvature should decay faster than 1/*σ* at infinity, so that the full angle of the bend, , is finite.

We study time-harmonic motion in the plate, and the common factor exp(−i*ωt*), where *ω* in the angular frequency, will be considered understood, and is henceforth suppressed. The two-dimensional displacement vector *u* can be represented in terms of two scalar potentials, *ϕ* and *ψ*, that satisfy the Helmholtz equations(2.1)where *k*_{L}=*ω*/*c*_{L} and *k*_{T}=*ω*/*c*_{T} are the bulk longitudinal and transverse wavenumbers, with respective wavespeeds and . The Laplacian operator in our curvilinear coordinates is(2.2)

In terms of *σ* and *η*, the displacement components along the *σ* and *η* directions, *u* and *v*, respectively, are given by(2.3)and for the stress tensor we have(2.4)where 11, 12 and 22 correspond to the *σσ*, *ση* and *ηη* components. Here, we have introduced , where *ν* is Poisson's ratio. Traction-free boundary conditions are assumed at the plate faces, so that we have(2.5)

Let us now introduce dimensionless variables and by(2.6)and re-write equations (2.1) as(2.7)and the boundary conditions (2.5) as(2.8)respectively. In (2.7) and (2.8), is a square of the dimensionless frequency, and the Laplacian becomes(2.9)

In this paper, we consider the following problem: we seek values of *Λ* such that there exist non-trivial functions *ϕ* and *ψ* that satisfy equations (2.7) and (2.8) and the following decay condition at infinity:(2.10)

Such a value of *Λ* corresponds to a trapped mode with frequency *ω*, which we will call a resonant frequency. Note that if we do not impose the decay condition (2.10), then non-trivial *ϕ* and *ψ* exist for any positive *Λ* 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 *ξ*=*ϵσ*, where *ϵ* is a small dimensionless parameter, and assume that the angle *α* is a smooth function of *ξ*. The small parameter *ϵ* may be thought of as a ratio of the half-thickness *h* to a typical radius of curvature, and the limiting case *ϵ*=0 corresponds to a flat plate.

In terms of and , equations (2.7) and (2.8) become(3.1)and(3.2)respectively, where we have(3.3)

Note that , , etc., are all of the order one. In fact, the displacements below will also be functions of , with all derivatives of order one. This is because 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(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.

### (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:(3.5)and(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 *ϵ*, we obtain a hierarchy of equations for *ϕ*^{(i)}, *ψ*^{(i)} and . 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 *ϵ*^{0}

The *ϵ*^{0} equation of the hierarchy is(3.7)subject to the boundary conditions(3.8)

Given (3.7), equation (3.8) is re-written as(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 , and the other is antisymmetric (odd). The symmetric solution is(3.10)with(3.11)and the antisymmetric solution is(3.12)with(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 *ϵ*^{1}

Let us proceed with the antisymmetric case; final formulae will be given for both cases. The hierarchy at order *ϵ*^{i} for each *i* ≥ 1 contains equations for both *ϕ*^{(i)} and *ψ*^{(i)}, but they conveniently split into two uncoupled problems. At *ϵ*^{1}, the equation for *ϕ*^{(1)} is(3.14)subject to the boundary conditions(3.15)

Substituting the general solution of (3.14), which is(3.16)into the boundary conditions (3.15), we find that and(3.17)so that *ϕ*^{(1)} becomes(3.18)where both and are still unknown.

The equation for *ψ*^{(1)} is(3.19)subject to the boundary conditions(3.20)for which the solution is(3.21)

#### (iii) Equations of motion and boundary conditions at *ϵ*^{2}

The equation for *ϕ*^{(2)} is(3.22)subject to the boundary conditions(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 :(3.24)where we use notation(3.25)

The ODE (3.24) together with the decay condition(3.26)constitutes an eigenproblem for *f*^{(0)} 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 *Λ*^{(L,a)} is(3.27)and the eigenfunctions are(3.28)

The analysis for the symmetric case is similar, and the corresponding ODE is(3.29)where we use notation(3.30)and the approximations of the eigenvalue and eigenfunctions are(3.31)and(3.32)respectively.

### (b) Near T cut-off frequencies

Now the wavefield is predominantly shear, and we assume the following asymptotic expansions:(3.33)and(3.34)

The analysis is similar to the longitudinal case; the following two ODEs result for the symmetric and antisymmetric leading components, respectively:(3.35)and with eigenfunctions(3.36)and(3.37)and with eigenfunctions(3.38)

The coefficients and are given by(3.39)

### (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 *γ*>1/4; this condition holds for most naturally occurring materials. Then, the coefficient of the second term in the left-hand side (LHS) of (3.29) is positive. Now, if 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 leaving on ODE with oscillatory solutions). If, however, is positive, that is,(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 group velocity near the flat-plate cut-off frequencies . Let us consider the symmetric Rayleigh–Lamb equation for a flat plate,(3.41)(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(3.42)where is the difference between (dimensionless) frequency and the cut-off frequency. Equation (3.42) is the usual square dependence near cut-off. Thus, the group velocity *v*_{g}=∂*Ω*/∂*k* near cut-off is(3.43)

If is negative then the corresponding dispersion curve is ‘regular’, that is, it has a positive group velocity. If 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 of group velocity near the flat-plate cut-off frequency .

When *γ*<1/4, which is not practically interesting, the situation reverses, and the condition allowing the existence of trapped modes near is , that is, it requires the group velocity to be positive. The other ODEs 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, *c*_{L}=5960 and *c*_{T}=3260 m s^{−1}, so that *γ*≈0.547, and plot the fourth and fifth modes for a flat plate (dashed line) and for a circular annulus with a centreline of radius *R*=5*h* (solid lines). For a flat plate these curves are solutions of the symmetric Rayleigh–Lamb equations (3.41)—they are the so-called *S*_{1} and *S*_{2} modes, respectively. Dispersion relations for a circular annulus can be found in, for instance, Liu & Qu (1998) or Gridin *et al*. (2003). The *S*_{1} cut-off frequency is , and the *S*_{2} cut-off frequency is . For *S*_{1}, we find from (3.30) that and indeed the group velocity is negative (see figure 2), while for *S*_{2}, we obtain and the group velocity is positive. The cut-off frequencies for the fourth and fifth annular modes are shifted to the right, that is they are larger than and , respectively; this also follows equations (3.29) and (3.35) with (constant curvature). We argue that a trapped mode may occur near the *S*_{1} 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 ‘forward’ *S*_{2} 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(3.44)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 small-curvature 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 , that is, the plate is symmetric with respect to the *z*-axis (see figure 1). We now solve the problem for a half-plate , with suitable boundary conditions at , and build the full solution from symmetry considerations. Since the plate is semi-infinite along the direction, it is convenient to use a Laguerre method (with *M* grid points) in this direction, which automatically incorporates the decay condition at infinity. Moreover, the Laguerre grid points cluster near the maximal curvature at , where they are required for accurate resolution of the curvature function . A Chebyshev method (with *N* grid points) is used in the direction, and a demonstration of incorporating the stress-free boundary conditions at can be found in, for instance, Adamou & Craster (2004), and we use their approach here.

There are two possible sets of symmetry conditions:(4.1)and(4.2)

These lead to two respective sets of boundary conditions at :(4.3)and(4.4)

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 , 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 *α*:(5.1)so that *α*_{0} is the full angle of the bend. The slowness parameter *ϵ* can be varied, so we can make the dimensionless curvature,(5.2)small if desired and then compare with the results from the asymptotics. The curvature has a maximum at and decays exponentially at infinity.

For the numerical computations shown here, we choose *γ*≈0.547, corresponding to the wavespeeds of steel, *c*_{L}=5960 and *c*_{T}=3260 m s^{−1}, and the right-angle bend *α*_{0}=π/2. The slowness parameter *ϵ* is chosen to be 0.25, so that the maximal curvature is approximately 0.2. In the numerical scheme, the numbers of grid points are *M*=80 and *N*=20; 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) is . It is close to and corresponds to the longitudinal symmetric case, when we have . 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, we find an approximate value of the resonance frequency as . This gives a relative error of just 0.05%, which, particularly considering the relatively large value of *ϵ* 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 *σ* this time; it is computationally efficient since we are now dealing with an ODE. In figure 4, the displacement *ν* at the inner side of the plate, , 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 time-dependent problems. In figure 5*a*,*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 . The problem is time-dependent, so that instead of (2.7) we have(5.3)where the dimensionless time is , with *t*_{0}=*h*/*c*_{T}. The plate face is assumed to be traction-free, that is,(5.4)while on the face a transducer with a Gaussian distribution of normal stresses centred at is applied. We model the transducer by(5.5)where is given by(5.6)The function is chosen so that the stresses and their derivatives are not discontinous as and ; 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 direction (with *N*=30 grid points), and a Fourier method in the direction (with *M*=200 grid points). The Fourier method is chosen because it automatically incorporates periodicity conditions at , 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. 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 time-derivatives.

In figure 5*a*, the energy density is shown at , which corresponds to approximately 200 cycles of cosine in . Since the structure is excited at the resonant frequency , the energy build-up at the bend is clearly seen owing to 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 5*b*, the transducer operates at a non-resonant frequency , and the energy distribution at the same time is almost 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 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.

## Acknowledgments

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.

## Footnotes

- Received June 23, 2004.
- Accepted October 20, 2004.

- © 2005 The Royal Society