## Abstract

This paper is concerned with elucidation of the general properties of the bending edge wave in a thin linearly elastic plate that is supported by a Winkler foundation. A homogeneous wave of arbitrary profile is considered, and represented in terms of a single harmonic function. This serves as the basis for derivation of an explicit asymptotic model, containing an elliptic equation governing the decay away from the edge, together with a parabolic equation at the edge, corresponding to beam-like behaviour. The model extracts the contribution of the edge wave from the overall dynamic response of the plate, providing significant simplification for analysis of the localized near-edge wave field.

## 1. Introduction

It is well known that the bending edge wave localized near the edge of a thin elastic plate is dispersive in contrast to the classical Rayleigh wave on an elastic half-space. The bending edge wave has been studied for more than half a century, and has a remarkable history. Konenkov [1] was the first to consider such a wave on a Kirchhoff plate and derive the associated dispersion relation. However, some initial ideas giving preliminary insights within the framework of stability of elastic plates, had in fact appeared in the earlier work of Ishlinsky [2]. Unfortunately, the contribution of Konenkov [1] was seemingly unnoted; indeed, after 14 years, the bending edge wave was rediscovered independently in [3,4]. Some further historical details, and periodic overviews of the state of art, may be found in [5] and in the review [6]. Among other recent contributions on the subject, we mention [7–10].

It is clear that edge waves are dispersive analogues of better known surface waves. However, the number of contributions investigating the general properties of the bending edge wave is considerably less. Construction of an appropriate mathematical theory for the Rayleigh wave resulted in the general representation of the wave field in terms of harmonic functions, originated by Friedlander [11] and followed by Chadwick [12]. Recent developments of this approach, including generalizations to anisotropy, laterally dependent surface waves and three-dimensional surface and interfacial waves of arbitrary profile and direction, may be found in [13–18]. Alternative approaches to the general description of surface waves were presented in [19,20].

A step forward in mathematical modelling of the Rayleigh wave was provided by Kaplunov *et al.* [21]. In this study, a slow time perturbation of the solution previously obtained by Chadwick [12] allowed treatment of non-homogeneous transient boundary conditions. The derived explicit model for the Rayleigh wave uncouples the contribution of the surface wave from the overall dynamic response. In addition to simple approximate formulations for the wave field, some fundamental features were noted, for example the dual hyperbolic–elliptic nature of the Rayleigh wave. Similar observations have been made in [16] for Schölte waves of arbitrary profile and direction.

The approach of [21] has been extended to the bending edge wave on a free Kirchhoff plate, with some results presented in [22], revealing a parabolic–elliptic formulation for the wave field. Other recent developments in the area of edge waves in elastic plates are related to plates supported by elastic foundations, see [9] studying edge wave on a plate resting on a Winkler foundation, and also [10], extending the results of [9] to more sophisticated two-parametric foundations.

In this paper, we extend the previous results to the bending edge wave on a plate supported by a Winkler foundation, modelling a deformable half-space as a set of elastic springs. Our aim is twofold, including establishment of the general time-dependent solution as well as development of a specialized formulation for non-homogeneous edge boundary conditions.

The key to construction of the bending edge wave eigensolution of arbitrary profile is the natural, though not readily straightforward, idea of an effective beam on an elastic foundation acting as a ‘basic object’ for the wave. Some physically intuitive reasoning for this kind of assumption, presented in [22], is clarified in this paper. In particular, it is shown that the promoted assumption of the beam-like behaviour allows construction of the wave field in terms of an arbitrary single harmonic function, actually generalizing the known bending edge wave of a sinusoidal profile.

The derived eigensolution of arbitrary shape is then perturbed in slow time, providing a path to an explicit model for the bending edge wave, approximating the wave field. The results contain a parabolic beam-like equation on the edge and an elliptic equation over the interior, governing decay away from the edge. Thus, the solution can be found in the form of a plane harmonic function in spatial coordinates, satisfying the parabolic equation on the edge.

The model is first derived for a particular type of loading evolving in slow time and corresponding to near-resonant behaviour. However, it is shown that the formulation always enables evaluation of the contribution of the bending edge wave to the overall dynamic response. This is demonstrated by calculation of the related residues, presented in appendix A. Thus, we may expect the model to provide leading-order approximation in the far-field near-edge zone for an arbitrary load, similar to the Rayleigh wave. The proposed approach leads to insightful general observations, for example it underlines the dual parabolic–elliptic nature of the dispersive bending edge wave on an elastically supported plate contrasting with the hyperbolic–elliptic nature of the Rayleigh wave.

The paper is organized as follows. A general review of the governing equations and statement of the problem are presented in §2. Bending edge wave of general time dependence is analysed in §3, with the representation of the wave field in terms of a single harmonic function obtained. A multi-scale approach, using slow time perturbation of this solution, is performed in §4, and an explicit model for the bending edge wave is formulated in §5. Both cases of boundary conditions, namely the prescribed bending moment and shear force, are investigated. In §6, we consider a model example, illustrating the proposed approach and focusing attention on the near-resonant excitation of the bending edge wave by a harmonic moment imposed at the edge.

## 2. Statement of the problem

Consider a semi-infinite isotropic elastic plate of thickness 2*h*, supported by a Winkler foundation (figure 1). The plate occupies the region *x*_{3}≤2*h*.

Within the framework of the classical Kirchhoff theory, the approximate two-dimensional equation of bending of a thin elastic plate, supported by a Winkler foundation, is given by [23]
*W*(*x*_{1},*x*_{2},*t*) denotes the deflection of the plate, Δ is a two-dimensional Laplace operator in variables *x*_{1} and *x*_{2}, *β* is the Winkler foundation modulus, *ρ* denotes the volume mass density, *D* is the bending stiffness given by
*E* and *ν* are the Young modulus and Poisson's ratio, respectively. This model of a plate on an elastic foundation is widely applied in engineering for analysis of thin structures interacting with environment (see [24]). We remark that the validity of the governing equation (2.1) is restricted to long-wave low-frequency motion [25].

The boundary conditions at the edge *x*_{2}=0 are adopted in the form of
*M*_{0}=*M*_{0}(*x*_{1},*t*) and *N*_{0}=*N*_{0}(*x*_{1},*t*) are the prescribed bending moment and shear force, respectively. We also impose the two initial conditions
*A* and *B* are given initial data.

## 3. Homogeneous wave of arbitrary profile

The dispersion relation for the free bending edge wave associated with a Kirchhoff plate resting on the Winkler foundation has the form
*k* and *ω* denote wavenumber and frequency, respectively, and the coefficient
*γ*_{e}<1 (figure 2).

It has been shown in the aforementioned contribution of Kaplunov *et al.* [9] that the presence of the Winkler foundation brings in a few novel features. In particular, it leads to a cut-off frequency, along with a local minimum of the phase velocity. This minimal phase speed coincides with the group velocity and corresponds to the critical speed of an edge moving load.

Still the eigensolution of a sinusoidal profile analysed in [9] is not general enough. Following Chadwick [12], we now derive the sought for ansatz of a homogeneous edge wave of arbitrary shape, presenting the associated wave field in terms of a single plane harmonic function.

We begin by rewriting the plate equation (2.1) in terms of the dimensionless variables
*ξ* and *η*, and *β*_{0}=*βh*^{4}/*D*.

Next, we proceed with a serendipitous assumption
*γ*_{e} is defined after (3.2). This is actually the key to generalizing the conventional sinusoidal wave

Indeed, it is clear that this sinusoidal solution (3.7) satisfies (3.6), actually leading to the same dispersion relation (3.1). On the other hand, intuition suggests an elastically supported beam as a one-dimensional model for our plate bending edge wave, in line with a similar analogy between the Rayleigh wave and a string, which was pointed out in [26]. Unfortunately, in contrast to the well-known Rayleigh wave of arbitrary shape *f*(*x*±*c*_{R}*t*,*y*), see [12], where *c*_{R} is the Rayleigh wave speed, in case of the bending edge wave, there is seemingly no explicit functionally invariant solution. Thus, (3.6) is an implicit counterpart of the explicit ansatz for the Rayleigh wave introduced by Friedlander [11] and Chadwick [12].

Now, using (3.6), the second-order time derivative in (3.5) may be eliminated, leading to
*γ*_{e}<1, *M*_{0}=*N*_{0}=0), rewritten in terms of the dimensionless variables and employing the Cauchy–Riemann identities [27], we obtain

The representation for the bending edge wave field in terms of a single harmonic function may now be established by using the boundary conditions (3.13), giving

The deflection *W* may thus be expressed through (3.16) as a solution of the following initial value problem for any of the harmonic functions *W*_{j} (*j*=1,2)
*j*≠*m*≤2. Note that (3.17) necessitates that *A*_{j} and *B*_{j} are harmonic functions.

Applying the integral Fourier transform with respect to the variable *ξ* to the elliptic equations (3.17), and imposing the decay conditions *W*_{j}→0 as *f*_{j} given by
*W*_{j} may be expressed as
*γ*_{e} in (3.6) may be confirmed by analysis of the boundary conditions. Indeed, in case of an arbitrary *γ* in (3.6) substitution of solution (3.12) into the homogeneous boundary conditions (2.2) leads to the dispersion relation, from which *γ*=*γ*_{e}.

## 4. Perturbation scheme

Once representation in terms of a single plane harmonic function is established, we proceed with the development of an explicit model for the bending edge wave. Our intention is to extract the edge wave contribution from the overall dynamic response in a similar manner to [21] for the surface waves. In parallel with the cited paper, our starting point is a multiple scale procedure, perturbing equation (2.1) around the eigensolution constructed in §3. Accordingly, fast (*τ*_{f}=*τ*) and slow (*τ*_{s}=*ετ*) time variables are introduced, where *ε*≪1 is a small parameter, indicating the underlying assumption that the deviation of the phase speed from that of the homogeneous edge wave is small. An example of a near-resonant motion evolving in slow time (*τ*_{s}=*ετ*) is considered later in §6.

We begin the perturbation procedure by representing equation (3.5) in terms of the introduced scaling as
*W* is then expanded as
*λ*_{j} (*j*=1,2) are defined by (3.11) assuming decay as

At the next order, we obtain from (4.1) that
*j*=1. Employing the properties of harmonic functions, one may deduce that
*ξ*, and using (4.9), we infer that
*Φ*_{1}=*Φ*_{1}(*ξ*,*λ*_{1}*η*,*τ*_{f},*τ*_{s}) is an arbitrary harmonic function of the first two arguments.

Similar consideration for *j*=2 yields
*Φ*_{2}=*Φ*_{2}(*ξ*,*λ*_{2}*η*,*τ*_{f},*τ*_{s}) is also an arbitrary harmonic function.

We may thus obtain the following two-term asymptotic expansion for the third derivative

## 5. Parabolic equation on the edge

We proceed further with the analysis of the non-homogeneous edge boundary conditions (2.2). Owing to the linearity of the problem, it may be decomposed into two separate problems, imposing a prescribed edge bending moment or shear force, respectively.

### (a) Bending moment

Consider the case of an edge bending moment, that is when *N*_{0}=0,*M*_{0}≠0. The boundary conditions (2.2) are rewritten in terms of dimensionless variables at *η*=0 as
*m*_{0}=*m*_{0}(*ξ*,*τ*_{f},*τ*_{s}) satisfying the beam-like assumption, i.e.
*B* is a constant related to the amplitude, *α* is defined by (3.4), *ω*_{0} is an eigenfrequency satisfying the dispersion relation (3.1), and *ω*_{1} is a perturbation term, as we shall see later in §6.

Substituting the asymptotic expansion (4.16) into the (5.1), we obtain at leading order

At the next order, the boundary conditions (5.1) yield
*ξ*, we deduce, using (4.10) and (4.14), that

Using the Cauchy–Riemann identities, taking harmonic conjugation of the second equation and integrating with respect to *ξ*, we may establish that
*W*^{(0)} on the edge *η*=0, yielding
*Q* to the form
*Q* depends on the Poisson's ratio only. The graphical illustration is presented in figure 3, revealing a monotonic increase of *Q* with the Poisson's ratio *ν*.

Employing the leading-order approximation
*O*(*ε*^{2}) terms and returning to the original variables results in the parabolic equation on the edge, given by

Within the obtained approximate formulation, the decay away from the edge is described by the elliptic equation
*M*_{0} the solution of (5.13), and therefore, the parabolic–elliptic model (5.19) and (5.20) provides a correct evaluation of the edge wave contribution to the overall dynamic response. This is not surprising, because the procedure does in fact involve approximation in the vicinity of edge wave poles. More details may be found in appendix A, see (A 9) and (A 10).

We remark that the analysed wave usually dominates in the far-field near-edge zone for a general load, and also in case of near-resonant regimes of moving loads. These have been studied in the context of an explicit hyperbolic–elliptic model for the Rayleigh wave (see [26]). We also note a model example for a moving load on the edge of a Kirchhoff plate resting on the Winkler foundation considered in Kaplunov *et al*. [9], providing a hint of a beam-like behaviour at the edge.

The explicit model for the bending edge wave for moment edge loading is formulated as a Dirichlet problem for any of the following two pseudo-static elliptic equations
*y*=0 sought from the parabolic equation (5.19), taking into account (3.16) and (5.22). In other words, the solution of the dynamic parabolic equation (5.19) is used together with relation (5.22) as a boundary condition for the pseudo-static elliptic equation (5.24). The resulting plane harmonic function is then substituted into the relation (3.16) in order to restore the deflection of the plate. Thus, the *dual parabolic–elliptic nature* of the bending edge wave is established.

### (b) Shear force

A similar formulation may be derived for the second type of boundary conditions (2.2), with now *M*_{0}=0,*N*_{0}≠0. This corresponds to shear force excitation, taking the form
*W*. Instead, it provides an equation for the rotation angle *y*=0, namely
*Q* defined in (5.15).

The resulting explicit model for the shear edge force is similar to that obtained in respect of a bending moment. It contains the elliptic equation,

## 6. Near-resonant harmonic excitation

Let us consider an example illustrating the implementation of the model for a near-resonant edge loading. Consider inhomogeneous boundary conditions, when the bending edge moment is given by
*V* (*y*), namely
*κ*_{i} will coincide with *λ*_{i} provided the frequency *ω* and the wavenumber *k* satisfy the dispersion relation (3.1). The constants *C*_{i} may be determined from the boundary conditions. The exact solution at the edge is then given by

Let us now compare the last formula with that obtained within the approximate formulation derived in §5. In the case of the specified boundary conditions, the related particular solution of equation (5.19) is given by
*Q* defined in (5.15). It may be observed that both exact and approximate formulae, (6.6) and (6.7), respectively, display resonant behaviour whenever the frequency *ω* and the wavenumber *k* satisfy the dispersion relation (3.1).

We will now compare solutions (6.6) and (6.7) when the wave speed of the excitation is close to that of the bending edge wave. Consider a frequency perturbation of the form
*t*_{s}=*εt* as the asymptotic theory in §4 and §5 requires. Indeed, it may be shown that in view of (6.8) and (5.2) the form of the near-resonant excitation (6.1) will coincide with (5.4) provided that *A*=−*Bh*^{2}*α*^{2}*ω*_{0}*ω*_{1}. First, we obtain

## 7. Concluding remarks

Two main goals have been achieved in this paper. First, using the beam-like assumption, a general representation for the bending edge wave field has been obtained in terms of a single harmonic function in §3. Then, perturbing this solution in slow time, an explicit model for the bending edge wave has been constructed in §5. This model consists of a pseudo-static elliptic equation over the interior, governing the decay away from the edge, together with a parabolic equation on the edge describing wave propagation. The model reveals the dual parabolic–elliptic nature of the bending edge wave on a plate supported by a Winker foundation. Considerable simplifications in the analysis of dynamic phenomena associated with edge wave propagation are shown to arise. The model enables the contribution of the bending edge wave field to be separated from the overall dynamic response of the plate. It is therefore envisaged that application of the model would be efficient for analysis of dynamic resonant-type problems when the wave field associated with the bending edge wave is dominant, as is the case of an example considered in §6. The model also provides a leading-order approximation in the near-edge far-field region, where the bending edge wave is usually dominant.

The formulation presented in this paper may be developed for bending edge waves in the case of refined plate theories [28], with the approach relying on the plate theories with modified inertia (see [29] and references therein). Another direction of extension is related to edge waves in anisotropic plates [30,31], laminated structures [32] and pre-stressed plates [33]. More elaborate algebra is required to consider curved plates [34,35], shells [36,37] and interfacial edge waves [38]. Finally, we mention considerations of more advanced models of elastic foundation [10]. These problems provide further possible applications of the developed theory.

## Data accessibility

The paper does not report primary data.

## Authors' contributions

All authors have contributed substantially to the paper.

## Competing interests

We have no competing interests.

## Funding

The authors received no funding for this work.

## Acknowledgements

Fruitful discussions with Prof Y. Fu are gratefully acknowledged.

## Appendix A. Integral transform solution

The resulting parabolic–elliptic formulation may also be derived through integral transforms. Indeed, applying the Laplace transform to (3.5) with respect to scaled time *τ*, and the Fourier transform along the scaled longitudinal coordinate *ξ* (see (3.3)), we have
*p* and *s* denote the parameters of Fourier and Laplace transforms, respectively, and *W*^{FL} is the transformed deflection *W*. The decaying solution of (A 1) is given by
*C*_{1} and *C*_{2} are arbitrary constants, and
*η*=0, then the boundary conditions (5.1) are transformed to
*M*_{0}. Substituting the solution (A 2) into the boundary conditions (A 4), it is possible to determine the constants *C*_{1} and *C*_{2}. The result for the deflection transform *W*^{FL} may be expressed as
*K* may be written explicitly as
*η*=0. Indeed, using (A 6), it follows from (A 5) that
*ξ* and *τ*.

It is now evident that the presented parabolic–elliptic formulation (5.19) and (5.20) corresponds to the contribution of the bending edge wave field to overall dynamics response. Analogous consideration for a free Kirchhoff plate may be found in [39].

- Received March 11, 2016.
- Accepted May 5, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.