## Abstract

High-frequency vibrations of a semi-infinite elastic strip with traction-free faces are considered. The conditions on end data that are derived do not allow non-radiating in Sommerfeld's sense of polynomial modes at thickness resonance frequencies. These represent a high-frequency analogue of the well-known decay conditions in statics that agree with the classical Saint-Venant principle. The proposed radiation conditions are applied to the construction of boundary conditions in the theories of high-frequency long-wave vibrations describing slow-varying motions in the vicinity of thickness resonance frequencies. The derivation is based on the Laplace transform technique along with the asymptotic methodology that is typical for thin plates and shells.

## 1. Introduction

Static decay conditions for an elastic semi-infinite strip were derived in papers by Gusein-Zade (1965) and Gregory & Wan (1984) by making use of the Laplace transform technique and the reciprocity theorem in elasticity, respectively. They express the self-equilibrium of stress resultants and stress couples at the strip edge and fully agree, therefore, with the classical Saint-Venant principle.

In the low-frequency domain *λ*≪1 (see formula (2.5) for the dimensionless frequency parameter *λ*), the refined decay conditions recently proposed by Babenkova & Kaplunov (2004) demonstrate a deviation from the Saint-Venant principle in higher order terms. As a result, they involve small-amplitude propagating modes induced by statically self-equilibrated edge loads (see also Babenkova & Kaplunov 2003).

In this paper we attempt to establish an analogue of the static decay conditions applicable over the high-frequency domain *λ*∼*O*(1). To this end, we recall that the Saint-Venant principle is most useful when formulating boundary conditions in the elementary theories of plate bending and extension. It allows, in particular, extracting from edge loads the components corresponding to non-decaying polynomial modes. In low-frequency dynamics, the latter transfer to slow-varying vibration modes that may demonstrate sinusoidal or exponential behaviour.

This paper assumes that the separation of polynomial modes occurs similarly to that in statics. In dynamics, these are associated with the double-zero roots of the Rayleigh–Lamb dispersion relations occurring at thickness shear and stretch resonance frequencies, and may be described by low-dimensional long-wave theories that represent, in a sense, high-frequency counterparts of the above-mentioned elementary theories for plates. They are discussed in greater detail in §5 (see also Kaplunov *et al*. 1998). In this case, we could also expect that the proposed approach should result in approximate boundary conditions for high-frequency long-wave models.

As in statics, we impose certain conditions on end data that ensure prescribed behaviour at infinity. We adopt the term ‘radiation condition’, which appears to be more appropriate than ‘decay condition’. In fact, by exploiting the aforementioned analogy with static problems, we do not require a total decay, but simply intend to remove non-decaying polynomial modes that do not satisfy the Sommerfeld radiation condition. At the same time, we allow radiation to infinity that is associated with short-wave propagating modes typical for the high-frequency domain. To the best of the authors' knowledge, such a formulation was not a feature of existing publications on the subject (e.g. Karp 2004 and references therein).

The consideration below utilizes the Laplace transform technique previously developed for a semi-infinite strip in Gusein-Zade (1965) and Babenkova & Kaplunov (2004). As in statics, the polynomial modes of interest correspond to multiple zero poles, which now occur at thickness resonance frequencies. It is demonstrated that explicit radiation conditions expressed in terms of two prescribed end quantities are possible only for special mixed boundary conditions in plane elasticity, or, for thickness stretch motion under prescribed edge stresses, provided that the Poisson ratio equals zero.

The proposed radiation conditions are applied to the derivation of boundary conditions for the 1D equations governing long-wave vibrations in the vicinity of thickness resonance frequencies. In particular, the boundary conditions at free and fixed edges are established. It is interesting that these two sets of boundary conditions coincide in the leading order. To construct the higher order boundary conditions, we formulate canonical 2D problems for a semi-infinite strip. The solutions to these problems may be expanded in terms of Lamb modes. It should be mentioned that the approach making use of canonical problems is typical for asymptotic analysis of the boundary conditions in the general theory for plates and shells (see Goldenveiser 1969, 1998; Gregory & Wan 1984, 1985).

## 2. Statement of the problem

Consider harmonic vibrations of a semi-infinite elastic strip of the thickness 2*h*. Specify the dimensionless (related to the strip half-thickness *h*) Cartesian coordinates (*x*, *y*) such that 0≤*x*<∞, −1≤*y*≤1. The equations of motion in the theory of plane strain can be written as(2.1)Here, *v _{x}*(

*x*,

*y*) and

*v*(

_{y}*x*,

*y*) are longitudinal and transverse displacements that have been normalized by

*h*, respectively,

*ω*is circular frequency,

*v*is Poisson's ratio,

*ρ*is mass density, and

*G*=

*E*/2(1+

*v*) is shear modulus, where

*E*is Young's modulus.

The stress tensor components satisfy the ‘stress–displacement’ formulae(2.2)where *c*_{1} and *c*_{2} are the dilatation and distortion wave speeds, respectively.

Below we consider a strip with traction-free faces:(2.3)

For the initial data at the edge *x*=0, we introduce the notation(2.4)

Our main goal is to obtain an analogue of the Saint-Venant principle for the high-frequency domain(2.5)where *λ* is dimensionless frequency.

We recall that the classical Saint-Venant principle allows us to separate polynomial terms in the static solution of the plane problem in elasticity for a semi-infinite strip by imposing conditions on end data. These conditions are known as decay conditions and result, in particular, in self-equilibrium of stress resultants and stress couples. The mathematical justification of the decay conditions is given in Gusein-Zade (1965) and Gregory & Wan (1984).

It is well known that the aforementioned polynomial terms may be described by the elementary theories of plate extension and plate bending. The Saint-Venant principle is generally applied to the formulation of the boundary conditions in these theories.

In this paper, we develop a similar idea for the high-frequency domain. To this end, we derive conditions on the end data, equation (2.4), ensuring the absence of polynomial terms. The latter are then described using high-frequency counterparts of the elementary plate theories, which are usually referred to as theories of high-frequency, long-wave vibrations (e.g. Kaplunov *et al*. 1998).

However, in contrast to statics, high-frequency behaviour is often characterized by short-wave propagating sinusoidal modes that do not decay along with polynomial terms. These propagating modes have to satisfy the Sommerfeld condition at infinity. Thus, we do not require a total decay. We require only the absence of polynomial modes that do not satisfy the radiation condition at infinity. Hence, the conditions sought on the end data, equation (2.4), seem to be radiation conditions rather than decaying ones.

## 3. Analysis of Laplace transforms

We begin with analysis of the Laplace transforms for the formulated problem. These are reproduced in a recent paper by Babenkova & Kaplunov (2004); see also Gusein-Zade (1965). In particular, for strip extension (symmetric strip motion with respect to the midline *y*=0), transformed displacements become(3.1)

while for strip bending (antisymmetric motion), they are(3.2)In the above formulae, *p* is Laplace transform parameter. In addition,withand(3.3)where *ϕ*(*p*) and *ψ*(*p*) denote the symmetric and antisymmetric Rayleigh–Lamb denominators, respectively:(3.4)(3.5)All the other functions in the formulae (3.1) and (3.2) are defined in Appendix.

By using Fourier–Mellin integrals, we may express unknown displacements as(3.6)with *δ*>0.

At this stage, we allow only polynomial solutions and radiating solutions, i.e. those having the e^{i(cx−ωt)} form at infinity for some *c*>0. It follows from the considerations in Kirrmann (1995) and Ustinov (1991) that the original problem can be resolved in terms of these functions for any given frequency.

It is well known that the Rayleigh–Lamb equations *ϕ*(*p*)=0 and *ψ*(*p*)=0 have many complex roots. However, not all of them create poles of the Laplace transforms and , since the associated numerators *A*_{0}(*p*), *B*_{0}(*p*) and *C*_{0}(*p*), *D*_{0}(*p*) are equal to zero for some of these roots.

In particular, there are no poles corresponding to the roots in the right complex half-plane (Re *p _{n}*>0). These would result in growing exponential terms that are not taken into consideration in the desired solutions. It may be readily verified that the end data, equation (2.4), automatically satisfies conditions ensuring zero numerators at the roots with a positive real part (see Babenkova & Kaplunov 2004; Gusein-Zade 1965 for more detail).

For the same reason, the pure imaginary roots in the lower complex half-plane (Re *p _{n}*>0, Im

*p*>0) also do not generate poles. They are associated with waves coming from infinity, and relevant numerators are equal to zero owing to the radiation condition imposed at infinity (see figure 1).

_{n}It can be easily verified that the values *p*=±*λk*i and *p*=±*λ*i of the transform parameter do not create branch points for the integrands in equation (3.6). As a result, the residue theorem leads to the infinite series for unknown displacements(3.7)There are three different types of poles *p _{n}* in formula (3.7). The first is represented by the large poles located in the left complex half-plane (Re

*p*<0, |

_{n}*p*|≳1). These poles are associated with exponentially decaying terms in the solutions sought, i.e. with boundary layers. The second type is formed by the pure imaginary poles with Im

_{n}*p*>0, which correspond to radiating modes. Finally, the pole at the origin creates the polynomial terms in the solutions sought.

_{n}The pole *p*=0 exists only if *λ* satisfies the Rayleigh–Lamb equation *ϕ*(0)=0 or *ψ*(0)=0. Such special values are referred to as thickness stretch and shear resonance frequencies and are denoted by *λ*=*Λ*_{st} or *λ*=*Λ*_{sh}, respectively. They are the natural frequencies of an infinitely thin transverse fibre of the strip. They are separated into symmetric and antisymmetric families and are defined as follows (*m*=1,2,…):

for antisymmetric stretch vibrations,(3.8)

for symmetric stretch vibrations,(3.9)

for antisymmetric shear vibrations,(3.10)

and for symmetric shear vibrations,(3.11)

At and , the antisymmetric Rayleigh–Lamb equation *ψ*(*p*)=0 has the second-order root *p*=0 provided that the quantities(3.12)and(3.13)take non-zero finite values, respectively.

Similarly, for and , we require finite non-zero quantities(3.14)and(3.15)ensuring a double-zero root of the symmetric Rayleigh–Lamb equation *ϕ*(*p*)=0.

## 4. Radiation conditions

We first extract the polynomial terms associated with the pole *p*=0 for various types of thickness resonance frequencies defined in §3.

### (a) Stretch resonance frequencies

#### (i) Antisymmetric stretch modes

Let an antisymmetric stretch resonance frequency be defined by formula (3.8), and let the quantity take finite non-zero values. Then, the Laplace transforms (3.2) possess the second-order pole *p*=0. By calculating the corresponding residue, we have(4.1)(4.2)where(4.3)and(4.4)

Using equations (4.1) and (4.2), we can derive the desired expressions for stresses(4.5)

(4.6)

#### (ii) Symmetric stretch modes

For symmetric stretch modes, all the results are similar to those in the antisymmetric case. In particular, displacements at have the following form(4.7)(4.8)where(4.9)and(4.10)

The stresses then become(4.11)

(4.12)

### (b) Shear resonance frequencies

#### (i) Antisymmetric shear modes

Let be an antisymmetric shear resonance frequency defined by equation (3.10) and the quantity takes non-zero finite value. Then, by calculating the residue at the second-order pole *p*=0, we obtain(4.13)(4.14)where(4.15)and(4.16)

From equations (4.13) to (4.14), we express the stresses as(4.17)

(4.18)

#### (ii) Symmetric shear modes

Similarly, for displacements we have(4.19)(4.20)where(4.21)and(4.22)with defined by equation (3.11).

As before, we obtain for the stresses(4.23)

(4.24)

Inspection of the polynomial modes above shows that the desired radiation conditions may be deduced by equating to zero the integrals and with the suffices *n*=1,2 and *f*=(*a*,*s*), where *a* and *s* refer to antisymmetric and symmetric problems, respectively. In this case, resulting expressions will generally include all the initial data from equation (2.4) specified at the edge *x*=0. At the same time, the boundary conditions in plane elasticity usually operate with only two of the four quantities in equation (2.4).

As an example, consider mixed boundary conditions(4.25)for the thickness stretch resonance frequencies .

By analysing the formulae for shear modes, we obtain the radiation conditions expressed in terms of two prescribed quantities on the right-hand side of equation (4.25). For antisymmetric and symmetric motions, it is given by and , respectively, where and are defined by equations (4.4) and (4.10).

These conditions allow removing the polynomial terms, equations (4.1) and (4.2) and (4.7) to (4.8), for antisymmetric and symmetric problems, respectively, to within their eigencomponents, thus satisfying the homogeneous boundary conditions associated with equation (4.25). Eigenmotions of a semi-infinite strip represent, in a sense, a dynamic analogue of rigid body motions in statics, as discussed in Babenkova & Kaplunov (2004).

To exclude the aforementioned eigenmotions, we should impose the additional radiation condition . Apparently, this is not too important for further applications since eigenmotion is a feature of semi-infinite bodies only.

Similar radiation conditions can also be formulated for vibrations with the thickness shear resonance frequencies . In particular, for another type of mixed boundary conditions(4.26)we arrive at the radiation conditions expressed in terms of the end data in equation (4.26). For antisymmetric and symmetric motions, they are given by and respectively, where and are defined by equations. (4.16) and (4.22).

For more general boundary conditions at the edge of a semi-infinite strip, explicit radiation conditions are possible only for vibrations with the thickness stretch resonance frequencies in the case of prescribed edge stresses(4.27)provided that Poisson's ratio is equal to zero. For *v*=0, the longitudinal stress *σ _{xx}* equals zero in equations (4.5) and (4.11). As a result, the radiation conditions take now an explicit form: for antisymmetric motion,(4.28)and for symmetric motion,(4.29)

Additional conditions responsible for eigenmotions become .

## 5. Boundary conditions for high-frequency long-wave models

The radiation conditions in §4 may be used for formulating boundary conditions in asymptotic theories of high-frequency long-wave vibrations (e.g. Kaplunov *et al*. 1998).

Let us introduce, for convenience, a new notation for the end data in equation (2.4), assuming that these do not generally satisfy the radiation conditions and, therefore, may excite high-frequency polynomial modes. Accordingly, we introduce(5.1)

As an example, consider antisymmetric motion in the vicinity of thickness stretch resonance frequencies, so that(5.2)where *ϵ*≪1 is a small frequency deviation.

In this case, the 1D high-frequency long-wave equation corresponding to the original plane problem in elasticity, equations (2.1) and (2.3), for a semi-infinite strip with traction-free faces can be written as(5.3)where and are given by equations (3.8) and (3.12), respectively, and *w*(*x*) is a slowly varying transverse amplitude.

For the sake of definiteness, we assume that . Then, equation (5.3) possesses slow-varying radiating and decaying at infinity solutions(5.4)and(5.5)for positive and negative *ϵ*, respectively, where , and the constant *C* has to be determined from the boundary conditions at the edge *x*=0. In the limiting case *ϵ*=0 (), both solutions in equations (5.4) and (5.5), transform to the uniform distribution *w*(*x*)=*C*.

In equation (5.2), all 2D displacement and stress long-wave components can be expressed in terms of the function *w*(*x*). To within the error *O*(*ϵ*), we get(5.6)where *ξ*=*ηx* is a scaled longitudinal coordinate. In doing so, we assume that d*w*/d*ξ*∼*w*(*ξ*).

Boundary conditions for equation (5.3) may be derived by substituting the discrepancy between the analysed long-wave modes, equation (5.6), and the prescribed end data, equation (5.1), into appropriate radiation conditions. Below we start from the radiation conditions in §4 related to the discrete stretch resonance frequencies,. However, it may be readily deduced that these conditions are also valid over the vicinity (5.2) to within the error *O*(*ϵ*).

In particular, for the mixed end data(5.7)we set in the radiation condition , where is defined by equation (4.4):(5.8)and(5.9)

Finally, we get the following boundary condition for *w*(*x*) at *x*=0(5.10)

For the prescribed edge stresses expressed in terms of the functions Σ_{0}(*y*) and *T*_{0}(*y*) in (5.1), we obtain from the formula (4.28) the boundary condition in the case of zero Poisson's ratio (*k*^{2}=1/2). It is(5.11)

Now, we discuss boundary conditions on free and fixed strip edges. First, let end data be given by(5.12)

Since the distribution in equation (5.12) along the thickness, coincides with that for the principal longitudinal stress *σ _{xx}* in formula (5.6), we may expect a small (of order

*O*(

*η*)) contribution of the asymptotically secondary transverse displacement , into the radiation condition . Thus, in the leading order we may utilize the substitution(5.13)

As a result, we arrive at the boundary condition(5.14)

We also note that the latter may be immediately deduced by substituting the formula (5.6) for the stress *σ _{xx}* into the first boundary condition (5.12).

To derive the first-order corrector from the leading order boundary conditions (5.14), we should consider the terms of order *O*(*η*) in the radiation condition . To this end, we set in equation (5.13)(5.15)where the function *M*(*y*) has to be found.

Let us determine the discrepancy in the original boundary condition (5.12) arising from the solution of the leading order problem in equation (5.3) and (5.14). From equations (5.6) and (5.12), we have(5.16)where the quantities marked with asterisk correspond to the discrepancy sought and(5.17)

To calculate the function *M*(*y*), we formulate a canonical plane problem for the initial 2D equation (2.1) with , boundary conditions in equation (2.3) on faces and boundary conditions(5.18)on the edge motivated by the relations in equation (5.16). In doing so, we impose the traditional radiation condition at infinity for propagating modes, but also allow the uniform in *x* terms corresponding to the solutions in equations (5.4) and (5.5) at *ϵ*=0 (*η*=0).

This canonical problem can be uniquely resolved using, for example, expanding in Lamb modes (e.g. Kirrmann 1995; Ustinov 1991). Canonical problems are also useful for asymptotic refinement of the boundary conditions in the general static plate-and-shell theories (e.g. works by Goldenveiser 1969, 1998; Gregory & Wan 1984, 1985).

Let the function *M*_{0}(*y*) be the short-wave component of the transverse displacement in the solution of the canonical problem at the edge *x*=0. Thus, the contribution of the uniform mode (where *C* is a constant) has to be extracted from this solution. Below, we assume the function *M*_{0}(*y*) to be known. Then, we have in equation (5.15) by comparing boundary conditions in equations (5.16) and (5.18)(5.19)Next, by substituting equation (5.13) with , as defined by equation (5.15), into the radiation condition , we obtain(5.20)where the coefficient *D* depends on the thickness resonance frequency number *m* and is given by(5.21)At *A*=0 formula (5.20) represents a refined boundary condition for a free edge.

We also note that a similar problem with the prescribed edge displacements(5.22)results in the boundary condition(5.23)with(5.24)where *S*_{0}(*y*) is the short-wave component of the longitudinal displacement *v _{x}* at the edge,

*x*=0 in the solution of a 2D canonical problem with the boundary conditions(5.25)whereand all the other relations are the same as for the canonical problem above.

The homogeneous case *B*=0 corresponds to a fixed edge. Thus, the asymptotic boundary conditions for high-frequency long-wave motions of a semi-infinite strip with free and fixed edges differ only in *O*(*η*) terms.

## 6. Concluding remarks

The proposed radiation conditions have a clear physical meaning. The end data that fulfil these conditions excite only short-wave modes over the high-frequency domain.

Slowly, varying vibration modes in the vicinity of thickness resonance frequencies, generalize the polynomial modes in question. They are of interest for many applications (see Kaplunov *et al*. 1998, and references therein) and are governed by high-frequency, long-wave theories. The examples of boundary conditions in the abovementioned theories in §5 for free and fixed edges of a semi-infinite strip may allow important extensions to a shell of general shape similarly to asymptotic analysis in statics (see Goldenveiser 1969, 1998).

The derivation of appropriate radiation conditions would also be useful in the case of fixed faces that involves only high-frequency motions (see also Kaplunov 1995).

## Acknowledgments

The work of the first author is supported by the UK Overseas Research Student Award and by the University of Manchester. These awards are very gratefully acknowledged.

## Appendix

The numerators in the Laplace transforms have the following form:

for symmetric motion,for antisymmetric motion,and particular solutions in equations (3.1) and (3.2) are given by

In the above,where

## Footnotes

- Received May 4, 2004.
- Accepted September 16, 2004.

- © 2005 The Royal Society