Inspired by soliton theory and exploiting the conservation law of wave momentum, it is shown that one can associate with the surface Rayleigh wave of macroscopic elasticity a quasi-particle, a ‘surface phonon’, which is in inertial motion for the standard boundary conditions. The ‘mass’ of this ‘particle’ is determined in terms of the wave properties. Different types of alteration in the boundary conditions are shown to result in perturbations of this inertial motion in various ways. The essential tool in the presented derivation is the exploitation of the canonical equations of conservation, which are consequences of the celebrated Noether theorem of field theory. The results obtained may be useful in the mechanics of surface waves at the nanoscale, in particular in treating perturbations of various kinds.
After a long period of competition, the wave-like and particle-like visions of some dynamical theories seem to have reached agreement in their useful complementarity. Both serve to describe propagating information via their well-founded duality. The wave modelling favours a description of the propagation of information in terms of wavenumber and frequency. The particle model pertains to a diffusion of information through certain interactions in terms of momentum and energy. Classically, the duality between the two models is built quantum mechanically.
Here, we are interested in elastic vibrations in a solid; the relevant particles therefore deserve to be called ‘quasi-particles’, the most popular ones being the phonons (Feynman et al. 1964; Kittel 2005). In a crystalline lattice not at absolute zero, random motion takes place that corresponds to heat. In a crystalline medium subjected to boundary conditions, the phonon is associated with a modal vibration characterized by its frequency (see Maynard 2000). The ‘particles’ just described, e.g. phonons, are objects introduced to easily model interactions at this micro-level. But Nature also offers a more global scale of wave phenomena involving interactions that can essentially be understood in terms of a particle or ‘quasi-particle’ description. Such waves are called solitary waves. These, briefly described, have the shape of a unique strongly localized ‘wave’ of large amplitude moving over long distances at the surface of a fluid (Dauxois & Peyrard 2006), and also at the surface of a deformable solid (Maugin 2007a). The study of the interactions between the two elements of such a global solution shows that no change in shape or speed is exhibited, save a possible change in phase. This resembles the elastic ‘interaction’ of two material objects exchanging momentum without loss of energy. The quasi-particle interpretation then is particularly well adapted and has rapidly given rise to the notion of a soliton (Dauxois & Peyrard 2006). This notion is the materialization of the wave that carries the information. It has a meaning only in so far as it correctly models eventual interactions. Seen as a virtual object it can be equipped with a mass and a momentum. Various definitions are possible for these that depend on the system of partial differential equations considered to start with (see Maugin & Christov 2002).
In the present work, exploiting the analogy with solitons, we study the quasi-particles that are dual—in the sense of the above emphasized duality—of surface waves or guided waves that are known solutions of elastic wave problems associated with specific boundary conditions such as Rayleigh waves, Lamb waves, etc. Remember that such waves have an energy confined to the vicinity of the appropriate guiding interface. We are thus led to a clear introduction of the notion of a surface-acoustic phonon, although usually the definition of such quasi-particles relies essentially on a lattice-dynamics approach (Ludwig 1991; Henrich & Cox 1994). The interest in this study stems from the potentially associated simple interpretation of the interaction between fellow waves or of the interaction of such a wave with material objects (defects, inclusion, etc.). The proposed original approach consists, once we know a macroscopic surface-wave solution, in exploiting the conservation equations of canonical momentum and energy as revisited in continuum mechanics (Maugin 1993).
This methodology is easily understood. Standard equations (here, field equations of elasticity and associated boundary conditions) are used to obtain the surface-wave solution, e.g. the celebrated Rayleigh wave (we do not need to repeat this step). Another set of continuum equations (so-called conservation laws in the sense of Noether’s theorem of field theory) are used—such as in post-processing—to build some other quantities; here, those that will appear in the conservation of so-called wave momentum (the latter notion in agreement with Brenig (1955)). It is that equation, and possibly in parallel the energy equation, which is integrated over a vertical band of the sagittal plane of width equal to the wavelength, to provide the sought equations (momentum and energy) of the associated quasi-particle; here, exactly in the form of a Newtonian ‘point mechanics’. This strategy is the one that underlies the whole of ‘configurational mechanics’ in modern continuum thermomechanics (Maugin 2010) where, once we know a problem solution at each regular point, the associated conservation laws are exploited in a second stage to evaluate such quantities as driving forces acting on field singularities, material defects, inclusions, phase-transition fronts, etc.
The conservation of canonical wave momentum in small-strain elasticity is introduced in §2. A brief reminder on the standard Rayleigh surface-wave solution (pure isotropic homogeneous elasticity, free surface) is given in §3 with a view to defining all symbols and expressions to be exploited in further sections. Its implementation in the wave-momentum equation is effected in §4. Section 5 deals with a more involved case when the surface of the substrate is endowed with a surface energy (surface tension), while §6 approaches the problem of a leaky Rayleigh wave when a fluid occupies the external space. The conclusion in §7 points at various future studies.
2. The conservation law of canonical wave momentum
The general approach is unambiguously explained in the finite-strain framework in which a clear distinction is made between the relevant field—here the elastic displacement—and the spatial parametrization (usually, in solid mechanics, material coordinates; see Maugin (1993) for this and the application of Noether’s theorem). But, here, we are concerned with the simple dynamics of elastic materials in small strains in bodies where the local balance of linear momentum at regular material points is written, in Cauchy’s format, in the form (no body force for the sake of simplicity)2.1where σ is Cauchy’s symmetric stress of Cartesian components σji=σij and p=ρ0v is the linear momentum in which the density ρ0 may be considered as a constant and with elastic displacement u of Cartesian components ui. The stress is derivable from a potential W(ε), where εij=(1/2) (ui,j+uj,i) is the strain. Then equation (2.1) can be deduced as the Euler–Lagrange field equations from a Lagrangian density per unit volume L=K−W, where K is the kinetic energy K=ρ0v2/2. Application of Noether’s theorem for material translations yields the conservation law of wave momentum in the form (Maugin 1993)2.2with (T denotes transposition)2.3which are, respectively, the wave momentum and the so-called Eshelby stress. In Cartesian tensor notation, these read2.4Equation (2.2) is accompanied by the conservation of energy, which reads here in local form2.5where the Hamiltonian density H=K+W is such that .
Had we not known Noether’s theorem, we could have simply taken the product of equation (2.1) to the right by (∇u)T and obtained equation (2.2) after some manipulations assuming the dependence of the energy W on strains. Similarly, equation (2.5) is obtained by the application of Noether’s theorem for time translations or by direct computation following the scalar multiplication of equation (2.1) by the velocity.
3. Reminder on the Rayleigh surface-acoustic wave
We consider a linear elastic semi-infinite medium limited by the plane z=0. Its mechanical constitutive equation is the well-known Hooke Law given in Cartesian tensorial form by3.1where λ,μ are the Lamé coefficients. The Helmholtz decomposition of a vector field allows us to write3.2Then uL and uT satisfy wave equations3.3where3.4are, respectively, the speeds of longitudinal and transverse waves.
In the case of plane strains in the (x,z)-plane, and if we assume that the fields uL and uT are two harmonic plane waves, we have3.5with wave numbers (ω is the angular frequency)3.6Then the Rayleigh wave is an acoustic eigenmode corresponding to a particular coupling between longitudinal and transverse waves. This mode is a surface wave propagating at the free surface of an elastic solid semi-infinite space and polarized in the sagittal plane.
The boundary conditions3.7allow one to obtain the corresponding dispersion equation (but here there is no real dispersion). The Rayleigh wave is the real solution of this equation of dispersion.
This mode propagates in the x direction with a speed noted cR and is evanescent in the z vertical direction. So, it is a surface wave. Its speed is given by the real solution of the so-called Rayleigh equation (Achenbach 1973; Dieulesaint & Royer 2000)3.8The displacement of the particles during the Rayleigh wave propagation is ellipsoidal with a magnitude decreasing in depth with the variable z. The two components (ux,uz) of the displacement vector in the so-called sagittal plane are given by3.9with3.10where A is a real constant that has the dimension of a squared length (relatively small compared with the wavelength in linear elasticity), and3.11are, respectively, the wave numbers of a Rayleigh wave, of longitudinal waves and of transverse waves. In addition,3.12are representative of the evanescence of longitudinal and transverse waves, which compose the Rayleigh wave. Let us remark that the evanescence of longitudinal and transverse waves is a necessary condition for the existence of the Rayleigh wave. With this condition we have the inequalities3.13
4. Conservation of wave momentum for Rayleigh surface-acoustic waves
We consider a part of the previous semi-infinite elastic medium of unit thickness defined in the sagittal plane by (figure 1). So the width of this domain is one (Rayleigh) wavelength. Using the divergence theorem, the conservation law (2.2) of wave momentum integrated over the material domain D gives4.1where is an element of volume, dS an element of surface, ∂D the boundary of D and n the external unit vector to ∂D.
As the surface integral is simplified by the free boundary conditions, the vanishing condition at infinity and the x-periodicity of the solution, equation (4.1) reduces to (here n=−ez at z=0)4.2The evaluation of the volume integral (in fact a surface integral with unit thickness in the direction orthogonal to the sagittal plane) is easy but lengthy (see sketchy proof below) and yields4.3In this equation, cR, which is the Rayleigh wave speed, is now viewed as speed of ‘information’. Then, MR (which is homogeneous to a mass per unit of length in the direction ey) is viewed as a mass, not of matter but of information. This is representative of an ‘average-acoustic energy’ or of a certain amount of ‘acoustic information’, and so defines the quasi-particle of Rayleigh.
It is found to be given by4.4with (here, kRA will be the amplitude of the displacement, while A was that of the potential)4.5The parameter f depends on the mechanical characteristics of the elastic medium and bears the print of the boundary conditions. It is the quantity that truly characterizes the quasi-particle of Rayleigh.
The right-hand side of equation (4.2), which is proportional to the Rayleigh wave equation, vanishes identically (see §4a). The equation4.6represents the inertial motion (conservation of momentum in Newton’s sense) of the quasi-particle of mass MR along the surface of the limiting plane. So, we have obtained a Newtonian formalism of the Rayleigh wave propagation viewed as a particle. We can speak of a true surface-acoustic quasi-particle. We elaborate a little more on the evaluation of the two sides of equation (4.2) and of MR.
(a) Evaluation of equation (4.2)
The left-hand side of equation (4.2) reads4.7The first term in the right-hand side of this vectorial expression contains terms in and , so that its contribution is not zero by integration along one wavelength. The evaluation of this term is lengthy but it allows one to obtain expression (4.4) of MR. The second term in the right-hand side contains only terms in , so its contribution is zero by integration along one wavelength. As a result, equation (4.7) has only a non-zero component along ex. Furthermore, it remains to evaluate the right-hand side of equation (4.2). This contribution has no contribution along ex; hence equation (4.6). There remains to evaluate the ez component, i.e.4.8
It is easy to show that this whole term vanishes as a consequence of the ‘dispersion’ relation (3.8). This coincides with a result of the kinematic theory of waves (Lighthill–Whitham) in which it is shown that the averaged value of the Lagrangian L=K−W over a period of motion vanishes when the linear dispersion relation is satisfied as the above expression is essentially , in a classical formalism (). In this regard, compare this with Whitham (1974) and the more recent Maugin (2007b). We have thus checked that the two sides of equation (4.2) projected along ez vanish identically in the considered semi-infinite band.
The canonical conservation laws form a true thermo-mechanics, in the sense that the energy conservation should always be considered in parallel with that of canonical momentum. We did not do this here. However, with integration over the semi-infinite band of equation (2.5), we would show that the corresponding total energy E takes a Newtonian form such as4.9That is, the energy of the ‘point-like’ Rayleigh quasi-particle appears to be purely kinetic, in agreement with equation (4.6), while in the continuum description the energy is made of kinetic and potential (elastic) parts; but MR still contains all the relevant information. The situation is even more spectacular in the electroelastic case treated in a companion paper (Maugin & Rousseau 2010), where the continuum energy is made of kinetic, potential (elastic), electric (quasi-electrostatic) and interaction (electromechanical) parts, but still the energy of the associated quasi-particle appears to be purely kinetic. We refer the reader to that more interesting case for the proof.
5. Surface energy and quasi-particle of Rayleigh
If we now take into account a surface free energy F at z=0, the Rayleigh wave becomes dispersive because a characteristic length (the ratio of surface and bulk elasticities (l=F/μ)) is then associated with the semi-infinite elastic medium (Vlasie-Belloncle & Rousseau 2006).
The boundary conditions at z=0,5.1allow one to obtain the corresponding dispersion relation that explicitly depends on the angular frequency ω. The Rayleigh wave is then the real solution of the following perturbed equation of dispersion (Vlasie-Belloncle & Rousseau 2006)5.2where cRF is the speed of the Rayleigh wave with surface free energy and kRF=ω/cRF is the corresponding wavenumber. If the non-dimensional coefficient εF=FkT/μ or εF(ω)=(F/μcT)ω is very small compared with unity, the speed cRF itself is written in perturbed form as5.3with (Vlasie-Belloncle & Rousseau 2006)5.4The parameter αF is a negative constant—of order unity—that depends on the speeds of the Rayleigh wave, the transverse wave and the longitudinal wave. Then, we observe that cRF increases from cR to cT when the angular frequency ω varies between 0 and ωCF, the latter being defined as5.5This characteristic value of the angular frequency corresponds to the limit for the Rayleigh wave to exist, as the transverse wave still is evanescent. In the frequency interval (0,ωCF) the approximation (5.3) is valid if and only if ω≪(μcT/F)<ωCF. Outside this, one needs a numerical implementation of equation (5.2) in order to obtain the value of cRF.
We now consider a part of the previous semi-infinite elastic medium of unit thickness defined in the sagittal plane by . The conservation law (2.2) of wave momentum integrated over the material domain D is given by equation (4.1).
The evaluation of the volume integral gives5.6The right-hand side of equation (4.2), which is proportional to the Rayleigh wave equation (see §4), vanishes identically.
Under these conditions, cRF is the speed of the quasi-particle of Rayleigh with surface free energy and MRF is its mass.
The equation5.7stands for the inertial motion (conservation of momentum in Newton’s sense) of the quasi-particle of mass MRF along the surface of the limiting plane. By continuity with respect to F going to zero, the conservation of momentum cannot depend on the angular frequency. Consequently, equation (5.7) can be written as5.8and then, at the first order, the mass of the quasi-particle of Rayleigh with free surface energy is necessarily given by5.9Since the speed of the quasi-particle of Rayleigh with surface free energy is larger than that of the quasi-particle of the pure Rayleigh wave, its mass is smaller. These two results can be explained independently of their derivation.
First, the higher the angular frequency, the smaller the wavelength. Accordingly, the action range of the surface energy seems to be enlarged. In this range, the energy surface acts as rigidifying the medium. So, the longitudinal and transverse speeds in the apparent medium are higher. As the Rayleigh wave speed is proportional to the transverse speed, it is easy to deduce that cRF is larger than cR.
Second, concerning the mass, as previously mentioned, it is representative of the amount of information carried by the surface wave. Since the apparent medium is more rigid, it deforms less, the amount of information is less and, thus, the mass is smaller.
6. Leaky Rayleigh wave and quasi-particle of Rayleigh
We now study another perturbation of boundary conditions, considering that the elastic medium is in contact with a fluid medium that occupies the half-space z<0. The Rayleigh wave associated with the semi-infinite elastic medium is then perturbed by the existence of the fluid and it becomes a leaky Rayleigh wave.
The boundary conditions at z=0 are now written as6.1where p is the acoustic pressure in the fluid and […] denotes the jump through the fluid/solid interface. Only ux verifies a boundary condition, because uz is arbitrary since the fluid considered is not viscous.
The conditions (6.1) allow one to obtain the well-known dispersion relation6.2where εl=ρ1/ρ0 (the ratio between the mass densities of the fluid and of the solid) and c1 is the speed of sound in the fluid medium. In the case of a water/metal interface, εl≪1, and the second term in equation (6.2) appears as a perturbation of the Rayleigh equation (3.8).
The leaky Rayleigh wave is then the complex solution of the dispersion equation (6.2), which can be written in perturbed form as6.3with6.4This can be obtained by differentiating (6.2) and identifying αl. The latter depends on the speeds of the Rayleigh wave, the transverse (cT) wave and the longitudinal (cL) wave in the solid and of the sound speed c1 in the fluid. This parameter is a positive constant. So, the imaginary part of clR indicates that, in the solid, the wave propagates obliquely from the solid to the fluid and vanishes at infinity. At the same time, in the fluid the wave propagates obliquely and increases at infinity (which is nonsense). This defines the notion of a leaky Rayleigh wave.
We now consider a part of the previous semi-infinite elastic medium of unit thickness defined in the sagittal plane by . The conservation law (2.2) of wave momentum integrated over the material domain D is given by equation (4.2).
The evaluation of the volume integral gives6.5The right-hand side of equation (4.3), which is proportional to the leaky Rayleigh wave equation (see §4), vanishes identically.
In these conditions, clR is the speed of the leaky Rayleigh wave and MlR is its mass.
The equation6.6represents the inertial motion (conservation of momentum in Newton’s sense) of this quasi-particle. The conservation of momentum is verified for any εl and especially when εl→0 (i.e. the mass density of the fluid goes to zero); consequently, by continuity, equation (6.6) can be written as6.7and then, at the first order, the mass of the quasi-particle of the leaky Rayleigh wave is given by6.8Thus, the mass of the quasi-particle is a complex number. The imaginary part of this mass is negative, which describes the re-emitting of a part of the information from the solid to the fluid. In the fluid we obtain an accumulation of information at infinity, what is again nonsense. The nonsensical situation reached can be solved only by the theory of bounded beams in the framework of wave theory. The question remains of how to remedy this nonsense in the quasi-particle framework. This matter is not dealt with here.
If we can say that photons are grains of light and acoustic phonons are grains of acoustic vibrations, both in an unbounded space, the particle-like objects studied in this paper are grains of surface-acoustic waves guided by the upper boundary of the substrate. They account in their definition for the fact that the continuous waves are confined along a guiding surface and for a type of boundary condition. This explains the complexity required to prove their existence and to define their effective mass. Passing from the surface-wave concept to that of surface quasi-particle can be summarized in the illustration in figure 2.
We have considered the case of a Rayleigh wave in isotropic bodies for different boundary conditions that perturb the pure quasi-particle of Rayleigh. It is possible to consider other surface waves with specific boundary conditions. The strategy will be just the same. Then it is probable that we shall obtain analogous results after a somewhat lengthy derivation. A more original study could be to consider a case where the momentum is not conserved, such as for a viscoelastic medium, where the perturbation in the constitutive behaviour will result in a non-inertial motion of the associated quasi-particle (see final remark below). Yet another rewarding study would be to describe the interaction between the obtained Rayleigh quasi-particle and a localized inhomogeneity, i.e. a defect along the path of the guided surface wave. The consideration of a Rayleigh surface-acoustic wave in anisotropic crystals (Dieulesaint & Royer (2000) for a classical continuum treatment) and of Stoneley waves at the interface of elastic materials with appropriate values of elasticity coefficients (Achenbach 1973; Louzar et al. 1994) is just a matter of more lengthy computations. Of greater interest should be the consideration of pure shear-horizontal waves (but with multi-modes and dispersion) as in the case of Love waves in future works.
One final remark concerns the appearance of a Lagrangian density. This could make the reader think that the method used applies only in the absence of dissipation, i.e. when the whole construct is based on a variational formulation and the exploitation of Noether’s theorem in its standard form. But this is not true because it was shown (Maugin 2006) that an equation such as (2.2) can also be obtained in the presence of dissipative phenomena, but at the price of a non-vanishing source term in the right-hand side. This, of course, will in turn affect the motion of associated quasi-particles. This will be the case for the propagation of a surface wave on top of a viscoelastic substrate when an attenuation of the wave, and thus a slowing down of the associated quasi-particle, must be observed. Anyway, the exploited technique is theoretically applicable to all surface-wave problems in continuum physics when we know the continuum analytical solution.
- Received May 3, 2010.
- Accepted July 13, 2010.
- This journal is © 2010 The Royal Society