Maugin G.A., Rousseau M. Wave Momentum and Quasi-Particles in Physical Acoustics

World Scientific, 2015. — 250 p.
After a long period of competition, the wave-like and particle-like visions of some dynamical theories seem to have reached an 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 wave number and frequency. As to the particle model, it 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. Regarding this competition the interested reader will find it rewarding to peruse the book of B. R. Wheaton1. In the present book, we are interested in elastic vibrations in a deformable solid; the relevant particles then deserve the christening of “quasi-particles”, the most popular ones being the phonons. In a crystalline lattice outside the 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. The “particles” just conceived, 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. In recent times, such waves are called solitary waves. These, briefly described, have the shape of a unique strongly localized “wave” of unusually large amplitude moving over long distances at the surface of a fluid or a deformable solid2. The allied quasi-particle interpretation is then particularly well adapted and rapidly gives rise to the notion of soliton. This notion is a materialisation of the wave that carries the information. In the present work, we study the quasi-particles that are dual — in the sense of the above emphasized duality — of elastic waves that are known solutions of elastic wave problems, i.e., in physical acoustics. 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 (discontinuity surfaces, defects, inclusions...) The proposed original approach consists, once we know a macroscopic wave solution, in exploiting the conservation equations of canonical (or wave) momentum and energy, as recently revisited by us in continuum mechanics. This methodology is easily understood. Standard equations (here field equations of elasticity and associated boundary conditions) are used to obtain the dynamical solution (e.g., the celebrated Rayleigh wave for surface propagation). Then another set of continuum equations (so-called conservation laws in the sense of Noether’s theorem of field theory) is used — such as in a post-processing — to build some other quantities, here those that will principally appear in the conservation of so-called wave-momentum. It is that equation, and possibly in parallel the energy equation, which is integrated over a volume element that is representative of the considered wave motion (e.g., a vertical band of width equal to one wavelength in the sagittal plane for surface waves). This integration will provide the looked for equations (momentum and energy) of the associated quasi-particle, often exactly in the form of a Newtonian “point mechanics”, but with a “mass” that may depend on the velocity. This strategy is the one that pervades the whole of “configurational mechanics” in modern continuum thermomechanics3— where it is applied to the evaluation of driving forces acting on field singularities, material defects, inclusions, phase-transition fronts, etc.