Wave Momentum and Quasi-Particles in Physical Acoustics

250 Pages·2015·1.74 MB·English

by  Gerard A Maugin

Wave Momentum and Quasi-Particles
in Physical Acoustics
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WORLD SCIENTIFIC SERIES ON
NONLINEAR SCIENCE Series A Vol. 88

Series Editor: Leon O. Chua
Wave Momentum and Quasi-Particles
in Physical Acoustics
Gérard A Maugin
Martine Rousseau
Université Pierre et Marie Curie, Paris, France
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World Scientific Series on Nonlinear Science Series A — Vol. 88
WAVE MOMENTUM AND QUASI-PARTICLES IN PHYSICAL ACOUSTICS
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Preface
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.
Inacrystallinelatticeoutsidetheabsolutezero,randommotiontakesplace
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
thatcanessentiallybeunderstoodintermsofaparticleor“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
fluidoradeformablesolid2. The alliedquasi-particleinterpretationisthen
1B. R. Wheaton (1983). The tiger and the shark: Empirical roots of wave-particle
dualism,CambridgeUniversityPress,UK.
2Theinterestedreadermayconsulttheshorthistoricalperspectivegivenbyoneofthe
authors: G.A.Maugin(2011),Solitonsinelasticsolids(1938–2010),MechanicsResearch
Communications,38,pp. 341–349.
v
February18,2015 11:39 WorldScientificBook-9inx6in MauginRousseau Chp0 pgvi
vi Wave Momentum and Quasi-particles inPhysical Acoustics
particularly well adapted and rapidly gives rise to the notion of soliton.
This notion is a materialisation of the wave that carries the information.
Inthepresentwork,westudythequasi-particlesthataredual—inthe
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 fromthe potentially associatedsimple interpretationof
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 mechan-
ics. This methodology is easily understood. Standard equations (here field
equations ofelasticity andassociatedboundary conditions)areused to ob-
tain the dynamical solution (e.g., the celebrated Rayleigh wave for surface
propagation). Then another set of continuum equations (so-called conser-
vationlawsinthesenseofNoether’s theorem offieldtheory)isused—such
as in a post-processing — to build some other quantities, here those that
will principallyappearinthe conservationofso-calledwave-momentum. It
isthatequation,andpossiblyinparalleltheenergyequation,whichisinte-
grated overa volume element that is representativeof the consideredwave
motion(e.g., a verticalbandofwidth equalto one wavelengthinthe sagit-
tal 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 wholeof “configurationalmechanics”inmoderncontinuumthermome-
chanics3— where it is applied to the evaluation of driving forces acting on
field singularities, material defects, inclusions, phase-transitionfronts, etc.
The first chapter should be viewed as a prolegomenon introducing the
inclusivenotionsofwavemomentum andradiativestresses inonedimension
of space in the line of L´eon Brillouin. This provides an occasion to pay a
tribute to this brilliant, sometimes underrated, physicist4. The notions of
3Seethebookbyoneoftheauthors: G.A.Maugin(2010). ConfigurationalMechanics,
(CRC-Chapman&Hall,BocaRaton,FL).
4L´eon Brillouin(1889–1969) isaFrench-American physicistof the Nobel-prizecalibre
whose name is attached to many remarkable advances (quantum theory of solids, Bril-
louin scattering, Brillouin zones, WKB method, Brillouin–Wigner formula, notion of
neg-entropy). He is the physicist who thoroughly clarified the notion of group velocity
indispersivesystems (Cf. hisbookwithA.Sommerfeld,WavePropagation andGroup
Velocity, New York, Academic Press, 1960). He isalso the author of aremarkable pio-
February18,2015 11:39 WorldScientificBook-9inx6in MauginRousseau Chp0 pgvii
Preface vii
EulerianandLagrangiandescriptionsare introducedfor this purpose. The
contentsofthischapterowemuchtoourfruitfulcollaborationwiththelate
Alexander Potapov from Nizhny–Novgorod, a famous school of nonlinear
physics. Thebasesofcontinuumthermomechanicsaredealtwithingreater
detail and more precise mathematical format in three-dimensions of space
in Chapter 2 with the necessary reminder on finite-strain and small-strain
elasticity theories. Following along the same line, Chapter 3 introduces
the general notions of pseudo-momentum and Eshelby stress — i.e., the
time-like component and the spatial part of an energy-momentum tensor
— by application of Noether’s theorem of field theory when a variational
formulation is at disposal (e.g., in pure elasticity). However, a procedure
mimickingNoether’sidentityisalsoproposedinthepresenceofdissipation
(such as viscosity). Chapter 4 deals with the notion of action (energy
multiplied by time) — seldom considered locally in continuum mechanics
— and elements of the wave mechanics expanded by Lighthill, Whitham
and Hayes, and recently revisited by one of the authors. This exploits
perturbation schemes.
Trueapplicationsofthenotionofquasi-particlesassociatedwithelastic
wave modes startin Chapter 5 with the problem of transmission-reflection
at material interfaces, whether perfect or imperfect. Chapter 6 follows
with the application to so-called dynamic materials. The latter have the
essential property of presenting heterogeneities in both space and time.
That is, in addition to the presence of material discontinuities reflecting
abrupt changes in inertial and elastic properties, these materials are also
subjected to more or less abrupt changes in time such as due to the action
of a practically sudden import of energy through an external action. Such
systems are thermodynamically open. The phenomenon of interest is very
much like the pumping of energy in lasers. The point of view of quasi-
particles sheds an interesting light on this phenomenon where the wave-
like picture is somewhat artificial. Chapters 7 and 8 deal with a subject
of importance in seismology and electromechanical devices: surface waves
of both elastic and electro-elastic nature. Special emphasis is placed in
Chapter 7 onthe casesof Rayleigh waves (evenwhen perturbed by surface
energyoranexternalfluid), Lovewaves(multimode dispersivewaves),and
so-called Murdoch surface waves which correspond to the propagation of
neering book on Tensors in mechanics and elasticity (FirstFrench edition, Paris, 1938;
Dover reprint, New York, 1946). A detailed biography and list of publications were
givenbyL.HillethThomas,“L´eonNicolasBrillouin1889–1969”, BiographicalMemoir,
NationalAcademyofSciences oftheUSA,pp. 68–89,Washington, DC,1985.
February18,2015 11:39 WorldScientificBook-9inx6in MauginRousseau Chp0 pgviii
viii Wave Momentum and Quasi-particles inPhysical Acoustics
a mono-mode shear surface wave. The latter mode can be viewed as a
limit of one of the Love surface acoustic modes when the superimposed
layer has a thickness that tends to zero while keeping its essential kinetic
properties. This limit property is also kept for the view of the associated
quasi-particle properties. The inclusion of Chapter 8 may be due to an
immoderate tasteofthe authorsfor electro-magneto-mechanicalcouplings.
But this chapter, by selecting a rather simple case of coupling (so-called
Bleustein–Gulyaev waves), provides an occasion to illustrate the possible
influence of a nonlinearity in the elasticity of the substrate, and also of a
possible viscosity of this substrate. This last case is of particular interest
becauseityieldsanon-inertialmotionoftheassociatedquasi-particlewitha
drivingforceinterpretedasfriction. Globally, thesetwochaptersintroduce
the notion of quasi-particles guided by a bounding surface.
Chapter9offersadifferentperspectivebyunfoldingwhathappenstothe
notions of wave-momentumand associatedquasi-particlesin whatare now
referred to as generalized continua. These include three different types of
generalization of standard (Cauchy–Navier–Green) elasticity: a weak non-
localityaccountingforhigherordergradientsoftheelasticdisplacement,an
internal microstructure that accounts for additional degrees of freedom of
rotation(suchasin Cosseratcontinua), andstrongnonlocalityrepresented
byspace-functionalelasticityconstitutiveequations. Thislastpointinfact
bringsusbacktothediscretevisionofcrystallinestructures. Finally,Chap-
ter10, althoughnotintendedasanadditionaltreatiseonsolitons,presents
a few originalcaseswhichthen includesimultaneously bothdispersionand
nonlinearity, by generalizing some of the cases already exposed in Chap-
ters 7 and 8. The corresponding results have often been obtained in close
co-operationwithotherresearchersintheperiod1985-2002,especiallyJo¨el
PougetandHichemHadouaj(inParis),BorisMalomed(nowinIsrael),and
the late Christo I. Christov (Bulgaria, and Lafayette, Louisiana, USA).
Inall,weexpectthatthecontentsofthiswork,quiteoriginalandsome-
whatthought-provoking,shouldbeofinteresttocuriousphysicists,applied
mathematicians, and mechanicians, as we think that they broaden insuffi-
cientlystudiedavenuesinthephysicsandmathematicsofwavepropagation
in deformable bodies.
The final presentation of the text and the beautiful figures are due to the
expertise of Ms Janine INDEAU,whom we thank from the bottom of our heart
for herkind and efficient co-operation.
G´erard A. Maugin and Martine Rousseau
February18,2015 11:39 WorldScientificBook-9inx6in MauginRousseau Chp0 pgix
Contents
Preface v
1. Prolegomena: wave momentum and radiative stresses in
1D in the line of Brillouin 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 One-dimensional motion in the Eulerian description. . . . 3
1.2.1 Basic equations . . . . . . . . . . . . . . . . . . . 3
1.2.2 Method of perturbations . . . . . . . . . . . . . . 5
1.2.3 First-order approximation. . . . . . . . . . . . . . 5
1.2.4 Second-order approximation . . . . . . . . . . . . 6
1.2.5 Example of momentum and radiative stress in a
thin rod . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 One-dimensional motion in the Lagrangiandescription . . 11
1.3.1 Basic equations . . . . . . . . . . . . . . . . . . . 11
1.3.2 Perturbationanalysis at the first-orderof approxi-
mation . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Perturbation analysis at the second order of ap-
proximation . . . . . . . . . . . . . . . . . . . . . 13
1.4 Summary and concluding remarks . . . . . . . . . . . . . 14
2. Elements of continuum thermomechanics 19
2.1 Material body . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Balance laws of the thermomechanics of continua . . . . . 23
2.2.1 Global balance laws in the Euler–Cauchy format . 23
2.2.2 Euler–Cauchy format of the local balance laws of
thermomechanics. . . . . . . . . . . . . . . . . . . 25
ix