Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 3, pp. 797-805, Warsaw 2012

50th Anniversary of JTAM

WAVE-QUASI-PARTICLE DUALISM IN THE TRANSMISSION-REFLECTION

PROBLEM FOR ELASTIC WAVES

Gérard A. Maugin, Martine Rousseau

Université Pierre et Marie Curie – Paris 6, Institut Jean Le Rond d’Alembert, Paris, France

e-mail: gerard.maugin@upmc.fr

Following along the line of recent works in which the notion of quasi-particles associated
with surface acoustic waves of different types was introduced via canonical conservation
laws, here the emphasis is placed on the possible exploitation of this dualism in the classical
problem of the transmission and reflection of waves by a discontinuity between two media
(in perfect contact orwith possible delamination) and themore general case of amono-layer
or multi-layer sandwiched slab.

Key words: elastic waves, quasi-particle, transmission-reflection

1. Introduction

In recent works (Maugin and Rousseau, 2010a,b; Rousseau andMaugin, 2011), a kind of wave-
-particle dualism has been constructed for elastic waves in acoustic physics. The idea is close
to that of phonons, but we only need to know the analytical wavelike solution to the continu-
um problem in order to build this dualism. For practical reasons related to a potential use in
Non Destructive Evaluation techniques, the emphasis was placed on SAWs (Surface Acoustic
Waves) propagating on the top of an elastic half-space with various boundary conditions on the
limiting plane and various physico-mechanical couplings (e.g., in electroelasticity). What must
be retained from these studies is (i) the exploitation of the notion of pseudo-momentum (in the
sense of Peierls (1991, pp. 30-42) orwave momentum in themanner of Brenig (1955) and in the
spirit of the dynamical theory of configurational forces (Maugin, 2011) – and the accompanying
application of Noether’s theorem for conservation laws, (ii) the fact that the quasi-particle thus
associatedwith thewave-likemotionhas amomentumobtainedby integration of thisBrenigmo-
mentum (or its canonical generalization in amulti-field theory) over a volume representative of
thewavemotion (for a SAW,onewavelength in thepropagation direction, onepenetrationdepth
in depth in the substrate, and one unit laterally), and (iii) only wave modes were considered,
i.e., independently of any initial-condition or interaction problem. Themotion of quasi-particles
thus obtained is “Newtonian” and inertial in the absence of dissipative effects. The associated
energy is purely kinetic in form, while in the original continuum problem it could be of mixed
origin, including, e.g., kinetic, elastic, electromagnetic, and electro-magneto-elastic interactions.
Still the “mass” associated with the said quasi-particle accounts for all these origins as well as
for the type of boundary condition and depth behaviour.
In the present work, relying on a simple case of the wave – one elastic component of the SH

(shear-horizontal) type – and considering the normal incidence of such a wave on the interface
between two elastic continua characterised by their own elastic (shear) coefficient and mass
density, we revisit the problem of the transmission and reflection of such awave at this interface
in terms of the wave properties (this is only a reminder of well known results) or those of the
associated quasi-particle properties (this is the new contribution). The situation considered is
that which prevails in the potential exploitation in nanoscale systems (for macro-systems this



798 G.A.Maugin, M. Rousseau

would be the realm of potentially dangerous seismic waves). In the present case, discarding
the variation of amplitude with depth we are satisfied with considering a plane face wave. The
representative volume element of the wave modes then is one wavelength in the propagation
direction and a unit square in the transverse plane.

2. Reminder on the wavelike picture

For the sake of simplicity, we consider a one-dimensional wave problemalong the x-direction for
an elastic displacement u that may be a transverse shear-horizontal (SH)mode. The governing
wave equation is given by

∂p

∂t
−
∂σ

∂x
=0 p= ρ

∂u

∂t
σ=µ

∂u

∂x
(2.1)

where ρ and µ are a prescribedmatter density and fixed elasticity (e.g., shear) coefficient µ in
a given region of space. Space and time partial derivatives are alternately denoted by a comma
followed by x or a t. The first of (2.1) follows in a variational format from the Lagrangian
density per unit volume

L=
1

2
ρu2,t−

1

2
µu2,x (2.2)

We consider a harmonic wave motion

u=U cos(kx−ωt) (2.3)

so that from (2.1) we have the trivial dispersion relation

D(ωk)=ω2−c2k2 =0 c=

√

µ

ρ
(2.4)

2.1. Transmission-reflection problem for a perfect interface

Thestandard transmission-reflectionproblemhere consists in consideringanormally incident
wave inmedium1(properties ρ1 and µ1) on the interface at x=x0 =0that separatesmedium1
frommedium 2 (properties ρ2 and µ2). Thewave solution inmedium 1 is sought as the sum of
an incident component and a reflected component, while the solution inmedium 2 consists in a
transmitted component.With an obvious notation, we have thus

u1 =uI +uR u2 =uT (2.5)

with

u1 =U cos(k1x−ωt)+R0U cos(k1x+ωt) u2 =T0U cos(k2x−ωt) (2.6)

where R0 and T0 are the reflection and transmission coefficients for this perfect interface case
for which we assume the continuity of displacement and stress, i.e.

u1 =u2 µ1u1,x =µ2u2,x at x=0 (2.7)

The solution of this system is, for any amplitude U, the value of the reflection and transmission
coefficients as

R0 =
z1−z2
z1+z2

T0 =
2z1
z1+z2

(2.8)

where zα = ραcα,α=1,2 are impedances.With these we check the conservation of energy flux
in the well known form

F0 =1−R
2
0−
z2
z1
T20 ≡ 0 (2.9)



Wave-quasi-particle dualism in the transmission-reflection problem... 799

2.2. Transmission-reflection problem for an interface with delamination

In view of applications to non-destructive (NDT) evaluation methods, it is of interest to
consider the case of an imperfect interface that admits the possibility of delamination and for
which matching conditions (2.7) are replaced by the conditions (known as Jones’ conditions
(Jones andWhittier, 1967))

σ1 =σ2 ≡K[[u]] (2.10)

where K is a positive coefficient characterising the degree of delamination and the symbol [[·]]
means the jumpof its enclosure, i.e., [[u]] =u2−u1 at x=0.Wemust look for complex solutions
of the type u=Aexp[i(kx−ωt)]. Conditions (2.10) yield the system

z1(1−R)= z2T z2T =−i
K

ω
(1+R−T) (2.11)

where R and T are the complex reflection and transmission coefficients corresponding to this
imperfect case. The solution to (2.11) reads

R=
z1z2− i(K/ω)(z1 −z2)

z1z2− i(K/ω)(z1 +z2)
T =

−2i(K/ω)z1
z1z2− i(K/ω)(z1 +z2)

(2.12)

By computing the squares of themoduli of the complex quantities R and T , we check that (2.9)
is replaced by

FK =1−|R|
2−
z2
z1
|T |2 ≡ 0 (2.13)

We note that

FK =F0
(

1−
z21z
2
2

z21z
2
2 +(K/ω)

2(z1+z2)2

)

(2.14)

The solution of this imperfect interface case is characterised by the parameter K/ωwhich shows
the role played by the frequency ω. The limit case K →∞ corresponds to the perfect interface
for which (2.9) holds true. The limit case K → 0 corresponds to full delamination (no more
transmission and complete reflection: T =0,R=1).

Remark 2.1. It is possible to obtain an estimate of the K coefficient bymeasuring the ampli-
tude of reflected or transmittedwaves. In particular, whenevermedia 1 and 2 are identical,
but K still is not zero, this measure provides a means of determining the presence of an
internal delamination in the body.

3. Associated quasi-particle picture

With field equation (2.1), there are associated (via the application of Noether’s theorem or by
direct manipulation) conservation laws of energy and wave momentum in the local form in a
homogeneous medium

∂H

∂t
−
∂Q

∂x
=0

∂P

∂t
−
∂b

∂x
=0 (3.1)

where the energy orHamiltonian per unit volume H, the energy flux Q, thewavemomentum P
and (here reduced to a scalar) the Eshelby stress b are defined by (see Maugin and Rousseau
(2010a) for the canonical definitions in three dimensions)

H =E+W =
1

2
ρu2,t+

1

2
µu2,x Q=σut =µu,xu,t

P =−ρu,tu,x b=−(L+σux)
(3.2)



800 G.A.Maugin, M. Rousseau

The first of these last two follows Brenig’s definition. It is a remarkable – but sometimes
misleading- fact that in this one-dimensional case there hold the following identities

Q=−c2P b=−H (3.3)

Now the concept of quasi-particle is introduced by integrating conservation equations (3.1) over
a volume that is representative of the present wave motion, i.e., over one wavelength in the x
propagation direction and a square section of sides equal to unity in the transverse direction.
This introduces averages notedwith the symbolism 〈·〉. For solutions of type (2.3) this procedure
yields the following results

〈L〉=0
d

dt
〈P〉=0

d

dt
〈H〉=0 (3.4)

where, in a homogeneous medium

〈P〉= ρωπU2 ≡Mc 〈H〉=
1

2
Mc2 (3.5)

where the first of these defined the “mass” M = ρkπU2. Simultaneously, equations (3.4)2,3
show that the said quasi-particle has an inertialNewtonianmotion. It also satisfies theLebnizian
conservationof kinetic energy (or vis-viva). Indeed,while therearebothkinetic andelastic energy
at the continuum level, the energy of the associated quasi-particle is purely kinetic on account
of the given definition of “mass”. Note that homogeneous equations (3.4)2,3 hold good because
of the periodicity at the two ends of the integration interval that makes the contributions from
Q and b to vanish. Equation (3.4)1 holds true because it is shown in computing the average
for solutions (2.3) that the result is proportional to “dispersion” relation (2.4) and therefore
vanishes identically, a result known in the kinematic-wave mechanics of Lighthill andWhitham
(see, e.g., Whitham, 1974). Note finally that all quantities introduced in (3.4) and (3.5) are
proportional to the square of the amplitude of the wave, hence are all of energetic nature.

3.1. Transmission-reflection problem in the quasi-particle picture

Considering first the case of the perfect interface at x = 0, we can associate one quasi-
particle with each wave component of the problem. With an obvious notation, we have the
following “masses”

MI = ρ1k1πU
2 MR = ρ1k1πR

2
0U
2 MT = ρ2k2πT

2
0U
2 (3.6)

The corresponding averaged wave momenta are given by

PI ≡〈PI〉= ρ1ωπU
2 (3.7)

and

PR ≡〈PR〉=−ρ1ωπR
2
0U
2 PT ≡〈PT〉= ρ2ωπT

2
0U
2 (3.8)

where we account for the fact that the averaged wave momentum PR is oriented towards nega-
tive x’s.
We note ∆M and ∆P the possible misfits in mass andmomentum defined by

∆M := (MR+MT)−MI (3.9)

and

∆P =(|PR|+ |PT |)−|PI| (3.10)



Wave-quasi-particle dualism in the transmission-reflection problem... 801

where the symbolism | · | refers to the absolute value of its enclosure. That is, we are comparing
the strengths of the momenta and, therefore, we are not performing a vectorial balance.
Similarly, for the kinetic energy of the associated quasi-particles:

∆H =(HR+HT)−HI (3.11)

We say that a quantity is conserved during the transmission-reflection problem if the correspon-
dingmisfit vanishes.
By applying definition (3.5)2, one immediately shows that

∆H =
(1

2
z1ωπU

2
)

F0 (3.12)

where F0 has been defined in (2.9). But the latter vanishes. Accordingly, ∆H ≡ 0: kinetic
energy is conserved in the transmission-reflection problem seen as a quasi-particle process that
may be qualified of Leibnizian (conservation of vis-viva). Historically, it was in fact the collision
problem between particles that led the late 17th and early 18th century scientists (prominently
Leibniz in 1686; see Smith, 2006) to introduce the vis-viva – twice the actual kinetic energy –
as the “physical quantity” to be conserved in such interactions and not Descartes’ “quantity of
motion” as suggested by a different school (see below for the particle moment in the present
problem).
Indeed, “mass” is not generally conserved in the present problem as it is immediately shown

that

∆M = ρ2k2πT
2
0U
2
(c21− c

2
2

c21

)

(3.13)

Similarly, on computing ∆P it is obtained that

∆P = ρ2k2πT
2
0U
2
(c1c2− c

2
2

c1

)

(3.14)

This shows that

∆P =
( c1c2
c1+ c2

)

∆M (3.15)

so that ∆P and ∆M always are in the same sign. Inparticular, ∆M > 0 if c1 >c2 and ∆M < 0
if c1 <c2; ∆M =0 if and only if c1 = c2, that is, if there is no interface or for the extraordinary
case where the two media have elasticity coefficients and densities in the same ratio; ∆P and
∆M also vanish in the same conditions, i.e., we can say that P and M are conserved in the
same conditions, in particular when media 1 and 2 have the same characteristic elastic speed.
This is also true whenmedium 2 is a vacuum (ρ2 → 0) or is perfectly rigid (µ2 →∞), cases for
which there is perfect reflection (R20 = 1) from the wavelike viewpoint or perfect (i.e., elastic)
rebound from the particle-like viewpoint.

3.2. Case of an imperfect interface with possible delamination

Remarkably enough, there is no need to redo the computations. It suffices to replace the
transmission and reflection coefficients of the perfect case by the moduli of the new complex
coefficients. Thus, (3.12) is replaced by

∆H =
(1

2
z1ωπU

2
)

FK (3.16)

and this again vanishes. Similarly, (3.13) and (3.14) hold with T20 replaced by |T |
2 while (3.15)

remains unchanged, noting that the coefficient c1c2/(c1+c2) does not depend on K.



802 G.A.Maugin, M. Rousseau

Remark 3.1. In the case when K 6=0 butmedia 2 and 1 are identical, the presence of the K
spring distribution can simulate a homogeneous surface damage. In this case, both ∆M
and ∆P vanish so that K is no longer involved.Thedependence on K shows only through
the value of any of MR,MT , PR and PT .

Remark 3.2. In the case when K 6=0 but media 1and 2 are different, both ∆M and ∆P do
notvanish.This result is notdueto thepresenceof the K springdistribution; it canonlybe
attributed to the difference in characteristic wavelength in bothmedia. Thesewavelengths
depend on the integration length of propagation in each medium. This hypothesis being
accepted, the value of K can also be evaluated from ∆M and ∆P. As the surface damage
increases (i.e., as K decreases), themass of the reflected quasi-particle increases and that
of the transmitted quasi-particle decreases compared to the fully undamaged case.

4. Case of a sandwiched slab

Here we consider the case of an elastic slab (medium 2) of thickness d situated between x=0
and x= d and sandwiched between twomedia of elastic type 1. The propagation is from left to
rightwith the reflection coefficient R in leftmedium1and the transmission coefficient T in right
medium 1.We assume that d≫λ2, where λ2 is the (elastic wave) characteristic wavelength of
medium 2, so that the association of quasi-particle properties makes sense in the slab.We need
not to reconsider thewavelike solution.Weare satisfiedwith applying the results of the foregoing
section to the two interfaces at x=0 (transition 1→ 2) and at x= d (transition 2→ 1). Just
as before MI,MR,MT ,PI,PR and PT aremasses andmomenta granted to the quasi-particles
in left and right regions 1.We obviously have (compare (3.6))

MI ∝U
2 MR ∝ |R|U

2 MT ∝ |T |U
2 |R|2+ |T |2 =1 (4.1)

and in an obvious notation

∆M11 =(MR+MT)−MI =0 (4.2)

As to the momenta, we have (cf. Section 3)

∆P11 =(|PR|+ |PT |)−|PI| ≡ 0 (4.3)

Within the slab we distinguish between the particle momentum P+ with mass M+ associated
with the right motion and particle momentum P− with mass M− associated with the left
motion.

Thus, for the interface x=0we can write

∆M1→2 := (M
++M−+MR)−MI =(M

++M−)−MT 6=0

∆P1→2 =(|P
+|+ |P−|+ |PR|)−|PI|=(|P

+|+ |P−|)−PT 6=0
(4.4)

while for the interface x= dwe put down similarly

∆M2→1 =MT − (M
++M−)=−∆M2→1 6=0

∆P2→1 = |PT |− (|P
+|+ |P−|)=−∆P1→2 6=0

(4.5)

Note that

(|P+|+ |P−|)=∆P1→2+ |PT | (4.6)



Wave-quasi-particle dualism in the transmission-reflection problem... 803

and, obviously

R≡R1→2→1 T =T1→2→1 (4.7)

These two coefficients can be computed in terms of themechanical properties of media 1 and 2.
A standard calculation involving the matching conditions at the crossing of the two interfaces
separated by the distance d yields

T =
1

cos(k2d)−
i
2

(

z1
z2
+ z2
z1

)

sin(k2d)

R=
− i
2

(

z1
z2
− z2
z1

)

sin(k2d)

cos(k2d)−
i
2

(

z1
z2
+ z2
z1

)

sin(k2d)

(4.8)

Remark 4.1. It is readily checked that these two formulas comply with the conservation law
|R|2+ |T |2 =1. Of course we also check that T =1 and R=0 if d≡ 0 (perfect contact).
As to the condition d≪λ2, it does notmake sense for the introduction of a quasi-particle
in the slab.

Remark 4.2. The typeof formalism,bookkeeping and“algebra” just introduced canbeapplied
to the more complicated case where the sandwiched slab is made of a number n− 1 of
perfectly elastic layers (each with its own elastic properties) numbered i = 2, . . . ,n, in
perfect contact. That is, themedium left to the slab and themedium right to it aremade
of the same material (material 1). The estimates in (4.1) still hold good. This is also the
case of (4.2). Same as in (4.4)1, we will write

∆M1→2 =(M
+
2 +M

−

2 +MR)−MI 6=0

∆M2→3 =(M
+
3 +M

−

3 )− (M
+
2 +M

−

2 ) 6=0
(4.9)

and so on

∆Mi→i+1 =(M
+
i+1+M

−

i+1)− (M
+
i +M

−

i ) 6=0

· · ·

∆Mn−1→n =(M
+
n +M

−

n )− (M
+
n−1+M

−

n−1) 6=0

∆Mn→1 =MT − (M
+
n +M

−

n ) 6=0

(4.10)

Here superscripts + and − indicate masses associated with particle motions to the right
and left, respectively. On summing over all equations (4.9) through (4.10) we re-obtain
(4.2). A similar bookkeeping can be done for themomentum transfers ∆P11 (cf. equation
(4.3)) and ∆Pi→i+1. We also note that, globally

R=R1→...→1 T =T1→...→1 (4.11)

where the dots stand for the (n− 1) layers numbered (2, . . . ,n). In principle, global
reflection and transmission coefficients (4.11) can be computed in terms of the individual
properties of the layers, but this is not of an immediate interest.

5. Conclusion

Theabove reportedanalysis shows that, in the transmission-reflectionproblemwithin the frame-
work of purely elastic material regions, the resultingmechanics of the associated quasi-particles



804 G.A.Maugin, M. Rousseau

results in an exchange of linearmomentumwithoutanyglobal loss of itwhile there also is nomis-
fit in “mass”, but themasses depend on the region of propagation, i.e., the various propagation
properties.

Interestingly enough, there is no misfit in the kinetic energies of the quasi-particles – cf.
equations (3.12) and (3.16). This is a direct consequence of the acoustic (wavelike) energy con-
servation represented by equations (2.9) and (2.13). There is no principle difficulty in dealing
with the slab problemand its extension to amulti-layered interface. But theremay be computa-
tional difficulties. The case of delamination at an interface fits in the picture, and the properties
of this delamination (spring coefficient K) can be deduced from the “misfits” in mass and
momentum of the associated quasi-particles.

Onemaywonderwhat happens if some of the involvedmedia of propagation are viscoelastic.
The wavelike problem of an interface between two media was studied a long time ago (see
Mackenzie, 1960, and references therein), at least for a small perturbing viscosity. Insofar as
the concept of associated quasi-particle is concerned, for a propagation of surface SH waves in
an infinite continuum along the propagation direction we have shown elsewhere (Rousseau and
Maugin, 2012) the complexity and originality of the result. Not only does the viscous damping
result in a friction force as a source in the equation ofmotion of the associated quasi-particle, but
the “mass” itself of this so-called particle is found to vary in time. One can easily imagine the
analytical difficulties to be met in treating the case of a viscoelastic slab sandwiched between
two identical elastic or viscoelastic media. The limit case of the slab thickness going to zero
should provide the case of an absorbing interface of vanishing thickness thus re-instating the
notion of coefficient of restitution.

References

1. Brenig W., 1955, Bessitzen Schallwellen einen Impulz, Zeit. Phys., 143, 168-172

2. Jones J.P.,Whittier J.S., 1967,Waves at a flexibly bonded interface, J. Appl. Mech. (ASME).,
34, 4, 905-908

3. MackenzieK.V., 1960,Reflectionof sound fromcoastalbottoms,Journal of theAcoustical Society
of America., 32, 2, 221-231

4. Maugin G.A., 2011, Configurational Forces: Thermomechanics, Physics, Mathematics and Nu-
merics, CRC/Chapman&Hall/Taylor and Francis, Boca Raton, FL

5. Maugin G.A., Rousseau M., 2010a, Bleustein-Gulyaev SAW and its associated quasi-particle,
Int. J. Engn. Sci., (Special issue honouring K. Rajagopal)., 48, 1462-1469

6. Maugin G.A., Rousseau M., 2010b, Grains of SAWs: Associating quasi-particles to surface
acoustic waves, Intern. Symp. in the Honour of V.L. Berdichevsky, Bochum, July 2010; to be
published in Int. J. Engng. Sci., special issue, 2012

7. Peierls R., 1991,More Surprises in Theoretical Physics, Princeton University Press, Princeton,
N.J.

8. Rousseau M., Maugin G.A., 2011, Rayleigh surface waves and their canonically associated
quasi-particles,Proc. Royal Soc. Lond., A467, 495-507

9. Rousseau M., Maugin G.A., 2012, Influence of viscosity on the motion of quasi-particles asso-
ciated with surface acoustic waves, Int. J. Engng. Sci., 50, 10-21

10. Smith G.E., 2006, The vis viva dispute: A controversy at the dawn of dynamics,Physics Today,
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11. Whitham G.B., 1974,Linear and Nonlinear Waves, Interscience-Wiley, NewYork



Wave-quasi-particle dualism in the transmission-reflection problem... 805

Dualizm falowo-quasi-korpuskularny w zagadnieniu przesyłania i odbicia fal sprężystych

Streszczenie

Idąc w ślad za ostatnimi publikacjami, w których użyto pojęcia tzw. quasi-cząstek wprowadzone-
go poprzez zastosowanie kanonicznych praw zachowania w odniesieniu do powierzchniowych fal aku-
stycznych różnego rodzaju, w prezentowanej pracy położono nacisk na możliwe wykorzystanie dualizmu
falowo-quasi-korpuskularnegow klasycznym zagadnieniu przesyłania i odbicia fal na nieciągłości pomię-
dzy dwoma ośrodkami (doskonale połączonych lub zdelaminowanych) i w ogólniejszymprzypadku jedno-
lub wielowarstwowej płycie typu sandwich.

Manuscript received January 7, 2012; accepted for print February 21, 2012