Wave Motion 39 (2004) 93–110Non-dis

Wave Motion 39 (2004) 93–110
Non-dispersive and dispersive electromagnetoacoustic
SH surface modes in piezoelectric media
Maurizio Romeo∗
D.I.B.E. Università, via Opera Pia 11/a, 16145 Genoa, Italy
Received 26 March 2003; received in revised form 28 May 2003; accepted 11 July 2003
Abstract
This paper deals with a detailed analysis of the electromagnetic problem for the propagation of shear horizontal (SH) surface
waves in a non-conducting piezoelectric half-space. The compatibility equations are solved in both cases of grounded surfaces
and surfaces matched with an external potential. It is shown that a unique quasi-acoustical mode exists in non-dispersive media
and the corresponding solution is worked out in a closed form having recourse to a method of complex analysis. Dispersive
half-spaces are considered accounting for a one-resonance model and the dispersion equation is numerically solved for a
matched surface. Two types of admissible surface waves are found in different frequency ranges. One mode generalises the
well-known Bleustein–Gulyaev wave, while the second one is a quasi-electromagnetic mode occurring above the characteristic
frequency of the model. Results are compared with those obtained by the quasi-static approximation.
© 2003 Elsevier B.V. All rights reserved.
1. Introduction
Since their theoretical prediction[1,2], shear horizontal (SH) surface waves on a piezoelectric half-space have been
treated under the quasi-static approximation. This amounts to neglect the effects of rapidly varying electromagnetic
fields, thus allowing to write the electric field as the gradient of a scalar potential (see, for example, Chapter 4 in[3]).
Such an approach is widely justified in a large part of actual applications concerning surface wave technology since
electromagnetic effects are often irrelevant in determining the propagation properties within the frequency ranges
of interest[4]. Of course, the quasi-static approximation results into a simplification of the governing equations for
boundary values problems in both linear and non-linear electroelasticity[5,6]and, in particular, it allows to obtain
explicit expressions for wave speed and attenuation within the linear theory.
However, recently it has been pointed out that the quasi-static approximation is not necessary in order to derive
explicit solutions to the dispersion equation[7]. This is surely true in the case of an half-space whose boundary
is electrically grounded. For boundaries matched with an external potential, full electromagnetic solutions can be
obtained as the roots of an irrational equation. The same holds for solutions which generalise the classical interfacial
wave problem[8]. In the work of Li[7], it is argued that more than one surface wave can exist and that electromagnetic
modes could be found as solutions of the irrational equation. Nevertheless, until now, no attempts have been made
to derive such solutions. In particular, the aforementioned dispersion equations must be complemented with the
∗
Fax:+39-010-353-2777.
E-mail address:romeo@dibe.unige.it (M. Romeo).
0165-2125/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.wavemoti.2003.07.005
94 M. Romeo / Wave Motion 39 (2004) 93–110
additional requirement that the amplitude decay away from the surface. Hence, the roots of the dispersion equation
do not necessarily correspond to admissible surface waves.
In addition, a relevant question arises in connection with the more realistic setting of dispersive dielectric materials.
In this case memory effects due to non-instantaneous electric response of the solid medium may be relevant (see
[9,10]). Accounting for time dispersion, one expects transition regions to occur in the frequency domain, where
speed, attenuation and penetration depth of the surface modes may suffer noticeable changes. As a consequence,
the admissibility condition could not be satisfied for some ranges of frequencies.
Due to these motivations, the present work is addressed to the solution of the dispersion equations in the full
electromagnetoelastic case for both electrically grounded or matched surfaces. The main task is a detailed analysis
of solutions compatible with the requirement of the amplitude’s decay. Concerning with matched surfaces, we
show that a closed form solution can be worked out in the case of non-dispersive media and that this solution
is unique. It generalises the well-known Bleustein–Gulyaev (B–G) wave and accounts for the slight influence of
the electromagnetic field. More interesting results are obtained in the dispersive case. Assuming that the dielectric
behaves as a Lorentz medium with a single resonance, we have found a quasi-acoustical SH surface wave at low
frequencies and a quasi-electromagnetic SH surface wave within a relatively narrow range of frequencies. Besides,
in contrast to the quasi-static approximation, no surface waves are admitted at high frequencies.
The plan of the paper is as follows. InSections 2 and 3, assuming that the sagittal plane is a plane of material
symmetry, we rederive, in the more general context of dispersive media, the compatibility conditions for SH surface
waves on a piezoelectric half-space obtained in[7]. In the case of matched surfaces, this conditions amounts to a set
of four irrational equations to be solved according to the admissibility requirements on the imaginary parts of the
wave vectors. A complex analysis method is used inSection 4to show the uniqueness of the solution in the case of
non-dispersive media and to express it in a closed form.Section 5is devoted to outline the procedure implemented in
the dispersive case, choosing a constitutive equation for the permittivity kernel based on the Lorentz one-resonance
model. Numerical results are then given inSection 6for both dispersive and non-dispersive half-spaces with different
boundary conditions. A comparison with dispersive quasi-static results is also discussed.
2. Piezoelectric dispersive half-space
We consider a homogeneous, anisotropic non-conducting and non-magnetisable dielectric solid which occupies
the half-space Bbounded by the plane surfaceS. We assume that, at least in a suitable range of frequencies, the
solid behaves like a dispersive electromagnetic medium where the dielectric response is non-local in time, while
the elastic response and the electromechanical coupling are free from time dispersive effects. More precisely we
account for a linear dependence of the electric displacement on the history of the electric field and suppose that,
for comparable times, the history of the mechanical field negligibly contributes to the current value of the electric
field. Accordingly we exploit the well-known linear theory of memory-dependent electromagnetic continua (see
Chapter 13 in[9]). Denoting, respectively, bye(x,t)andh(x,t)the electric field and the infinitesimal strain tensor,
we write the following constitutive equations for the electric displacementDand the Cauchy stress tensorT
D(x,t)=[e(x,·)](t)+Eh(x, t), (2.1)
T(x,t)=Gh(x,t)−e(x,t)E, (2.2)
whereGandEare, respectively, the elasticity (fourth-order) tensor and the piezoelectric (third-order) tensor. The
dielectric tensorin(2.1)is defined as the following linear integral operator on any vector fieldf(t)
[f(x,·)](t)=(0)f(x,t)+

∞
0


(τ)f(x,t−τ)dτ, (2.3)
where the dielectric permittivity kernel(t)is such that 

(t) →0 fort →+∞and where(0)represents the
instantaneous permittivity. The tensors G,E,comply with the usual symmetry conditions.Gis assumed to be
M. Romeo / Wave Motion 39 (2004) 93–110 95
positive definite in Lin(Sym) and(0)is taken to be positive definite in Sym (see [11]). Having recourse to the
second law of thermodynamics, it can also be shown that the half-range sine transform of the kernel derivative


(t)is positive definite in Sym onR
++
[12]. Since magnetisation is absent, the magnetic induction Bis given by
B(x,t)=µ0H(x,t), whereHis the magnetic field andµ0is the magnetic permeability.
In order to satisfy the divergence free condition onBand the Faraday’s law we introduce the scalar and vector
potentialsφ(x, t),A(x,t), such that
e=−∇φ−
∂A
∂t
, B=∇ ×A. (2.4)
According to(2.1), (2.2) and (2.4)the Ampère’s law and the equation of motion
∇×H=
∂D
∂t
,ρ¨ u=∇ ·T,
take the following form
∇(∇·A)−A=−µ0
∂
∂t



∇φ+
∂A
∂t

−E∇u

(2.5)
ρ¨ u=∇ ·(G∇u)+∇ ·

∇φ+
∂A
∂t

E

(2.6)
whereρis the mass density in the half-spaceB.
Now we assume that the solid admits a six-fold axis of material symmetryaparallel to the plane surfaceS.We
choose Cartesian reference axes{e1,e2,e3}such thate3lies alongaande2be normal to the surfaceSand directed
towards the interior of the half-space. As a consequence, the non-vanishing entries of the constitutive tensors reduce
to
G11=G22,G12,G13=G23,G33,G44=G55,G66=2(G11−G12),
E31=E32,E33,E24=E15,
11=22,33,
(2.7)
where the usual six-dimensional notationGαβ,Ekα,(α,β=1,2,... ,6;k=1,2,3), has been adopted for sym-metric pairs of indices.
In order to deal with waves propagating alonge1we assume thatu,φ,A, do not depend on x3. This means that
the sagittal plane coincides with the plane of material symmetry(e1,e2)and, in view of(2.7), Eqs. (2.5) and (2.6)
decouple into the following two set of integrodifferential equations
A2,12−A1,22=−µ0
∂
∂t

11

φ,1+
∂A1
∂t

−E15u3,1

,
A1,12−A2,11=−µ0
∂
∂t

11

φ,2+
∂A2
∂t

−E15u3,2

,
ρ
∂
2
u3
∂t
2
=G55u3+E15

φ+
∂A1,1
∂t
+
∂A2,2
∂t

,
(2.8)
A3=µ0
∂
∂t

33

∂A3
∂t

−E31(u1,1+u2,2)

,
ρ
∂
2
u1
∂t
2
=G11u1,11+G12u2,21+G66(u2,12+u1,22)+E31
∂A3,1
∂t
,
ρ
∂
2
u2
∂t
2
=G12u1,12+G11u2,22+G66(u2,11+u1,21)+E31
∂A3,2
∂t
,
(2.9)
96 M. Romeo / Wave Motion 39 (2004) 93–110
where commas denote partial differentiation with respect to the spatial variables.Eq. (2.9)represent the governing
system for the sagittal partus =u1e1+u2e2of the mechanical displacement an