The geometry of the guidein structure is shown in Fig. 1. A two-dimensional
Figure 1: Geometry of the guiding system in an adhesive layer of thickness 2h
with an interface of thickness 
.
case is considered, so that the guided wave has no x dependence leading to
. The wave is propagating along the z direction
through a three-layer sandwich with the middle layer representing the adhesive
layer, indicated by the index 0. The index m denotes the interface between
the adhesive and adherend layers.
The wave equation for elasticity theory, for homogeneous isotropic media, in
terms of the displacement 
, is [10]
where we have assumed that there are no body forces. 
and 
are
the Lamè elastic constants, and 
is the density. The displacement
field may be expressed as the sum of an irrotational component, grad 
,
and a solenoidal component, curl 
where 
. This solution contain one part, 
,
for longitudinal waves, and the other part, 
, for
transversal waves. Substituting Eq. (2)
into Eq. (1), it can be shown that and satisfy
the wave equation
where 
is the longitudinal wave velocity and 
is the shear wave
velocity in bulk media. The solutions are assumed to be time harmonic of the
form 
with z dependence of the form 
. The
propagation constant k has the same value for both potentials in order to
satisfy the boundary conditions along z. Those assumptions lead to an
incident wave with vertical polarization,
, thus
where 
, 
. 
and
are the bulk propagation constants given by 
and
respectively. The solution to the wave equation has the form
and 
. One seek for solutions of true guided
modes in the middle layer, which will be referred to as the adhesive layer.
The potentials take the form [9],
for |y|<h
for y>h
for y<-h
where the subscript denotes the region of interest. The time harmonic term is
omitted in the following discussion. The normalized velocity is defined as
where V is the phase velocity of the wave
( 
). The normalized velocity has an
upper limit determined by 
,
which in our case reduces to 
. and are
the bulk longitudinal and shear velocities respectively. The displacements
and stresses are described in terms of the displacement potentials so that the
solutions in each region can be calculated directly.
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