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Shear horizontal waves in a nonlinear elastic layer overlying a rigid substratum

01 Oct 2017-Hacettepe Journal of Mathematics and Statistics-Vol. 46, Iss: 5, pp 801-815

Abstract: In this work, the propagation of shear horizontal (SH) waves in a homogeneous, isotropic and compressible nonlinear hyper-elastic layer having finite thickness is studied. The upper surface of the layer is assumed to be free from traction and the lower boundary is rigidly fixed. These waves are dispersive like the Love waves. The problem is examined by a perturbation method that balances the nonlinearity and dispersion in the analysis. A nonlinear Schrodinger equation is derived describing the nonlinear self modulation of the waves. Then, the effect of nonlinear properties of the material on the propagation characteristics and on the existence of solitary waves are discussed.

Summary (2 min read)

1. Introduction

  • Elastic waves propagating in an unbounded media are non-dispersive i.e. phase velocities of waves are constants.
  • Below some of these works will be reviewed to relate the present work to them.
  • The materials of the layer and the half space are both assumed to be homogeneous, isotropic and compressible hyper-elastic.
  • In [17], Ahmetolan and Teymur studied the propagation of nonlinear SH waves and the formation of Love waves in a double layered plate each having nite thickness.
  • An NLS equation is derived which governs the nonlinear self modulation of waves asymptotically and the e ect of nonlinearity on the propagation characteristics is discussed.

2. Formulation of the Problem

  • Let (x1, x2, x3) and (X1, X2, X3) be, respectively, the spatial and material coordinates of a point referred to the same rectangular Cartesian system of axes.
  • The summation convention on repeated indices is implied in (2.1) and in the sequel of this section, and Latin and Greek indices have respective ranges (1, 2, 3) and (1, 2).
  • Let us now assume that the constituent material of the layer is nonlinear, homogeneous, isotropic and compressible hyper-elastic.
  • For any material de ned by (2.5), if the rst two equations in (2.2) are satis ed by a given solution of the third equation in (2.2), then the motion (2.1) can exist in the medium in the absence of body forces.
  • ∂I3 = 0, whenever the invariants are given by (2.7).

3. Asymptotic analysis of the nonlinear SH waves

  • 0 is a small parameter which measures the weakness of the nonlinearity, {x0, t0, y} are fast variables describing the fast variations in the problem while {x1, x2, . . . , t1, t2, . . .} are slow variables describing the slow variations.
  • Now, rst writing the equation of motion (2.17), the boundary conditions (2.18) and (2.19) in terms of the new independent variables (3.1) and then employing the asymptotic expansion (3.2) in the resulting expressions and collecting the terms of like powers of in ε , the authors obtain a hierarchy of problems from which it is possible to determine un, successively.
  • Note that the dispersion relation (3.19) (or (3.21)) is the same of the dispersion relation for the antisymmetric motion of SH waves in an elastic isotropic plate with the thickness 2h occupying the region between the planes Y = h and Y = −h.
  • Note that, the rst order solution given in (3.27) and the solution of the linear problem are of the same form(see [29]).
  • The explicit form of the vector b (3) 3 is not given here, since it represents the third harmonic interactions and therefore in the sequel it will not be required.

4. Concluding remarks

  • The variation of C, Vg, Γ, ∆, and Γ∆ with the non-dimensional wave number K = kh for the rst three branches of the dispersion relation (3.21) are calculated and they are plotted in Fig(1), Fig(2), and Fig(3) respectively.
  • The NLS equation (3.67), as in this work, asymptotically describes the self modulation of the monochromatic plane waves in a nonlinear dispersive medium[30, 31].
  • This solution is known as envelope soliton or bright soliton [30, 31, 32].
  • Several investigator have shown that the shear stress is a nonlinear function of the strain in certain soils and nT < 0, that is the response of the soil is softening in shear (see e.g.[33] and references given there).
  • An experimental result about the observation of solitons in soil mechanics was reported in [34] by Dimitriu.

Acknowledgment

  • The authors Would like to thank the referee for the invaluable comments leading to improvements to this paper.

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Hacettepe Journal of Mathematics and Statistics
Volume 46 (5) (2017), 801 815
Shear horizontal waves in a nonlinear elastic layer
overlying a rigid substratum
Dilek Demirkus
and Mevlut Teymur
†‡
Abstract
In this work, the propagation of shear horizontal (SH) waves in a homo-
geneous, isotropic and compressible nonlinear hyper-elastic layer having
nite thickness is studied. The upper surface of the layer is assumed
to be free from traction and the lower boundary is rigidly xed. These
waves are dispersive like the Love waves. The problem is examined by
a perturbation method that balances the nonlinearity and dispersion
in the analysis. A nonlinear Schrödinger equation is derived describing
the nonlinear self modulation of the waves. Then, the eect of nonli-
near properties of the material on the propagation characteristics and
on the existence of solitary waves are discussed.
Keywords:
Dispersive nonlinear shear horizontal waves, Multiple scale method,
Bright solitary waves, Dark solitary waves.
2000 AMS Classication:
AMS[2010], 74B20, 74J30, 74J35.
Received
: 19.10.2016
Accepted
: 06.12.2016
Doi
: 10.15672/ HJMS.2017.426
1. Introduction
Elastic waves propagating in an unbounded media are non-dispersive i.e. phase ve-
locities of waves are constants. On the other hand, in wave guides such as rods, plates,
layered half space, etc.,the phase velocities of the waves depend on wave number, hence
the waves are dispersive. Dispersive elastic waves have been studied extensively, because
of their application in geophysics, nondestructive testing of materials,electronic signal
processing devices, etc.. (see, e.g. Ewing et al. [1], Love [2], Achenbach [3], Graf [4],
Farnell [5], Maugin [6]).
In recent years, the eect of the nonlinear material parameters on the propagation
characteristics of dispersive elastic waves has been the subject of numerous investigations
Istanbul Technical University, Department of Mathematics, 34469 Maslak, Istanbul, Turkey,
Email:
demirkusdilek@itu.edu.tr
Istanbul Technical University, Department of Mathematics, 34469 Maslak, Istanbul, Turkey,
Email:
teymur@itu.edu.tr
Corresponding Author.

802
for similar reasons mentioned above. By employing asymptotic perturbation methods
many problems related with the propagation of nonlinear dispersive waves are examined.
In these works, as a result of balance between nonlinearity and dispersion various nonlin-
ear evaluation equations such as Korteweg-DeVries (KdV) equation, modied Korteweg-
DeVries (mKdV) equation, nonlinear Schrödinger (NLS) equation, Boussinesq equation
etc. have been derived to elucidate the nonlinear wave motion asymptotically. Then
various aspects of the problems such as the stability of modulated wave, the existence of
solitary waves, etc. were discussed on the basis of these equations. For an extensive review
of most of these works we refer to Parker and Maugin [7], Maugin [8], Parker [9], Mayer
[10], Norris [11], Porubov [12]. Among the works on nonlinear dispersive elastic waves,
the investigations of nonlinear shear horizontal(SH) waves occupies an important place.
Below some of these works will be reviewed to relate the present work to them. In [13],
Bataille and Lund considered the propagation of nonlinear Love waves in a layered half
space covered by a thin linear elastic layer. A modied Boussinesq equation is derived by
an intuitive approach guided by physical arguments which accounts the dispersive nature
of Love waves and the nonlinearity. This equation has an approximate modulated solitary
wave (an envelope solitary wave) solution which provides mechanisms for localized en-
ergy propagation along the surface of the layered medium. The propagation of nonlinear
Love waves in a half space covered by a layer of uniform nite thickness having dierent
mechanical properties, is investigated by a perturbation method in [14] by Teymur. The
materials of the layer and the half space are both assumed to be homogeneous, isotropic
and compressible hyper-elastic. Then, it is shown that the nonlinear self modulation of
Love waves is governed asymptotically by an NLS equation. The coecients of this NLS
equation are valid on all branches of the linear dispersion relation of Love waves for any
wave number. From the numerical evaluation of these coecients for various material
parameters it has been observed that the stability of modulated waves, the existence of
envelope ( bright) and dark solitary waves depend strongly on the nonlinear properties of
the layered media as well as the wave number. The problem is reconsidered by Maugin
and Hadouaj [15] where the nonlinear substrate covered by a linear thin elastic layer and
then by Teymur et al.[16] if the top layer is made of a thin nonlinear elastic material.
In [17], Ahmetolan and Teymur studied the propagation of nonlinear SH waves and the
formation of Love waves in a double layered plate each having nite thickness. In [18],
Ahmetolan and Teymur examined the propagation of nonlinear SH waves in a plate hav-
ing nite thickness and made of a generalized neo-Hookean material. An NLS equation
is derived which governs the nonlinear self modulation of waves asymptotically and the
eect of nonlinearity on the propagation characteristics is discussed. The propagation of
small but nite amplitude long SH waves in a double layered plate is examined in [25] by
Teymur . I! n that work by an asymptotic analysis, a modied KdV equation is derived
and then the dependence of various types of solitary wave solutions on the nonlinear
material parameters are discussed. The propagation of large amplitude Love waves in
a layered half-space made of dierent pre-stressed compressible neo-Hookean materials
(restricted Hadamard materials) is examined by Ferreira and Boluenger [20] and an exact
solution of the problem is given. Later, the anti-plane shear motions coupled with an
in-plane motion for a Hadamard materials are considered by Pucci and Saccomandi [21].
The pure anti-plane motion may be sustained in a Hadamard material in the absence
of body forces. When the constitutive parameter
β
is small, a perturbation analysis is
developed using this small parameter. Then this approach is also applied to the propaga-
tion of nite amplitude Love waves in a layered half space made of Hadamard materials.
And the solutions exhibiting a secondary in-plane motion caused by a principal anti-plane
motion are given.

803
In the present work, we consider the propagation SH waves in a homogeneous isotropic
compressible nonlinear elastic nite layer deposited on a rigid substratum. The upper
surface of the layer is assumed to be free from the stress. This problem may model some
real world problem. A uniform layer of a nonlinear soil overlying a rigid bedrock is an
example from soil dynamics (see for example[22]). Also a soft material layer overlying
an almost rigid material is an another example from the signal processing applications
(see for example [8] and [15]). The problem is examined by a perturbation method. By
balancing the nonlinearity and dispersion in the analysis, an NLS equation is derived
describing the nonlinear self modulation of SH waves. Then, the eect of nonlinearity
on the propagation characteristics of waves and on the existence of solitary waves are
discussed.
2. Formulation of the Problem
Let
(x
1
, x
2
, x
3
)
and
(X
1
, X
2
, X
3
)
be, respectively, the spatial and material coordinates
of a point referred to the same rectangular Cartesian system of axes. Consider an elastic
layer of uniform thickness
h
, occupying the regions between the planes
X
2
= 0
and
X
2
= h
in the reference frame
X
K
. It is assumed that the boundary
X
2
= h
is free of
traction and the displacements are zero at the rigid boundary
X
2
= 0
. Now, an SH wave
described by the equations
(2.1)
x
k
= X
K
δ
kK
+ u
3
(X
, t)δ
k3
is supposed to propagate along the
X
1
-axis in the layer, where
u
3
is the displacement
of a particle in the
X
3
direction,
t
is the time and
δ
kK
is the Kronecker symbol. The
summation convention on repeated indices is implied in (2.1) and in the sequel of this
section, and Latin and Greek indices have respective ranges
(1, 2, 3)
and
(1, 2)
. Since
det(x
k
/∂X
K
) = 1
; the deformation eld dened by (2.1) is isochoric and the density
ρ
of the layer in motion remain constant, i.e.
ρ = ρ
0
= constant
.
Let
T
Kl
be the rst Piola-Kircho stress tensor eld accompanying the deformation
eld (2.1); in the absence of body forces, the equation of motion in the reference state
take the following forms
(2.2)
T
β,
+ T
3β,3
= 0, T
∆3,
+ T
33,3
= ρ
0
¨u
3
where subscripts preceded by a comma indicate partial dierentiation with respect to
coordinates
X
K
and an over dot represents the partial dierentiation with respect to
t
[14].
The assumption of vanishing traction on the free surface of the layer imposes the
boundary condition
(2.3)
T
2k
= 0
on
X
2
= h,
and on the rigid boundary
(2.4)
u
3
= 0
on
X
2
= 0.
Let us now assume that the constituent material of the layer is nonlinear, homoge-
neous, isotropic and compressible hyper-elastic. Stress constitutive equations for such a
material may be expressed as
(2.5)
T
Kk
= (
Σ
I
1
δ
LK
+ 2
Σ
I
2
E
LK
+ 3
Σ
I
3
E
LM
E
MK
)x
k,L
where
E
KL
= (x
k,K
x
k,L
δ
KL
)/2
is the Lagrangian strain tensor and
Σ
is the strain
energy function (see e.g. Eringen and Suhubi [23]). For an isotropic material,
Σ
is an
isotropic function of the invariants of
E
dened as
(2.6)
I
1
= trE, I
2
= trE
2
, I
3
= trE
3
.

804
For the deformation eld (2.1), the invariants are found to be
(2.7)
I
1
= Q/2, I
2
= Q(1 + Q/2)/2, I
3
= Q
2
(3 + Q)/8.
where
(2.8)
Q = Q(u
3
) = u
3,
u
3,
.
The stress-strain relations (2.5) now read [14],
T
β
= [
Σ
I
1
δ
Ω∆
+ (
Σ
I
2
+
3
4
(1 + Q)
Σ
I
3
)u
3,
u
3,
]δ
β
, T
∆3
= 2
Σ
Q
u
3,
,
T
3α
= u
3,
T
α
+ δ
α
T
∆3
, T
33
=
Σ
I
1
+ (
Σ
I
2
+
3
4
(1 + Q)
Σ
I
3
)Q.
(2.9)
Note that for a specic material
Σ
is a prescribed function of
Q
through the invariants
(2.7). Hence the equations of motion (2.2) are three equations to be satised by a single
function
u
3
. For any material dened by (2.5), if the rst two equations in (2.2) are
satised by a given solution of the third equation in (2.2), then the motion (2.1) can
exist in the medium in the absence of body forces. This is only the case if
Σ = Σ(I
1
)
, i.e.
if the medium is made of a generalized Neo-Hookean material ( see e.g. Teymur [19] or
in more detail Saccomandi and Ogden [24] ). In general without any restriction on the
constitutive relation (2.5)(or (2.9)), the system of equations (2.2) is not compatible so that
the motion (2.1) cannot be maintained without body forces acting in the
(X
1
, X
2
)
-plane
(see for details Carroll [25], Pucci and Saccomandi [26], Rogers et al. [27]). Therefore as
in [14](or in [16]) we will assume that the motion (2.1) takes place in a material for which
the Cauchy stress components
t
αβ
are identically zero as in the case of linear problem.
Hence, since
T
β
= δ
α
t
αβ
then
T
β
=0 and as a consequence of this assumption, the
rst two equations in (2.2) are satised identically and the third equation becomes
(2.10)
2(
Σ
Q
u
3
,
),
= ρ
0
¨u
3
.
It is seen from (2.9) that for such a material the strain energy function
Σ
must satisfy
the following conditions
(2.11)
Σ
I
1
= 0,
Σ
I
2
+
3
4
(1 + Q)
Σ
I
3
= 0,
whenever the invariants are given by (2.7). We now employ the following fourth order
polynomial expansion of
Σ
in terms of the strain invariants
I
i
to deduce approximate
equations
(2.12)
Σ = α
1
I
2
1
+ α
2
I
2
+ α
3
I
3
1
+ α
4
I
1
I
2
+ α
5
I
3
+ α
6
I
4
1
+ α
7
I
2
1
I
2
+ α
8
I
1
I
3
+ α
9
I
2
2
+O(I
5
1
, I
5/2
2
, I
5/3
3
)
where
α
1
= λ/2, α
2
= µ
are second order (
λ
and
µ
are the usual Lamé constants),
α
3
, α
4
, α
5
are third order and
α
6
, α
7
, α
8
, α
9
are fourth-order elastic constants. The third
order elastic constants related to the Murnaghan`s constants
l, m, n
as (see Norris [11])
(2.13)
α
3
=
1
3
(l m +
1
2
n) , α
4
= (m
1
2
n), α
5
=
n
3
Then employing (2.12) in the stress-strain relation (2.9) and applying the restrictions on
Σ
dened in (2.11) we get
T
11
= T
12
= T
21
= T
22
= T
33
= 0,
T
∆3
= µu
3,
+ (α
9
+ m/2)u
3,
Q + O(Q
2
),
(2.14)
and the equation (2.10) becomes
(2.15)
¨u
3
c
2
T
u
3,∆∆
= n
T
(u
3,
Q),
+O(Q
2
)

805
where
(2.16)
c
2
T
= µ/ρ
0
, n
T
= (α
9
+ m/2)
0
Here,
c
T
is the linear shear velocity and
n
T
the nonlinear material constant which exhibit
the nonlinear characteristics of the constituent material. When
n
T
> 0
, the medium is
hardening in shear, but if
n
T
< 0
, then it is softening.
Hence, the SH wave motion (2.1) can be maintained in the restricted hyper-elastic
material dened by (2.14) without body forces acting in the
(X
1
, X
2
)
plane. Now,
let
X = X
1
, Y = X
2
, Z = X
3
and
u = u
3
. Then from (2.14) and(2.15) the following
approximate governing equation and boundary conditions involving terms not higher
than the third degree in the deformation gradients are written;
(2.17)
2
u
t
2
c
2
T
2
u
X
2
+
2
u
Y
2
= n
T
X
u
X
Q(u)
+
Y
u
Y
Q(u)

(2.18)
u
Y
1 +
n
T
c
2
T
Q(u)
= 0
on
Y = h,
(2.19)
u = 0
on
Y = 0,
3. Asymptotic analysis of the nonlinear SH waves
In this work, how the slowly varying amplitude of a weakly nonlinear SH wave is
modulated by nonlinear self interaction is investigated by a perturbation method. For
this purpose, the method of the multiple scales is employed by introducing the following
new independent variables
(3.1)
x
i
= ε
i
X, t
i
= ε
i
t, y = Y ; i = 0, 1, 2, . . .
instead of
X, Y, t
[28]. Here
ε > 0
is a small parameter which measures the weakness of
the nonlinearity,
{x
0
, t
0
, y}
are fast variables describing the fast variations in the problem
while
{x
1
, x
2
, . . . , t
1
, t
2
, . . .}
are slow variables describing the slow variations. Now,
u
is
considered to be a function of these new variables and it is expanded in the following
asymptotic series in
ε
;
(3.2)
u =
X
n=1
ε
n
u
n
(x
0
, x
1
, x
2
, . . . , y, t
0
, t
1
, t
2
, . . .)
In this work we aimed to obtain rst order uniformly valid asymptotic solution of the
problem. Therefore in the following part we will assume the dependence of
u
n
on the
slow scales
{x
1
, x
2
, t
1
, t
2
}
only. If one studies the contribution of higher order terms
then the third order, in the analysis the dependence on the slower scales
{x
3
, . . . , t
3
, . . .}
should also be considered as independent variables. Now, rst writing the equation of
motion (2.17), the boundary conditions (2.18) and (2.19) in terms of the new indepen-
dent variables (3.1) and then employing the asymptotic expansion (3.2) in the resulting
expressions and collecting the terms of like powers of in
ε
, we obtain a hierarchy of
problems from which it is possible to determine
u
n
, successively. Up to third order in
ε
these are given as follows;
O(ε) :
(3.3)
Lu
1
,
2
u
1
t
2
0
c
2
T
2
u
1
x
2
0
+
2
u
1
y
2
= 0,
(3.4)
u
1
y
= 0
on
y = h,
(3.5)
u
1
= 0
on
y = 0.

Citations
More filters


Journal ArticleDOI
Dilek Demirkuş1Institutions (1)
Abstract: In the present work, we search for the propagation of nonlinear shear horizontal waves (SH) in a finite thickness plate which consists of heterogeneous, isotropic, and generalized neo-Hookean materials. In the analysis, we apply the method of multiple scales and strike a balance between the nonlinearity and the dispersion. Then, the self-modulation of nonlinear SH waves can be given by a nonlinear Schrodinger equation which has the well-known dark solitary solution. Consequently, we show that the dark solitary SH waves can propagate in this plate. Moreover, we take the effects of heterogeneity and the nonlinearity into account for these waves.

13 citations


Journal ArticleDOI
Dilek Demirkuş1Institutions (1)
Abstract: This paper aims to make some comparative studies between heterogeneous and homogeneous layers for nonlinear shear horizontal (SH) waves in terms of the heterogeneous and nonlinear effects. Therefor...

2 citations


Cites background from "Shear horizontal waves in a nonline..."

  • ...Nonlinear SH wave (or surface SH wave) solutions in homogeneous media are also known in different waveguides, such as in a layer [17], a plate [16], a two-layered plate [15], or a layered half-space [27]....

    [...]

  • ...On the other hand, in the homogeneous problem, the propagation of SH waves in the geometrically same layer is considered for material properties such as hyperelastic, isotropic, homogeneous, and generalized neoHookean (for a similar problem, see [17])....

    [...]


Book ChapterDOI
Dilek Demirkuş1Institutions (1)
01 Jan 2020
Abstract: In this chapter, we compare the nonlinear bright solitary shear horizontal (SH) waves in heterogeneous and homogeneous layers in terms of the heterogeneous effect. Each layer has finite thickness overlying a rigid substratum. We assume that the layers are made up of isotropic, hyper-elastic and generalized neo-Hookean (similarly, compressible or incompressible) materials. Moreover, one layer contains heterogeneous materials and another contains homogeneous materials. The existence of nonlinear bright solitary SH waves in such layers can be found in the literature. Therefore, we aim to overcome the difficulty of a comparison of two nonlinear analyzes for this paper. Besides a comparison part, we add a discussion on some materials in homogeneous media.

2 citations


Journal ArticleDOI
Dilek Demirkuş1Institutions (1)
Abstract: In this article, the non-linear anti-symmetric shear motion for some comparative studies between the non-homogeneous and homogeneous plates, having two free surfaces with stress-free, is considered. Assuming that one plate contains hyper-elastic, non-homogeneous, isotropic, and generalized neo-Hookean materials and the other one consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the method of multiple scales, the self-modulation of the non-linear anti-symmetric shear motion in these plates, as the non-linear Schrodinger (NLS) equations, can be given. Using the known solitary wave solutions, called bright and dark solitary wave solutions, to NLS equations, these comparative studies in terms of the non-homogeneous and non-linear effects are made. All numerical results, based on the asymptotic analyses, are graphically presented for the lowest anti-symmetric branches of both dispersion relations, including the deformation fields of plates.

1 citations


References
More filters

Book
01 Jan 1962
Abstract: Preface Introduction 1 One-dimensional motion of an elastic continuum 2 The linearized theory of elasticity 3 Elastodynamic theory 4 Elastic waves in an unbound medium 5 Plane harmonic waves in elastic half-spaces 6 Harmonic waves in waveguides 7 Forced motions of a half-space 8 Transient waves in layers and rods 9 Diffraction of waves by a slit 10 Thermal and viscoelastic effects, and effects of anisotrophy and non-linearity Author Index Subject Index

4,124 citations


"Shear horizontal waves in a nonline..." refers background in this paper

  • ...(see, e.g. Ewing et al. [1], Love [2], Achenbach [3], Graf [4], Farnell [5], Maugin [6])....

    [...]

  • ...[1], Love [2], Achenbach [3], Graf [4], Farnell [5], Maugin [6])....

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07 Jan 1996
Abstract: Keywords: Tremblement de terre ; Danger naturel ; Propagation des ondes ; Probleme dynamique Reference Record created on 2004-09-07, modified on 2016-08-08

3,582 citations


"Shear horizontal waves in a nonline..." refers background in this paper

  • ...A uniform layer of a nonlinear soil overlying a rigid bedrock is an example from soil dynamics (see for example[22])....

    [...]


Book
01 Jun 1982
Abstract: A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.

1,707 citations


"Shear horizontal waves in a nonline..." refers background in this paper

  • ...5) φ(η) = φ0 tanh[(−∆/2Γ)φ0η], V0 = 2KΓ which represents the propagation of a phase jump [30, 31, 32]....

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  • ...67), as in this work, asymptotically describes the self modulation of the monochromatic plane waves in a nonlinear dispersive medium[30, 31]....

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  • ...This solution is known as envelope soliton or bright soliton [30, 31, 32]....

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  • ...On the other hand for disturbances that tend to a uniform state at in nity the envelope dark solitons exist for Γ∆ < 0 [30, 31]....

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Journal ArticleDOI
01 Jan 1958-Gff
Abstract: (1958). Elastic Waves in Layered Media. Geologiska Foreningen i Stockholm Forhandlingar: Vol. 80, No. 1, pp. 128-129.

1,621 citations


"Shear horizontal waves in a nonline..." refers background in this paper

  • ...[1], Love [2], Achenbach [3], Graf [4], Farnell [5], Maugin [6])....

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Book
08 Jul 2003
Abstract: * Basic Concepts * Mathematical Tools for the Governing Equations Analysis * Strain Solitary Waves in an Elastic Rod * Amplification of Strain Waves in Absence of External Energy Influx * Influence of Dissipative (Active) External Medium * Bulk Active or Dissipative Sources of the Amplification and Selection

170 citations


"Shear horizontal waves in a nonline..." refers result in this paper

  • ...For an extensive review of most of these works we refer to Parker and Maugin [7], Maugin [8], Parker [9], Mayer [10], Norris [11], Porubov [12]....

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20202
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19821