DSpace at VNU: Explicit secular equations of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity

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Applied Mathematics and Computation 215 (2010) 3515–3525

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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc

Explicit secular equations of Stoneley waves in a non-homogeneous
orthotropic elastic medium under the inﬂuence of gravity
Pham Chi Vinh a,*, Géza Seriani b
a
b

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam
Istituto Nazionale di Oceanograﬁa e di Geoﬁsica Sperimentale, Borgo Grotta Gigante 42/C, 34100 Sgonico, Trieste, Italy

a r t i c l e

i n f o

Keywords:
Stoneley waves
Stoneley wave velocity
Orthotropic
Secular equation
Non-homogeneous
Gravity

a b s t r a c t
The problem of Stoneley waves in a non-homogeneous orthotropic elastic medium under
the inﬂuence of gravity was studied recently by Abd-Alla and Ahmed [A.M. Abd-Alla, S.M.

Ahmed, Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic
medium under the inﬂuence of gravity, Appl. Math. Comput. 135 (2003) 187–200], who
derived the secular equation of the wave in the implicit form. In this paper, by using an
appropriate representation of the solution, we obtain the secular equation of the wave in
the explicit form. Moreover, considering its special cases, we derive explicit secular equations for a number of investigations of Stoneley waves under the inﬂuence of gravity, for
which only the implicit dispersion equations were previously obtained.
Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction
The propagation of Stoneley waves under the effect of gravity is a signiﬁcant problem in Seismology and Geophysics,
which has attracted the attention of researchers such as De and Sengupta [1], Dey and Sengupta [2], Das et al. [3]. These
authors, following Biot [4], all assumed the force of gravity to create a type of initial stress of hydrostatic nature, and derived
the secular equation of the wave in the implicit form. De and Sengupta [1] assumed the material is isotropic elastic, while
Dey and Sengupta [2] considered the case of transversely isotropic elastic materials. All supposed that the material is homogeneous. However, because any realistic model of the earth must take into account continuous changes of elastic properties
in the vertical direction, the problem was extended to the (exponentially) non-homogeneous case by Das et al. [3], who assumed that the material is isotropic. Recently, Abd-Alla and Ahmed [5] extended the problem to the orthotropic case; these
authors employed two displacement potentials for expressing the solution, and derived the secular equation of the wave in
the implicit form.
For Rayleigh and Stoneley waves, dispersion equations in the explicit form are very signiﬁcant in practical applications.
They can be used for solving direct (forward) problems, i.e. studying the effects of material parameters on the wave velocity;
and especially the inverse problems, i.e. determining material parameters from the measured values of the wave speed. The
main purpose of this paper is to obtain the explicit secular equation of Stoneley waves under the effect of gravity for inhomogeneous orthotropic elastic materials. This equation is identiﬁed in the explicit form by using an appropriate representation of
solution. From this we derive the explicit secular equations for the particular cases investigated by De and Sengupta [1], Dey
and Sengupta [2], Das et al. [3], and Pal and Acharya [6], in which only implicit dispersion equations were obtained.
Note that a secular equation F ¼ 0 is called explicit if F is an explicit function of the wave velocity c, the wave number k,
and parameters characterizing materials and external effects (see for example, [7–9]). Otherwise we call it an implicit secular
equation.
* Corresponding author.
E-mail address: (P.C. Vinh).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2009.10.047

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P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

Also note that, due to Abd-Alla and Ahmed’s incorrect representation of the solution (see [12]), the secular equation in the
implicit form derived by them in [5] for Stoneley waves is not valid.
2. Basic equations
Let us consider the two non-homogeneous orthotropic elastic bodies, X and XÃ , occupying the half-space x3 P 0; x3 6 0,
respectively, subject to gravity. They are in welded contact with each other at the plane x3 ¼ 0. These two media extend to an
inﬁnitely great distance from the origin and XÃ is to be taken above X. Same quantities related to X and XÃ have the same
symbol but are systematically distinguished by an asterisk if pertaining to XÃ .
We are interested in planar motion in the ðx1 ; x3 Þ-plane with displacement components u1 ; u2 ; u3 such that:

ui ¼ ui ðx1 ; x3 ; tÞ;

i ¼ 1; 3; u2 0:

ð1Þ

Then the components of the stress tensor rij ; i; j ¼ 1; 3 are related to the displacement gradients by the following equations
[5]:

r11 ¼ c11 u1;1 þ c13 u3;3 ;
r33 ¼ c13 u1;1 þ c33 u3;3 ;
r13 ¼ c55 ðu1;3 þ u3;1 Þ;

ð2Þ

where cij are the material constants.
Equations of motion are of the form [5]:

r11;1 þ r13;3 þ qgu3;1 ¼ qu€1 ;
r13;1 þ r33;3 À qgu1;1 ¼ qu€3

ð3Þ

in which q is the mass density of the medium, and g is the acceleration due to gravity, a superposed dot denotes differentiation with respect to t, commas indicate differentiation with respect to spatial variables xi . In matrix (operator) form, following the Stroh formalism (see [10,11]), the Eqs. (2) and (3) are written as follows:

u0

r

!
¼N

0

u

!

r

ð4Þ

;

where u ¼ ½u1 ; u3 T ; r ¼ ½r13 ; r33 T , the symbol T indicates the transpose of a matrix, the prime indicates the derivative with

respect to x3 and:

N¼
"
K¼

N1
K

!
N2
;
N3

N1 ¼

!
À@ 1
1=c55
; N2 ¼
0
0
#
Àqg@ 1
; N 3 ¼ NT1 :
q@ 2t

0
Àðc13 =c33 Þ@ 1

q@ 2t þ ½ðc213 À c11 c33 Þ=c33 @ 21
qg@ 1

!
0
;
1=c33

ð5Þ

Here we use the notations: @ 1 ¼ @=ð@x1 Þ; @ 21 ¼ @ 2 =ð@x21 Þ; @ 2t ¼ @ 2 =ð@t 2 Þ. In addition to Eq. (4), the displacement vector u and
the traction vector r are required to satisfy the decay condition:

u ¼ 0;

r ¼ 0 at x3 ¼ þ1:

ð6Þ

Ã

For X we have equations similar to (1)–(5) in which the quantities are asterisked, and the decay condition (6) is replaced by:

uÃ ¼ 0;

rÃ ¼ 0 on x3 ¼ À1:

ð7Þ

Since the half-spaces are in welded contact with each other at the plane x3 ¼ 0, the displacement vector and the traction

vector must satisfy the continuous condition:

u ¼ uÃ ;

r ¼ rÃ on x3 ¼ 0:

ð8Þ

3. Explicit secular equation
Assume that the half-spaces X and XÃ are made of materials with an exponential depth proﬁle:

cij ¼ c0ij e2mx3 ;
c0ij ;

0

Ã

Ã

2m x3
q ¼ q0 e2mx3 ; cÃij ¼ cÃ0
; qÃ ¼ qÃ0 e2m x3 ;
ij e

ð9Þ

where
q; m
q ; m are constants.

Now we consider the propagation of a wave, travelling with velocity c and wave number k in the x1 -direction, being
mostly conﬁned to the neighbourhood of the interface x3 ¼ 0. Then the components u1 ; u3 of the displacement vector
and r13 ; r33 of the traction vector at the planes x3 ¼ const are found in the form (see [13]):
cÃ0
ij ;

Ã0

Ã

P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

fuk ; rk3 gðx1 ; x3 ; tÞ ¼ feÀmx3 U k ðx3 Þ; iemx3 Rk ðx3 Þgeikðx1 ÀctÞ ;

k ¼ 1; 3:

3517

ð10Þ

Substituting (10) into (4) yields:

U0

!

¼ iM

R0

Q¼

!
ð11Þ

;

R

U 3 T ; R ¼ ½ R 1

where U ¼ ½ U 1

M¼

U

M1

M2

Q

M3

!

M1 ¼

;

kðX À dÞ

ia

Àia

kX

!
;

R3 T , and:
Àiðm=kÞ

À1

ÀD

Àiðm=kÞ

"

!
;

!
iðm=kÞ
ÀD

;
M3 ¼
À1
iðm=kÞ

1=c055
M 2 ¼ ð1=kÞ
0

#
0
;
1=c033

ð12Þ

ðc013 Þ2 Þ=c033 ;

c011

here d ¼
À
D ¼ c013 =c033 ; a ¼ q0 g; X ¼ q0 c2 , the prime indicates the derivative with respect to y ¼ kx3 .
Following the approach employed in [9,14,15], by eliminating U from (11), the traction vector RðyÞ is the solution of the
equation:

aR00 À ibR0 À cR ¼ 0;

ð13Þ

where the matrices a; b; c are given by:

!
X
Àia=k
1
; d ¼ XðX À dÞ À ða=kÞ2
kd ia=k ðX À dÞ
!
1 0 g1
; g 1 ¼ d À ð1 þ DÞX
b ¼ M 1 Q À1 þ Q À1 M3 ¼
kd g 1 0
!
h0
img 0 =k þ iag 2 =k
1
c ¼ M1 Q À1 M3 À M2 ¼
kd Àimg 0 =k À iag 2 =k
h1

a ¼ Q À1 ¼

ð14Þ
ð15Þ
ð16Þ

in which

g 0 ¼ d À ð1 À DÞX;

m2 X

g2 ¼ D À

m2
2

k
2ma

;

d
þ 2 ;
c055
k
d
2maD
h1 ¼ 2 ðX À dÞ þ D2 X À 0 À
:
2
c33
k
k

h0 ¼

2

k

m2

þ ðX À dÞ À

ð17Þ

The displacement vector U is determined in terms of R by:

U ¼ ÀiQ

À1

R0 À Q À1 M3 R:

ð18Þ

Now we seek the solution of the Eq. (13) in the form:

RðyÞ ¼ eipy R0 ;

ð19Þ

where R0 is a non-zero constant vector, p is a complex number which must satisfy the condition:

Ip > jmj=k

ð20Þ

in order to ensure the decay condition (6). Substituting (19) into (13) leads to:

ðp2 a À pb þ cÞR0 ¼ 0:

ð21Þ

As R0 is a non-zero vector, the determinant of the system (21) must vanish. This provides an equation for determining p,
namely:

p4 À Sp2 þ P ¼ 0;

ð22Þ

where

d
1
1
2m2
XÀ 2 ;
þ
þ
0
0
0
c55
c33 c55
k

!
ðc011 À XÞðc055 À XÞ m2
1
1
d
P¼
À 2
þ 0 X À 2D À 0
0 0
0
c33 c55
c33 c55
c55
k

S ¼ 2D À

þ

m4
4

k

À

a2 þ 2amðc055 À c013 Þ
2

k c033 c055

:

ð23Þ

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P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

It follows from (22) that:

p21 þ p22 ¼ S;
p21 ;

p21 p22 ¼ P;

ð24Þ

p22

T

2

where
are two roots of the quadratic Eq. (22) for p . It is not difﬁcult to demonstrate that the vector R0 ¼ ½A B , the
solution of (21), is given by:

ia 2

img 0
p þ g1 p À
À iag 2 =k;
k
k
B ¼ Xp2 þ h0 :
A¼

ð25Þ

Let p1 ; p2 be the two roots of (22) satisfying the condition (20). Then the general solution of the Eq. (13) is:

!
!
A1 ip1 y
A2 ip2 y
e þ c2
e ;
B1
B2

RðyÞ ¼ c1

ð26Þ

where Ak ; Bk ðk ¼ 1; 2Þ are given by (25), in which p is replaced by pk , and pk ; c1 , and c2 ðc21 þ c22 – 0Þ are constants to be
determined.
We have the following result:
Proposition. Suppose p1 ; p2 are the two roots of (22) satisfying the condition (20). Then we have:

P > 0;

pﬃﬃﬃ
2 P À S > 0;

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
p1 þ p2 ¼ i 2 P À S;

pﬃﬃﬃ
p1 p2 ¼ À P ;

ð27Þ

where S; P are deﬁned by (23).
Indeed, from (20) it follows that Imðpi Þ > 0. If the discriminant D of the quadratic Eq. (22) for p2 is non-negative, then its
two roots must be negative in order that Imðpi Þ > 0. In this case, P ¼ p21 p22 > 0 and the pair p1 ; p2 are of the form:
p1 ¼ ir 1 ; p2 ¼ ir 2 where r1 ; r2 are positive. If D < 0, the Eq. (22) for p2 has two conjugate complex roots, again
P ¼ p21 p22 > 0, and in order to ensure Imðpi Þ > 0, it must be p1 ¼ t þ ir and p2 ¼ Àt þ ir, where r is positive. In both cases,
P ¼ p21 p22 > 0; p1 p2 is a negative real number and p1 þ p2 is a purely imaginary number with a positive imaginary part, thus
are
ðp1 þ p2 Þ2 is a negative number. Therefore, with the help of (24), it follows that the relations (27) p
ﬃﬃﬃ true.
It is noted that the result (27)3, (27)4 were obtained in [13], but without showing that P > 0; 2 P À S > 0.
From (18) and (26) we have:

E1

UðyÞ ¼ c1

F1

!

E2

eip1 y þ c2

!

F2

eip2 y ;

ð28Þ

where

Ek ¼ e2 p2k À ie1 pk þ e0 ;
F k ¼ f3 p3k þ if2 p2k þ f1 pk þ if0 ;

ð29Þ

k ¼ 1; 2;

where ej ¼ bj =kd; j ¼ 0; 1; 2; fj ¼ wj =kd; j ¼ 0; 1; 2; 3, and:
2

2

b0 ¼ h0 ðDX À am=k Þ À ð1=k Þða þ XmÞðmg 0 þ ag 2 Þ;
b1 ¼ ð1=kÞ½g 1 ða þ XmÞ þ Xðmg 0 þ ag 2 Þ þ ah0 ;
2

2

b2 ¼ ð1=k Þaða þ XmÞ þ g 1 X þ XðDX À am=k Þ;
2

w0 ¼ ð1=kÞh0 ½aD À mðX À dÞ À ð1=kÞðmg 0 þ ag 2 ÞðX À d þ ma=k Þ;
2

ð30Þ

2

w1 ¼ g 1 ðX À d þ ma=k Þ þ ð1=k Þaðmg 0 þ ag 2 Þ þ ðX À dÞh0 ;
2

w2 ¼ ð1=kÞaðX À d þ ma=k Þ þ ð1=kÞag 1 þ ðX=kÞ½aD À mðX À dÞ;
w3 ¼ d:
Similarly, vectors uÃ and

rÃ are sought in the form:
Ãx

fuÃk ; rÃk3 gðx1 ; x3 ; tÞ ¼ feÀm

3

Ãx

U Ãk ðx3 Þ; iem

3

RÃk ðx3 Þgeikðx1 ÀctÞ ;

k ¼ 1; 3

ð31Þ

in which

U Ã ¼ cÃ1

!
!
EÃ
Ã
Ã
eip1 y þ cÃ2 2Ã eip2 y ;
F2

ð32Þ

!
!
AÃ1 ipÃ y
AÃ2 ipÃ y

1 þ cÃ
2 ;
2
Ã e
Ã e
B1
B2

ð33Þ

EÃ1
F Ã1

and

RÃ ¼ cÃ1

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P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

where cÃ1 ; cÃ2 are constants to be determined, AÃk ; BÃk ; EÃk ; F Ãk ðk ¼ 1; 2Þ are deﬁned by:
Ã

Ã

ia Ã 2
im g Ã0
Ã

ðpk Þ þ g Ã1 pÃk À
À ia g Ã2 =k;
k
k
Ã
BÃk ¼ X Ã ðpÃk Þ2 þ h0 ;

AÃk ¼

ð34Þ

EÃk ¼ eÃ2 ðpÃk Þ2 À ie1 pÃk þ eÃ0 ;
Ã

Ã

Ã

F Ãk ¼ f3Ã ðpÃk Þ3 þ if 2 ðpÃk Þ2 þ f1Ã pÃk þ if 0 ;

k ¼ 1; 2

Ã

Ã

Ã

Ã

Ã

in which aÃ ¼ qÃ0 g; g Ãj ; h0 are deﬁned by formulas similar to those for g j ; h0 , and eÃj ¼ bj =kd ; fjÃ ¼ wÃj =kd and bj ; wÃj are exÃ
pressed by those similar to (30); furthermore, d ðDÃ ; dÃ ; X Ã Þ are given by the expressions similar to those for d ðD; d; XÞ,
Ã
Ã
and p1 ; p2 are two roots of the equation:

p4 À SÃ p2 þ PÃ ¼ 0

ð35Þ

satisfying:

IpÃj < À

jmÃ j
;
k

j ¼ 1; 2;

ð36Þ

here SÃ ; P Ã are determined by those similar to (23).
Analogously as above, one can show that:

PÃ > 0;

pﬃﬃﬃﬃﬃ

2 P Ã À SÃ > 0;

pﬃﬃﬃﬃﬃ
pÃ1 pÃ2 ¼ À PÃ ;

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
pÃ1 þ pÃ2 ¼ Ài 2 PÃ À SÃ :

ð37Þ

From the continuity conditions (8) we have:

U k ð0Þ ¼ U Ãk ð0Þ;

Rk ð0Þ ¼ RÃk ð0Þ;

k ¼ 1; 3:

ð38Þ

Eq. (38) yield a homogeneous linear system for c1 ; c2 ; cÃ1 ; cÃ2 . The secular equation, determining the Stoneley wave velocity c,
is obtained by vanishing the determinant of the system:

E1

F
1

A1

B

E2

EÃ1

F2

F Ã1
AÃ1
BÃ1

A2
B2

1

EÃ2

Ã
F2
¼ 0:
AÃ2
BÃ2

ð39Þ

After some algebraic manipulations, taking into account (25), (29) and (34) and removing the factor ðp1 À p2 ÞðpÃ1 À pÃ2 Þ , the
dispersion Eq. (39) is equivalent to:

q11

q
21

q31

q
41

q12
q22

qÃ11
qÃ21

q32

qÃ31

q42

qÃ41

qÃ12

qÃ22
¼ 0;
qÃ32
qÃ

ð40Þ

42

where

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
2 P À S À b1 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
¼ b2 S þ b1 2 P À S þ 2b0 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
pﬃﬃﬃ
¼ w3 ðS À PÞ À w2 2 P À S þ w1 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
pﬃﬃﬃ
pﬃﬃﬃ
¼ Àw3 ðS þ PÞ 2 P À S À w2 S À w1 2 P À S À 2w0 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
!

pﬃﬃﬃ
¼ d Àða=kÞ 2 P À S þ g 1 ;

q11 ¼ b2
q12
q21
q22
q31

q32 ¼ d ð2=kÞðmg 0 þ ag 2 Þ À ða=kÞS À g 1
q41 ¼ d X

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
pﬃﬃﬃ
2 PÀS ;

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
pﬃﬃﬃ
2 PÀS ;

q42 ¼ d½XS þ 2h0 ;

ð41Þ

3520

P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pﬃﬃﬃﬃﬃ
Ã
2 PÃ À SÃ À b1 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
Ã
Ã
Ã
¼ b2 SÃ À b1 2 PÃ À SÃ þ 2b0 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
¼ wÃ3 ðSÃ À PÃ Þ þ wÃ2 2 PÃ À SÃ þ wÃ1 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
¼ wÃ3 ðSÃ þ PÃ Þ 2 PÃ À SÃ À wÃ2 SÃ þ wÃ1 2 P Ã À SÃ À 2wÃ0
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
!
pﬃﬃﬃﬃﬃ
Ã
¼ d ðaÃ =kÞ 2 P Ã À SÃ þ g Ã1
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
pﬃﬃﬃﬃﬃ
Ã
¼ d ð2=kÞðmÃ g Ã0 þ aÃ g Ã2 Þ À ðaÃ =kÞSÃ þ g Ã1 2 PÃ À SÃ ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
!

pﬃﬃﬃﬃﬃ
Ã
Ã
¼ d ÀX Ã 2 PÃ À SÃ ; qÃ42 ¼ X Ã SÃ þ 2h0 :
Ã

qÃ11 ¼ Àb2
qÃ12
qÃ21
qÃ22
qÃ31
qÃ32
qÃ41

ð42Þ

It is clear that qij ; qÃij are explicit functions of c; k; . . ., thus, the secular Eq. (40) is fully explicit.
4. Special cases
4.1. Explicit secular equation for inhomogeneous isotropic media subject to gravity
The propagation of Stoneley waves in inhomogeneous isotropic solids under the effect of gravity was investigated by Das
et al. [3], but the authors did not derived the dispersion equation in the explicit form due to the characteristic equations for p
and pÃ being fully quartic. When the half-spaces are isotropic we have:

c011 ¼ c033 ¼ k0 þ 2l0 ;
0Ã
0Ã
0Ã
c0Ã
11 ¼ c 33 ¼ k þ 2l ;

c055 ¼ l0 ;
0Ã
c0Ã
55 ¼ l ;

c13 ¼ k0 ;

ð43Þ

0Ã
c0Ã
13 ¼ k :

On view of (43), Eq. (40) becomes:

q11

q
21

q31

q
41

q12

qÃ11

q22

qÃ21

q32
q42

hqÃ31
hqÃ41

qÃ12

qÃ22
¼ 0;
hqÃ32
Ã
hq42

ð44Þ

where h ¼ lÃ0 =l0 . The elements qij are given by:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
2 P À S À b1 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
¼ b2 S þ b1 2 P À S þ 2b0 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pﬃﬃﬃ
pﬃﬃﬃ
¼ w3 ðS À PÞ À w2 2 P À S þ w1 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
pﬃﬃﬃ
pﬃﬃﬃ
¼ Àw3 ðS þ PÞ 2 P À S À w2 S À w1 2 P À S À 2w0 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
!
pﬃﬃﬃ
¼ d À 2 P À S þ g 1 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
pﬃﬃﬃ
0 þ g 2 Þ À S À g 1 2 P À S ;
¼ d 2ðmg
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !
pﬃﬃﬃ
¼ d x 2 P À S ; q42 ¼ d½xS þ 2h0 ;

q11 ¼ b2
q12
q21
q22
q31
q32
q41

where x ¼ X=l0 ðxÃ ¼ X Ã =lÃ0 Þ and:

Þ À ð þ xmÞð
mg
0 þ g 2 Þ;
b0 ¼ h0 ðDx À m
þ xðmg
0 þ g 2 Þ þ h0 ;
b1 ¼ ½g 1 ð þ xmÞ
þ g 1 x þ xðDx À mÞ;

b2 ¼ ð þ xmÞ
À dÞ À ðmg
0 þ g 2 Þðx À d þ m
Þ;
w0 ¼ h0 ½D À mðx

w1 ¼ g 1 ðx À d þ mÞ þ ðmg 0 þ g 2 Þ þ ðx À dÞh0 ;
Þ þ g 1 þ x½D À mðx
À dÞ;
w2 ¼ ðx À d þ m
g 0 ¼ 2½2ð1 À c0 Þ À c0 x;

w3 ¼ d;

ð45Þ

3521

P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

2;
g 1 ¼ 2ð1 À c0 Þð2 À xÞ; g 2 ¼ 1 À 2c0 À m
À 2
Á
0
2
x þ ½x À 4ð1 À c Þð1 À xÞ þ þ 2m
;
h0 ¼ m
À
Á
2;
d ¼ x½x À 4ð1 À c0 Þ À 2 ; S ¼ ð1 þ c0 Þx À 2 À 2m
2 ½2ð4c0 À 3Þ þ ð1 þ c0 Þx þ m
4 À c0
P ¼ ð1 À xÞð1 À c0 xÞ À m

c0 ¼

l

0

k 0 þ 2l 0

0

;

¼

qg
;
kl0

¼
m

2 2

c0 À 1Þ;
À 2mð3

m
;
k

ð46Þ

where

D ¼ 1 À 2c0 ;

d ¼ 4l0 ð1 À c0 Þ:

ð47Þ

ﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
pﬃﬃﬃ
The elements qÃij are expressed by formulas similar to (45) in which 2 P À S is replaced by À 2 P Ã À SÃ . The quantities
Ã
Ã
Ã
Ã ; DÃ ; dÃ are given by formulas similar to (46) and (47). Eq. (44), along with (45)–(47), estabbi ; wÃi ; g Ãi ; h0 ; d ; SÃ ; P Ã ; cÃ0 ; Ã ; m
lishes the explicit secular equation of Stoneley waves for the case of inhomogeneous isotropic elastic half-spaces subject
to gravity. Note that qij and qÃij are dimensionless quantities.
4.2. Explicit secular equation for homogeneous transversely isotropic half-spaces subject to gravity
When two half-spaces are homogeneous, i.e. m ¼ mÃ ¼ 0, one can see that the explicit secular equation of the wave is of
the form (40), in which the elements qij are simpliﬁed to:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
0
q11 ¼ À 2 P À S þ a=ðkc55 Þ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
a
q12 ¼ ÀS À 0
2 P À S þ 2Dð1 À X=c055 Þ;
kc
pﬃﬃﬃ 55
q21 ¼ S À P À D À ðX À dÞ=c055 ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

h
pﬃﬃﬃi
pﬃﬃﬃ
0
q22 ¼ D þ ðX À dÞ=c055 À S À P 2 P À S þ 2ðaDÞ=ðkc55 Þ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q31 ¼ Àða=kÞ 2 P À S þ d À ð1 þ DÞX;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q32 ¼ ða=kÞð2D À SÞ À ½d À ð1 þ DÞX 2 P À S;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
2
q41 ¼ X 2 P À S; q42 ¼ XS þ 2ðX À dÞð1 À X=c055 Þ þ 2a2 =ðk c055 Þ;

ð48Þ

where

S ¼ 2D À
P¼

d
1
1
þ 0 þ 0 X;
0

c55
c33 c55

ð49Þ

ðc011 À XÞðc055 À XÞ
a2
À 2
:
c033 c055
k c033 c055

ﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
pﬃﬃﬃ
Elements qÃij ; i ¼ 1; 2; 3; 4; j ¼ 1; 2 are deﬁned by formulas similar to (48) in which 2 P À S is replaced by À 2 P Ã À SÃ .
Note that, in this case, it can be shown that the Rayleigh wave velocity is limited by:
Ã0
Ã0
Ã0
0 < c2 < minðc055 =q0 ; c011 =q0 ; cÃ0
55 =q ; c 11 =q Þ:

ð50Þ

Indeed, in view of (49)1 we have:

h

i
S ¼ c033 ðX À c011 Þ þ c055 ðX À c055 Þ þ ðc013 þ c055 Þ2 =ðc033 c055 Þ:

ð51Þ

It follows from (27)1 and (49)2 that ðc011 À XÞ and ðc055 À XÞ must have the same sign. This yields:

0 < X < minðc011 ; c055 Þ or X > maxðc011 ; c055 Þ:

ð52Þ

2

Using (51) we see that the discriminant D ¼ S À 4P of Eq. (22) is given by:

n
Â
Ã Â
Ã2 o
D ¼ ðc013 þ c055 Þ4 þ 2ðc013 þ c055 Þ2 c033 ðX À c011 Þ þ c055 ðX À c055 Þ þ c033 ðX À c011 Þ À c055 ðX À c055 Þ =ðc033 c055 Þ2 þ

4a2
2
k c033 c055

:
ð53Þ

Now, if (52)2 exists, then it follows from (53) that D P 0, so Eq. (22) for this case has two real roots p21 ; p22 with the same sign,
according to (27)1. On the other hand, it is clear from (51) and (52)2 that S ¼ p21 þ p22 > 0. Thus, both p21 and p22 are positive.

This contradicts the fact that p1 ; p2 must have a positive imaginary part.

3522

P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

Thus, it must be:

0 < X ¼ q0 c2 < minðc055 ; c011 Þ:

ð54Þ

Similarly, we have:
Ã0
0 < X Ã ¼ qÃ0 c2 < minðcÃ0
55 ; c 11 Þ:

ð55Þ

Then, from (54) and (55) we deduce (50).
Eq. (40) in which qij ; qÃij given by (48) is the explicit secular equation of Stoneley waves for orthotropic homogeneous elastic half-spaces subject to gravity. It provides the explicit secular equation of Stoneley waves for the transversely isotropic
case which was investigated by Dey and Sengupta [2].
When the media are isotropic we have the relations (43). From (40), (43), (48) and (49) one can see that the explicit secular equation for this case is:

q11

q
21

q31

q
41

q12

qÃ11

q22

qÃ21

q32

qÃ31

q42

qÃ41

qÃ12

qÃ22
¼ 0;
qÃ32
qÃ42

ð56Þ

where qij are given by:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q11 ¼ À 2ð1 þ PÞ À ð1 þ c0 Þx þ ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q12 ¼ ð1 À c0 Þð4 À 3xÞ À 2ð1 þ PÞ À ð1 þ c0 Þx;
pﬃﬃﬃ
q21 ¼ 1 À 2c0 þ c0 x À P;
pﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q22 ¼ ð2c0 À 1 À c0 x À P Þ 2ð1 þ PÞ À ð1 þ c0 Þx þ 2ð1 À 2c0 Þ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q31 ¼ À 2ð1 þ PÞ À ð1 þ c0 Þx þ 2ð1 À c0 Þð2 À xÞ;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q32 ¼ ½4ð1 À c0 Þ À ð1 þ c0 Þx À 2ð1 À c0 Þð2 À xÞ 2ð1 þ PÞ À ð1 þ c0 Þx;
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
q41 ¼ Àx 2ð1 þ PÞ À ð1 þ c0 Þx;

ð57Þ

q42 ¼ x½ð1 þ c0 Þx À 2 þ 2½x À 4ð1 À c0 Þð1 À xÞ þ 22
in which:

P ¼ ð1 À xÞð1 À c0 xÞ À c0
Elements

qÃij

2 2

:

ð58Þ

are determined by similar formulas to (57), in which

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
pﬃﬃﬃ
2ð1 þ PÞ À ð1 þ c0 Þx is replaced by À 2ð1 þ PÃ Þ À ð1 þ cÃ0 ÞxÃ :
Eqs. (56)–(58) establish the explicit secular equation for the investigation [1].
4.3. Explicit secular equation for inhomogeneous transversely isotropic half-spaces
When gravity is absent, i.e. a ¼ aÃ ¼ 0, the dispersion equation of Stoneley waves is Eq. (40), in which qij are reduced to (qÃij
determined by the similar formulas):

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ

q11 ¼ À 2 P À S þ 2m;

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
2 P À S þ 2Dð1 À X=c055 Þ þ 2m
2;
q12 ¼ ÀS À 2m
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
pﬃﬃﬃ
2 PÀSÀDþm
2 À ðX À dÞ=c055 ;
q21 ¼ S À P þ m
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
h
pﬃﬃﬃi
pﬃﬃﬃ
Â
Ã
2 þ ðX À dÞ=c055 À S À P 2 P À S þ mS
2 À ðX À dÞ=c055 ;
þ ð2mÞ
Dþm
q22 ¼ D À m
q31 ¼ d À ð1 þ DÞX;

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
À ð1 À DÞX À ½d À ð1 þ DÞX 2 P À S;
q32 ¼ 2m½d
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ

2 X;
q41 ¼ X 2 P À S; q42 ¼ XS þ 2ðX À dÞð1 À X=c055 Þ þ 2m
where S is calculated by (23)1 and:

ð59Þ

3523

P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

P¼

ðc011 À XÞðc055 À XÞ
2
Àm
c033 c055

!
1
1
d
þ
D
À
X
À

2
:
c033 c055
c055

ð60Þ

Eqs. (40), (59) and (60) provide the explicit secular equation for the investigation [6].
4.4. Explicit secular equation for homogeneous isotropic half-spaces
Now we consider the case when two half-spaces are homogeneous isotropic elastic and they are not subject to gravity. For
this case we have:

c11 ¼ c33 ¼ k þ 2l;

c55 ¼ l;

cÃ11 ¼ cÃ33 ¼ kÃ þ 2lÃ ;

c13 ¼ k;
cÃ13 ¼ kÃ ;

cÃ55 ¼ lÃ ;

ð61Þ

m ¼ mÃ ¼ a ¼ aÃ ¼ 0;

where k; l; kÃ ; lÃ are Lame constants of the half-spaces. It is easy to verify that the roots pj ; pÃj ðj ¼ 1; 2Þ of the characteristic equations (22) and (35) are:

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

c2
p1 ¼ i 1 À 2 ;
c1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
c2
p2 ¼ i 1 À 2 ;
c2

pÃ1

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
c2
¼ Ài 1 À Ã2 ;
c1

pÃ2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
c2
¼ Ài 1 À Ã2
c2

ð62Þ

Ã
Ã
Ã
Ã2
Ã

Ã
in which c21 ¼ ðk þ 2lÞ=q; c22 ¼ l=q; cÃ2
1 ¼ ðk þ 2l Þ=q ; c 2 ¼ l =q . It is clear that the secular equation (39) is equivalent
to:

1
1
À1
À1

Ã
Ã
Ã
Ã

F =ðiE Þ
F
=ðiE
Þ
ÀF
=ðiE
Þ
ÀF
=ðiE
Þ

1
1
2
2

1
2
1
2

Ã
Ã
Ã
Ã ¼ 0:
A1 =ðiklE1 Þ A2 =ðiklE2 Þ ÀA1 =ðiklE1 Þ ÀA2 =ðiklE2 Þ

B =ðÀklE Þ B =ðÀklE Þ ÀBÃ =ðÀklEÃ Þ ÀBÃ =ðÀklEÃ Þ
1
1
2
2
1
1
2
2

ð63Þ

By employing (25), (29), (34), (61), (62) one can show that:

F 1 =E1 ¼ p1 ;

A1 =ðkE1 Þ ¼ 2lp1 ;

¼
¼

¼ Àlð2 À c2 =c22 Þ=p2 ;

B2 =ðkE2 Þ ¼ À2l;
lÃ pÃ1 ; BÃ1 =ðkEÃ1 Þ ¼ ÀlÃ ð2 À c2 =cÃ2
2 Þ;
Ã
Ã
Ã
2
Ã2
Ã
¼ Àl ð2 À c =c2 Þ=p2 ; B2 =ðkE2 Þ ¼ À2lÃ :

F 2 =E2 ¼ À1=p2 ;
F Ã1 =EÃ1
F Ã2 =EÃ2

B1 =ðkE1 Þ ¼ Àlð2 À c2 =c22 Þ;

A2 =ðkE2 Þ
Ã

pÃ1 ; AÃ1 =ðkE1 Þ ¼ 2
Ã
À1=pÃ2 ; AÃ2 =ðkE2 Þ

ð64Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ã
Ã
On use of (64) into (63) and setting pj ¼ Àbj =ðikÞ; pÃj ¼ bj =ðikÞ; j ¼ 1; 2 ðbj ¼ k 1 À c2 =c2j ; bj ¼ k 1 À c2 =cÃ2
j Þ we have:

1

b =k
1

2b1 =k

2 À c2 =c2
2

Ã

Ã

k=b2
b1 =k
k=b2

Ã
Ã ¼ 0
2
2
Ã
Ã
2
Ã2
ð2 À c =c2 Þðk=b2 Þ
2ðl =lÞðb1 =kÞ
ðl =lÞð2 À c =c2 Þðk=b2 Þ

2
ÀðlÃ =lÞð2 À c2 =cÃ2
À2ðlÃ =lÞ
2 Þ
1

À1

À1

ð65Þ

0.8

x=ρ0c2/c0

55

0.75

0.7

0.65

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

*

m

Ã . Here D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7; DÃ ¼ 0:4; dÃ1 ¼ 2:5; dÃ2 ¼ 0:6;
Fig. 1. Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on m
¼ À0:1; / ¼ 0; d3 ¼ 0:25; d4 ¼ 0:3.
m

3524

P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

This is the secular equation of Stoneley waves for the case of two homogeneous isotropic elastic half-spaces without the effect of gravity, that coincides with the one published in [16].
5. Numerical results and discussion
It is readily to see from (40) that the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 depends on 11 dimensionless
Ã
Ã
Ã0
Ã0
Ã0
Ã0
Ã0
¼ m=k; / ¼ g=ðkc22 Þ; DÃ ¼ cÃ0
parameters, namely, D ¼ c013 =c033 ; d1 ¼ c011 =c055 ; d2 ¼ c055 =c033 ; m

13 =c33 ; d1 ¼ c11 =c55 ; d2 ¼ c55 =c 33 ;
0
2
0
0
Ã ¼ mÃ =k; d3 ¼ qÃ0 =q0 ; d4 ¼ cÃ0
=c
,
here
c
¼
c
=
q
.
Given
11
these
dimensionless
parameters,
it
is
not
difﬁcult
to
m
55
55
2
55

numerically calculate the dimensionless Stoneley wave velocity x using the dispersion Eq. (40). It is well known that, for
the homogeneous isotropic half-spaces not being subject to the gravity, the Stoneley wave exists if shear velocities
c2 ¼ l=q and cÃ2 ¼ lÃ =qÃ differ only slightly (see, for example, [16,17]), we therefore take d3 ¼ 0:25; d4 ¼ 0:3.
Ã in three cases: XÃ is isotropic with
Ã . Fig. 2 shows the variation of x with m
Fig. 1 shows the dependence of x on m
Ã
Ã
Ã
Ã
DÃ ¼ 0:5; d1 ¼ 4; d2 ¼ 0:25 ðkÃ ¼ 2lÃ Þ; XÃ is orthotropic with DÃ ¼ 0:5; d1 ¼ 6; d2 ¼ 0:25; XÃ is orthotropic with
Ã
Ã
Ã
for different values of the parameter /, while Fig. 4
D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:25. Fig. 3 presents the dependence of x on m

shows the variation with / of x for distinct values of the parameter m.
It is clear from Figs. 1, 3 and 4 that the dimensionless Stoneley wave velocity x depends strongly on the inhomogeneity
and the gravity. Especially, the Fig. 2 shows that the orthotropy also strongly affects on the dimensionless Stoneley wave

0.9
0.85

x=ρ0c2/c0

55

0.8
0.75

0.7
0.65
0.6
0.55
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

*

m

Ã for three cases: XÃ is isotropic (dashed line) with
Fig. 2. Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on m
Ã
Ã
Ã
Ã
Ã
Ã
DÃ ¼ 0:5; d1 ¼ 4; d2 ¼ 0:25 ðkÃ ¼ 2lÃ Þ; XÃ is orthotropic with DÃ ¼ 0:5; d1 ¼ 6; d2 ¼ 0:25 (solid line); XÃ is orthotropic with DÃ ¼ 0:5; d1 ¼ 2; d2 ¼ 0:25
¼ À0:1; / ¼ 0; d3 ¼ 0:25; d4 ¼ 0:3.
(dash-dot line). For all cases X is orthotropic with D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7, and m

0 2 0

x=ρ c /c55

0.8

0.75

0.7

0.65
−1

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1

0

m

for different values of the parameter / : / ¼ 0 (dash-dot line), / ¼ 0:2
Fig. 3. Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on m
Ã
Ã
Ã ¼ 0:1; d3 ¼ 0:25; d4 ¼ 0:3.
(solid line), / ¼ 0:5 (dashed line). For three cases: D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7; DÃ ¼ 0:5; d1 ¼ 4; d2 ¼ 0:25; m

P.C. Vinh, G. Seriani / Applied Mathematics and Computation 215 (2010) 3515–3525

3525

0.9
0.8

x=ρ0c2/c0

55

0.7
0.6
0.5
0.4
0.3
0.2
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φ
: m
¼ 0 (solid line), m
¼ À0:1
Fig. 4. Dependence of the dimensionless Stoneley wave velocity x ¼ q0 c2 =c055 on / for different values of the parameter m
Ã ¼ 0:2; d3 ¼ 0:25; d4 ¼ 0:3.
¼ À0:4 (dash-dot line). Here D ¼ 0:5; d1 ¼ 2; d2 ¼ 0:7; DÃ ¼ 0:4; dÃ1 ¼ 2:5; dÃ2 ¼ 0:6; m
(dashed line), m

velocity x. This is unlike the conclusion of Alla-Abd and Ahmed [5] which stated that the effect of orthotropy on the Stoneley
wave is small and can be neglected.
6. Conclusions
In this paper, we have derived the explicit secular equation of Stoneley waves in a non-homogeneous orthotropic elastic

medium, under the inﬂuence of gravity, using an appropriate representation of the solution. From this secular equation, we
also derive explicit secular equations for a number of cases previously investigated for Stoneley waves under the inﬂuence of
gravity, for both homogeneous and non-homogeneous elastic media. These explicit secular equations will be useful in practical applications. The numerical results show that the Stoneley wave velocity depends strongly on the inhomogeneity, the
anisotropy and the gravity.
Acknowledgements
The ﬁrst author undertook this work during his visit to OGS (Istituto Nazionale di Oceanograﬁa e Geoﬁsica Sperimentale)
with the support of the ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy.
References
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