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

Wave Motion 46 (2009) 427–434

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Wave Motion
journal homepage: www.elsevier.com/locate/wavemoti

Explicit secular equations of Rayleigh 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

Article history:
Received 14 November 2008
Received in revised form 26 March 2009
Accepted 21 April 2009
Available online 3 May 2009

Keywords:
Rayleigh waves
Rayleigh wave velocity
Orthotropic
Secular equation

Non-homogeneous
Gravity

a b s t r a c t
The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium
under the influence of gravity is investigated. Using an appropriate representation of the
solution we derive the secular equation of the wave motion in the explicit form. Moreover,
following the same approach, we obtain the explicit secular equations for a number of previously investigated Rayleigh wave problems whose dispersion equations were obtained
only in the implicit form.
Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction
Elastic surface waves in isotropic elastic solids, discovered by Lord Rayleigh [1] more than 120 years ago, have been studied extensively and exploited in a wide range of applications in Seismology, Acoustics, Geophysics, Telecommunications
Industry and Materials Science, for example. It would not be far-fetched to say that Rayleigh’s study of surface waves upon
an elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take for
granted today, stretching from mobile phones through to the study of earthquakes, as addressed by Samuel [2].
For the Rayleigh waves, their dispersion equations in the explicit form are very significant in practical applications. They
can be used for solving the direct (forward) problems: studying effects of material parameters on the wave velocity, and
especially for the inverse problems: determining material parameters from the measured values of the wave speed. Thus,
the secular equations in the explicit form are always the main purpose of investigations related to Rayleigh waves.
The problem on the propagation of Rayleigh waves under the effect of gravity is a significant problem in Seismology and
Geophysics, and it has attracted attention of many researchers such as Bromwhich [3], Love [4], Biot [5], Gilbert [6], De and
Sengupta [7], Dey and Sengupta [8], Datta [9], Das et al. [10], Abd-Alla and Ahmed [12]. Bromwhich [3], Gilbert [6] and Love
[4] treated the force of gravity as a type of body force, while Biot [5] and the other authors, following him, assumed that the
force of gravity to create a type of initial stress of hydrostatic nature. Bromwhich [3] assumed that the material is incompressible for the sake of simplicity. Love [4] finished Bromwhich’s investigation by considering the compressible case. Biot
[5] also took the assumption of incompressibility in his study. Gilbert [6] used the Bromwich’s secular equation to

* Corresponding author. Tel.: +84 4 35532164; fax: +84 4 38588817.
E-mail address: (P.C. Vinh).
0165-2125/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.wavemoti.2009.04.003


428

P.C. Vinh, G. Seriani / Wave Motion 46 (2009) 427–434

investigate the influence of gravity on the Rayleigh wave. The material was assumed to be isotropic in the investigations [3–
7,9,10], transversely isotropic in [8]. Most of the investigations supposed that the material is homogeneous. However, because any realistic model of the earth must take into account continuous changes in the vertical direction of the elastic properties of the material, the problem was extended to the non-homogeneous case by Das et al. [10]. Das et al. assumed that the
material is isotropic and they obtained the implicit secular equation. Recently, Abd-Alla and Ahmed [12] extended the problem to the orthotropic case. Abd-Alla and Ahmed [12] employed two displacement potentials for expressing the solution, and
they have derived the secular equation of the wave in the implicit form.
In the present work we analyze the orthotropic case and using an appropriate representation of the solution we derive an
explicit form of the secular equation, which also provides the explicit secular equations for a number of previous investigations related to Rayleigh waves under the gravity, where only the 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 the parameters characterizing the material and external effects (see for example [13–15]). Otherwise we call it an implicit secular equation.
2. Basic equations
Consider a non-homogeneous orthotropic elastic body occupying the half-space x3 P 0 subject to the gravity. We are
interested in a plane motion in ðx1 ; x3 Þ-plane with displacement components u1 ; u2 ; u3 such that:

ui ¼ ui ðx1 ; x3 ; tÞ;

i ¼ 1; 3;

u2  0

Then the components of the stress tensor
tions [12]:

ð1Þ

rij ; i; j ¼ 1; 3 are related to the displacement gradients by the following equa-

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 [12]:

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, the Eqs. (2) and (3) are written as follows:

u0

r

!

0

¼N

u

!
ð4Þ

r

where: u ¼ ½u1 ; u3 ŠT ; r ¼ ½r13 ; r33 ŠT , the symbol T indicates the transpose of matrices, the prime indicates the derivative with
respect to x3 and:

N¼
"

N1

N2

K

N3

!
;

N1 ¼

0

À@ 1

Àðc13 =c33 Þ@ 1

0
#

!
;

N2 ¼

1=c55

0

0

1=c33

!

q@ 2t þ ½ðc213 À c11 c33 Þ=c33 Š@ 21 Àqg@ 1
; N 3 ¼ NT1
K¼
qg@ 1
q@ 2t

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 on x3 ¼ þ1

ð6Þ

and the free-traction condition at the plane x3 ¼ 0:

r ¼ 0 on x3 ¼ 0

ð7Þ

3. Secular equation
Assume that the half-space x3 P 0 is made of a material with an exponential depth profile:

cij ¼ c0ij e2mx3 ;

q ¼ q0 e2mx3

where c0ij ; q0 ; m are constants.

ð8Þ


P.C. Vinh, G. Seriani / Wave Motion 46 (2009) 427–434

429

Now we consider the propagation of a Rayleigh wave, travelling with velocity c and wave number k in the x1 -direction.
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 [16]):

fuj ; rj3 gðx1 ; x3 ; tÞ ¼ feÀmx3 U j ðx3 Þ;

iemx3 Rj ðx3 Þgeikðx1 ÀctÞ ;

j ¼ 1; 3

ð9Þ

Substituting (9) into (4) yields:

U0

!

R0

¼ iM

!

U

ð10Þ

R

where: U ¼ ½U 1 U 3 ŠT ; R ¼ ½R1 R3 ŠT , and:

M¼

M1

M2

Q

M3

!
;

M1 ¼

!
kðX À dÞ ia
Q¼
;
Àia
kX

Àiðm=kÞ

À1

ÀD

Àiðm=kÞ

M3 ¼

!
;

iðm=kÞ ÀD
À1

!

"
M2 ¼ ð1=kÞ

1=c055

0

0

1=c033

#

iðm=kÞ

ð11Þ

here d ¼ c011 À ðc013 Þ2 Þ=c033 , D ¼ c013 =c033 , a ¼ q0 g, X ¼ q0 c2 , the prime indicates the derivative with respect to y ¼ kx3 .
It is not difficult to verify, by eliminating U from (10), that the traction vector RðyÞ is the solution of the equation:

aR00 À ibR0 À cR ¼ 0

ð12Þ

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

!
Àia=k
1 X
; d ¼ XðX À dÞ À ða=kÞ2
kd ia=k ðX À dÞ
!
1 0 g1
b ¼ M 1 Q À1 þ Q À1 M3 ¼
; g 1 ¼ d À ð1 þ DÞX
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

ð13Þ
ð14Þ
ð15Þ

in which

g 0 ¼ d À ð1 À DÞX;

g2 ¼ D À

2

2

k
2ma

d
þ 2
c055
k
d
2maD
2
h1 ¼ 2 ðX À dÞ þ D X À 0 À
2
c33
k
k
h0 ¼

m X

m2

2

k
m2

þ ðX À dÞ À

ð16Þ

Now we seek the solution of Eq. (12) in the form:

RðyÞ ¼ eipy R0

ð17Þ

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

Ip > jmj=k

ð18Þ

in order to ensure the decay condition (6). Substituting (17) into (12) leads to:

ðp2 a À pb þ cÞR0 ¼ 0

ð19Þ

As R0 is a non-zero vector, the determinant of system (19) must vanish. This provides the equation for determining p,
namely:

p4 À Sp2 þ P ¼ 0

ð20Þ

where



d
1
1
2m2
þ
þ
XÀ 2
c055
c033 c055
k


!
ðc011 À XÞðc055 À XÞ m2
1
1
d
m4 a2 þ 2amðc055 À c013 Þ
P¼
À
þ

D
À
X
À
2
þ 4 À
0 0
2
0
0
0
2
c33 c55
c33 c55
c55
k c033 c055
k
k
S ¼ 2D À

ð21Þ

It follows from (20) that:

p21 þ p22 ¼ S;

p21 p22 ¼ P

ð22Þ

430

P.C. Vinh, G. Seriani / Wave Motion 46 (2009) 427–434

where p21 ; p22 are two roots of the quadratic Eq. (20) for p2 . It is not difficult to demonstrate that vector R0 ¼ ½A BŠT , the solution
of (19), is given by:

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

ð23Þ

Let p1 , p2 be the two roots of (20) satisfying the condition (18). Then the general solution of Eq. (12) is:

RðyÞ ¼ c1

A1
B1

!

A2

eip1 y þ c2

B2

!
eip2 y

ð24Þ

where Ak ; Bk ðk ¼ 1; 2Þ are given by (23) in which p is replaced by pk , c1 ; c2 (c21 þ c22 – 0) are constants to be determined from
the boundary condition (7) that reads:

Rð0Þ ¼ 0

ð25Þ

Making use of (24) into (25) yields two equations for c1 ; c2 , namely:

"

ðia=kÞp21 þ g 1 p1 À img 0 =k À iag 2 =k ðia=kÞp22 þ g 1 p2 À img 0 =k À iag 2 =k
Xp21 þ h0

Xp22 þ h0

#

!

c1

¼0
c2

ð26Þ

and vanishing the determinant of the system leads to the secular equation that defines the Rayleigh wave velocity. After
some algebraic manipulations and removing the factor ðp2 À p1 Þ, the secular equation results in:

g 1 Xp1 p2 À ði=kÞ½mg 0 X þ ag 2 X þ ah0 Šðp1 þ p2 Þ À g 1 h0 ¼ 0

ð27Þ

Suppose that p1 ; p2 are the two roots of (20) satisfying condition (18), we shall show that the following relations hold:

P > 0;

pffiffiffi
2 P À S > 0;

pffiffiffi
p1 p2 ¼ À P;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
p1 þ p2 ¼ i 2 P À S

ð28Þ

where S; P are defined by (21).
Indeed, from (18) it follows that Imðpi Þ > 0. If the discriminant D of the quadratic Equation (20) 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, Eq. (20) for p2 has two conjugate complex roots, again P ¼ p21 p22 > 0 and
in order to ensure Imðpi Þ > 0, it must be that p1 ¼ t þ ir; p2 ¼ Àt þ ir where r is positive, and t is a real number. 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 positive imaginary part, hence
ðp1 þ p2 Þ2 is a negative number. Therefore, with the help of (22), it follows that
pffiffiffithe relations (28) are true. It is noted that the
result (28)3, (28)4 were obtained in [16], but without showing that P > 0, 2 P À S > 0.
Taking into account (28), (27) becomes:

pffiffiffi
g 1 ðX P þ h0 Þ À ðm=kÞg 0 X

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
pffiffiffi
2 P À S À ða=kÞðh0 þ g 2 XÞ 2 P À S ¼ 0

ð29Þ

0 2

Since g i , h0 , P; S are explicitly expressed in terms of X ¼ q c , Eq. (29) is fully explicit in terms of the Rayleigh wave speed.
Eq. (29) is the (exact) secular equation, in the explicit form, of Rayleigh waves in non-homogeneous orthotropic elastic media
under the influence of gravity, where g 0 , g 1 , g 2 , h0 , S, P are defined by (16)1, (14)2, (16)2, (16)3, (21)1, (21)2, respectively.
Remark 1.
(i) One can obtain the quadratic Equation (20) for p2 by another way that has been used by Kulkarni and Achenbach [17].
First, by substituting (2) into (3) and taking into account the assumption (8), an equation for u is derived and we call it
 j eipy eikðx1 ÀctÞ ðj ¼ 1; 3), into this equathe equation for the displacement vector. Then, substituting u, defined as uj ¼ A
tion yields a homogeneous system of two linear equations for constants Aj . The vanishing of the determinant of the

 that is quite complicated as remarked by Kulkarni and Achenbach
system leads to a fully quartic equation for p
[17]. By a (linear) transformation that cancels the cubic term of the equation, the authors obtain the quadratic Equation (20) for p2 . This has also been pointed out in [16]. It should be noted that, the quantities cj (j ¼ 1; 2) of the paper
[17] are not always real number, they can be complex numbers. Thus, they are required to have positive real parts,
rather than to be positive numbers, in order that the decay condition is satisfied. Also note that the results (28)1,2 (with
a ¼ 0) ensure that the expressions in the square roots of formula (29) in [17] have positive values.
e j eipy ð A
e j being constantÞ, into the equation for the displacement vector will
(ii) Substituting u defined by (9), where U j ¼ A
also yields immediately the bi-quadratic Equation (20). However, the use of the equation for the traction vector (12) is
better than that of the equation for the displacement vector, since the boundary condition is expressed in terms of the
traction vector. The secular equation is derived more quickly if we use the equation for the traction vector. This can be
seen by comparing the secular Eq. (27) in which a ¼ 0 with the corresponding Equation (39) in [17].
(iii) The representation of solution (9) indicates clearly the decay behaviours of the displacement vector u and the traction
vector r. Unlike the homogeneous case, they are quite different from each other.


P.C. Vinh, G. Seriani / Wave Motion 46 (2009) 427–434

431

(iv) One can arrive at the explicit secular equation of the wave by following the procedure carried out by Kulkarni and
Achenbach [17].

4. Explicit secular equations of Rayleigh waves under the effect of gravity in special cases
4.1. Case of a non-homogeneous isotropic elastic half-space under gravity
This problem was consider Das et al. [10], but the authors only have derived the implicit form of the secular equation.
From (29) and (14)2, (16), (21), and taking into account that the material is isotropic, i.e.

c011 ¼ c033 ¼ k0 þ 2l0 ;

c013 ¼ k0 ;

c055 ¼ l0

here k0 ; l0 are constants, the explicit secular equation for this case is:

h pffiffiffi
i
2
2ð1 À cÞð2 À xÞ x P þ ðx þ 4c À 4Þð1 À xÞ þ m2 x=k þ 2m=k þ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
È
Â
ÃÉ
À 2mxð2 À 2c À cxÞ=k þ  Àx2 þ 6ðx À 1Þ þ 4ðc À 1Þ þ 2
2 PÀS¼0
2

2

02

2

2

ð30Þ

2

where x ¼ X=l0 ¼ c2 =c02 ; c ¼ c02 =c01 ;  ¼ g=kc2 ; c01 ¼ ðk0 þ 2l0 Þ=q0 ; c02 ¼ l0 =q0 , and

S ¼ ð1 þ cÞx À 2 À 2m2 =k

2
2

4

P ¼ ð1 À xÞð1 À cxÞ À m2 ½ð1 þ cÞx þ 2ð4c À 3ފ=k þ m4 =k þ 2mð1 À 3cÞ=k À c2

ð31Þ

4.2. Case of a non-homogeneous orthotropic elastic half-space without gravity
When the gravity is absent, i.e. a ¼ 0, Eq. (29) becomes:

pffiffiffi
g 1 ðX P þ h0 Þ À ðm=kÞg 0 X

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
2 PÀS¼0

ð32Þ

in which

g 0 ¼ d À ð1 À DÞX;

g 1 ¼ d À ð1 þ DÞX

X=c055 Þ

2

h0 ¼ ðX À dÞð1 À
þ m2 X=k


d
1
1
2m2
S ¼ 2D À 0 þ 0 þ 0 X À 2
c55
c33 c55
k


!
ðc011 À XÞðc055 À XÞ m2
1
1
d
m4
P¼
À 2
þ 0 X À 2D À 0 þ 4

0 0
0
c33 c55
c33 c55
c55
k
k

ð33Þ

Eq. (32) coincides with the secular equation derived recently by Destrade [16]. Note that in [16] it is not shown P > 0 and
pffiffiffi
2 P À S > 0.
4.3. Case of a non-homogeneous transversely isotropic elastic half-space without gravity
This problem was considered by Pal and Acharya [11], but only the implicit form of the secular equation has been derived
in their work. In their notations, the explicit secular equation for the problem is Eq. (32), where m is replaced by m=2, the
functions g 0 ; g 1 ; h0 ; S; P (in terms of X) are given by (33) in which c011 ; c013 ; c033 ; c055 ; q0 are replaced by A1 ; F 1 ; C 1 ; L1 ; q1 , respectively. Here A1 ; F 1 ; C 1 ; L1 are the material constants (see [18]).
4.4. Case of a homogeneous orthotropic elastic half-space under gravity
In this case m ¼ 0, Eq. (29) thus is simplified to:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h pffiffiffi
i
pffiffiffi
g 1 X P þ h0 À ða=kÞ½h0 þ DX Š 2 P À S ¼ 0

ð34Þ

in which:

S ¼ 2D þ ðX À dÞ=c55 þ X=c33 ;
g 1 ¼ d À ð1 þ DÞX;

P¼




c11
X
X
a2
1À
À 2
À
c55
c33 c33
k c33 c55
2

h0 ¼ ðX À dÞð1 À X=c55 Þ þ a2 =ðk c55 Þ

ð35Þ
ð36Þ


432

P.C. Vinh, G. Seriani / Wave Motion 46 (2009) 427–434

Eq. (34) is the explicit secular equation of Rayleigh waves in orthotropic elastic media under the effect of gravity. In this
case one can show that the Rayleigh wave velocity is limited by:

0 < X ¼ qc2 < minðc55 ; c11 Þ

ð37Þ

4.5. Case of a homogeneous transversely isotropic elastic half-space under gravity
This problem was considered by Dey and Sengupta [8], and only the implicit form of the secular equation was derived. In
their notations, the explicit secular equation for this problem is:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h pffiffiffi
i
pffiffiffi
g 1 X P þ h0 À ða=kÞ½h0 þ DX Š 2 P À S ¼ 0

ð38Þ

in which:

S¼



2F 2F 2 2A
2 1
X;
þ
þ

þ
À
C
L
L C
CL

g 1 ¼ Àð1 þ F=CÞX þ A À F 2 =C;



A X
À
C C

P¼



2X
2a2
1À
À
L
CL

h0 ¼ ðX À A þ F 2 =CÞð1 À 2X=LÞ þ 2a2 =L

ð39Þ
ð40Þ

here A; C; F; L are the material constants. It is noted that Eq. (38) is Eq. (34) in which c11 ; c33 ; c13 ; c55 are replaced by A; C; F; L=2,
respectively. The Rayleigh wave velocity is also subjected to the limitation (37) in which c11 ; c55 are, respectively, replaced by
A; L=2.
4.6. Case of a homogeneous isotropic elastic half-space under gravity
By putting m ¼ 0 in Eq. (30) and replacing c0ij ; q0 by cij ; q we obtain the explicit secular equation of Rayleigh waves in
homogeneous isotropic elastic half-space under gravity, namely:

!
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1 À cÞð2 À xÞ x ð1 À xÞð1 À cxÞ À c2 2 þ ðx þ 4c À 4Þð1 À xÞ þ 2
Â

2

þ ðx þ 4c À 4Þð1 À xÞ þ ð1 À 2cÞx þ 

Ã

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2 ð1 À xÞð1 À cxÞ À c  þ 2 À ð1 þ cÞx ¼ 0

ð41Þ

2

where x ¼ c2 =c22 ; c ¼ c22 =c21 ;  ¼ g=kc2 ; c21 ¼ ðk þ 2lÞ=q; c22 ¼ l=q and

S ¼ ð1 þ cÞx À 2;

P ¼ ð1 À xÞð1 À cxÞ À c2

ð42Þ

here k; l are Lame’s constants. The Eq. (41) provides the exact secular equation in the explicit form for the investigations by
De and Sengupta [7] and Datta [9].
4.7. Case of a homogeneous orthotropic elastic half-space without gravity
When the material is homogeneous and the gravity is absent we have: m ¼ a ¼ 0. Then Eq. (29) is simplified to (see also
[19,20]):

ðc55 À XÞ½c213 À c33 ðc11 À Xފ þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c33 c55 X ðc11 À XÞðc55 À XÞ ¼ 0

ð43Þ

In this case we can obtain the explicit formula for the Rayleigh wave velocity (see [20]), namely:

qc2 =c55 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
0 pffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffi 3

pffiffiffiffi
3
b1 b2 b3
ð b1 =3Þðb2 b3 þ 2Þ þ R þ D þ R À D

ð44Þ

where b1 ¼ c033 =c011 ; b2 ¼ d=c011 ; b3 ¼ c011 =c055 ; R and D are given by:

1
hðb1 ; b2 ; b3 Þ
54
i
1 h pffiffiffiffiffi
2 b1 ð1 À b2 Þhðb1 ; b2 ; b3 Þ þ 27b1 ð1 À b2 Þ2 þ b1 ð1 À b2 b3 Þ2 þ 4
D¼À
108

R¼À

ð45Þ

in which

hðb1 ; b2 ; b3 Þ ¼

pffiffiffi
b1 ½2b1 ð1 À b2 b3 Þ3 þ 9ð3b2 À b2 b3 À 2ފ

ð46Þ

and the roots in (44) taking their principal values. It is clear that the speed of Raleigh waves in homogeneous orthotropic
elastic solids is a continuous function of three dimensionless parameters b1 ; b2 ; b3 .


433

P.C. Vinh, G. Seriani / Wave Motion 46 (2009) 427–434
1

0.9

0.8

0.7

x=ρ c /c

0
55

0.6

0 2

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−m/k
Fig. 1. Dependence of squared dimensionless Rayleigh wave velocity x ¼ q0 c2 =c055 on the parameter Àm=k with different values of
 ¼ 0:3 (dashed line),  ¼ 0:5 (dash-dot line),  ¼ 0:8 (dotted line).

 :  ¼ 0 (solid line),

1

0.9

0 2

0

x=ρ c /c55

0.8

0.7

0.6

0.5

0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ε

0.8

Fig. 2. Dependence of squared dimensionless Rayleigh wave velocity x ¼ q0 c2 =c055 on the parameter  with different values of Àm=k : Àm=k ¼ 0 (solid line),
Àm=k ¼ 0:1 (dotted line), Àm=k ¼ 0:2 (dashed line), Àm=k ¼ 0:4 (dash-dot line).

5. A numerical example
As an example, we consider a non-homogeneous orthotropic elastic half-space whose elastic constants and mass density
are defined by (8), in which m 6 0 and (see [16]):

c011 =q0 ¼ 9 ðkm=sÞ2 ;

c013 =q0 ¼ 3:6 ðkm=sÞ2

c033 =q0 ¼ 9:89 ðkm=sÞ2 ;

c055 =q0 ¼ 2:182 ðkm=sÞ2

ð47Þ

Taking into account (47), it is easy to numerically solve the secular Equation (29), and the dependence of squared dimension0
less Rayleigh wave velocity x ¼ q0 c2 =c055 on Àm=k and  ¼ q0 g=kc55 are shown in Figs. 1 and 2. It appears that the influence of
the inhomogeneity on the Rayleigh wave velocity is stronger than that of the gravity.


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6. Conclusions
The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity is
considered and the secular equation of the wave motion in the explicit form is derived. Furthermore, by considering various
special cases, the explicit secular equations is obtained for the Rayleigh wave motions under the effect of inhomogeneity
and/or gravity, corresponding to a number of previous studies in which only the implicit dispersion equations were given.
The explicit secular equations derived in this work may be useful in practical applications.
Acknowledgements
The authors wish to thank Prof. J.D. Achenbach for helpful discussions. They also would like to give thanks to an anonymous reviewer for recommending the paper by F. Gilbert. The first author undertook this work during his visit to the
OGS (Istituto Nazionale di Oceanografia e Geofisica Sperimentale) with the support of the ICTP Programme for Training
and Research in Italian Laboratories, Trieste, Italy.
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