DSpace at VNU: Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress

Applied Mathematics and Computation 215 (2009) 395–404

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

Explicit secular equations of Rayleigh waves in elastic media under
the influence of gravity and initial stress
Pham Chi Vinh
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

a r t i c l e

i n f o

Keywords:
Rayleigh waves
Rayleigh wave velocity
Gravity
Initial stress
Orthotropic
Secular equation

a b s t r a c t
The problem of Rayleigh waves in an orthotropic elastic medium under the influence of
gravity and initial stress was investigated by Abd-Alla [A. M. Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl. Math. Comput. 99 (1999)
61–69], and the secular equation of the wave in the implicit form was derived. However,
due to the uncorrect representation of the solution, the secular equation is not right. The
main aim of the present paper is to reconsider this problem. We find the secular equation
of the wave in explicit form. By considering some special cases, we obtain the exact explicit

secular equations of Rayleigh waves under the effect of gravity of some previous studies, in
which only implicit secular equations were derived.
Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction
Elastic surface waves in isotropic elastic solids, discovered by 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 wave, its dispersion equations in the explicit form are very significant in practical applications. They can
be used for solving the forward (direct) problems, and especially for the inverse problems. Thus, the secular equations in the
explicit form are always the main purpose of investigations related to the Rayleigh wave.
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 [3–11].
Bromwhich [3] and Love [4] treated the force of gravity as a type of body force while Biot [5,6] 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 [5,6], Kuipers [10] also took the assumption of incompressibility in their studies. The
material is assumed to be isotropic in the investigations [3–7,9,10], transversely isotropic in [8], and orthotropic in [11].
Following Biot’s approach, Dey and Mahto [12] investigated the influence of gravity on the propagation of the Rayleigh
wave in an isotropic elastic medium, taking into account the effect of initial stress. The authors have derived the implicit
secular equation of the wave. Recently, Abd-Alla [13] extended this problem to the orthotropic case. He employed two displacement potentials for representing the solution, and has also derived the dispesion equation of Rayleigh waves in the implicit form. However, as will be shown, his represention of solution is uncorrect, the secular equation is, thus, not true.
E-mail address:
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2009.05.014


396

P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

The main purpose of the present paper is to re-investigate the problem on the propagation of Rayleigh waves in an orthotropic elastic medium under the effect of gravity and initial stress. Unlike Abd-Alla, we seek the solution directly, do not use
the displacement potentials. Interestingly that, we have found the dispersion equations of the wave in the explicit form.
From this we obtain the explicit secular equation for Dey and Mahto’s investigation [12]. When the initial stress is absent,
by considering its special cases, we derive the (exact) explicit secular equations of Rayleigh waves under the effect of gravity
of the previous studies [7–9], in which only the implicit secular equations have been found.
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 the material and external effects (see for example, [14–16]). Otherwise we call it an implicit
secular equation.
2. Basic equations
Consider a homogeneous orthotropic elastic body occupying the half-space x3 6 0 subject to the gravity and an initial
compression P 0 along the x1 -direction (see [13]). 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:

ð1Þ

Then the components of the stress tensor
[13]:

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

r11 ¼ ðc11 þ P0 Þu1;1 þ ðc13 þ P0 Þu3;3 ;
r33 ¼ c13 u1;1 þ c33 u3;3 ;
r13 ¼ c44 ðu1;3 þ u3;1 Þ;

ð2Þ

where cij are the material constants.
Equations of motion are [13]:

r11;1 þ r13;3 þ ðP0 =2Þðu1;3 À u3;1 Þ;3 À qgu3;1 ¼ qu€1 ;
r13;1 þ r33;3 þ ðP0 =2Þðu1;3 À u3;1 Þ;1 þ 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 signifies differentiation with respect to the time t, commas indicate differentiation with respect to the spatial variables xi . From (2)3 it
follows:

1
r31 À u3;1 :
c44

u1;3 ¼

ð4Þ

Analogously, from (2)2 we have:

1
c
r33 À 13 u1;1 :
c33
c33

u3;3 ¼

ð5Þ

Employing (3)2 and using (4) yield:

r33;3 ¼ qu€3 À qgu1;1 þ P0 u3;11 À /r13;1 ;

ð6Þ

where / ¼ 1 þ P0 =ð2c44 Þ. From (3)1 and taking into account (2)1, (4), (5) we have:

r13;3 ¼ ðq=/Þu€1 À ½ðd þ P0 Þ=/Šu1;11 þ ðqg=/Þu3;1 À ðD=/Þr33;1 ;
where d ¼ c11 À
are written as:

u0

!

r0

¼N

c213 =c33 ,

u

r

ð7Þ

D ¼ c13 =c33 . In matrix (operator) form, following the Stroh formalism (see [17,18]), the Eqs. (4)–(7)

!
;

ð8Þ

Â
ÃT
where: u ¼ u1 ; uÃ3 ; uÃ3 ¼ u3 =/; r ¼ ½r13 ; r33 ŠT , the symbol T indicates the transpose of matrices, the prime indicates the
derivative with respect to x3 and:

N¼
"
K¼

N1

N2

K

N3

!
;

N1 ¼

0

À/@ 1

ÀðD=/Þ@ 1

!

0

ðq=/Þ@ 2t À ½ðd þ P0 Þ=/Š@ 21

qg@ 1

Àqg@ 1

q@ 2t þ P0 @ 21

; N2 ¼
#
;

1=c44

0

0

1=ð/c33 Þ

!
;
ð9Þ

N 3 ¼ N T1 :

Here we use the notations: @ 1 ¼ @=ð@x1 Þ; @ 21 ¼ @ 2 =ð@x21 Þ; @ 2t ¼ @ 2 =ð@t2 Þ.
In addition to Eq. (8), the displacement vector u and the traction vector

r are required to satisfy the decay condition:


P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

rðÀ1Þ ¼ 0

uðÀ1Þ ¼ 0;

397

ð10Þ

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

rð0Þ ¼ 0:

ð11Þ

3. Secular equation
Now we consider the propagation of a Rayleigh wave, travelling with velocity c and wave number k in the x1 -direction.

The components u1 ; uÃ3 of the displacement vector and r13 ; r33 of the traction vector at the planes x3 ¼ const are found in the
form:

È
É
u1 ; uÃ3 ; rj3 ðx1 ; x3 ; tÞ ¼ fU 1 ðx3 Þ; U 3 ðx3 Þ; iRj ðx3 Þgeikðx1 ÀctÞ ;

j ¼ 1; 3:

ð12Þ

Substituting (12) into (8) yields:

U0

!

R0

¼ iM

where: U ¼ ½U 1

M¼

M1

U

R

!
;

ð13Þ

U 3 Š T ; R ¼ ½R 1

M2

!
;

R3 ŠT , and:

M1 ¼

0

À/

!
;

M 2 ¼ ð1=kÞ

Q M3
ÀD=/ 0
!
Àia

ðX À d À P0 Þ=/
; M 3 ¼ M T1 ;
Q ¼k
ia
/ðX þ P 0 Þ

1=c44

0

0

1=ð/c33 Þ

!
;
ð14Þ

here a ¼ qg=k, X ¼ qc2 , the prime indicates the derivative with respect to y ¼ kx3 .
Following the approach employed in [16,19,20], by eliminating U from (13), we obtain the equation for the traction vector
RðyÞ, namely:

a^ R00 À ib^R0 À c^R ¼ 0;

ð15Þ

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

!
ia
1 /ðX þ P0 Þ
;
kd
Àia
ðX À d À P 0 Þ=/
!
^ ¼ M 1 Q À1 þ Q À1 M3 ¼ 1 0 g 1 ;
b
kd g 1 0

a^ ¼ Q À1 ¼

d ¼ ðX þ P 0 ÞðX À d À P 0 Þ À a2 ;

ð16Þ
ð17Þ

where

g 1 ¼ ÀðX À d À P0 Þ À DðX þ P0 Þ;

c^ ¼ M1 Q

À1

1 h0 ÀiDa
M3 À M2 ¼

kd iDa
h1

!

ð18Þ
ð19Þ

in which

h0 ¼ /ðX À d À P 0 Þ À d=c44 ;
h1 ¼ D2 ðX þ P 0 Þ=/ À d=ð/c33 Þ:

ð20Þ

Now we seek the solution of the Eq. (15) in the form:

RðyÞ ¼ eipy R0 ;

ð21Þ

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

Ip < 0

ð22Þ

in order to ensure the decay condition (10). Substituting (21) into (15) leads to:

^þc

^ À pb
^ÞR0 ¼ 0:
ðp2 a

ð23Þ

As R0 is a non-zero vector, the determinant of the system (23) must vanish. This provides the equation for determining p,
namely:

p4 À Sp2 þ P ¼ 0;
where

ð24Þ


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P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

1
1
S ¼ 2D þ
ðX À d À P 0 Þ þ
ðX þ P 0 Þ;
/c44
c33



c11 X À P 0

X þ P0
a2
P¼
À
:
1À
À
c33
c33
/c44
/c33 c44

ð25Þ

It follows from (24) that:

p21 þ p22 ¼ S;

p21 p22 ¼ P;

ð26Þ

p21 ; p22

2

where
are two roots of the quadratic equation (24) for p . It is not difficult to demonstrate that vector R0 ¼ ½A
solution of (23), is given by:

T

BŠ , the

A ¼ g 1 p þ iaðD À p2 Þ;
B ¼ /ðX þ P 0 Þp2 þ h0 :

ð27Þ

Let p1 , p2 be the two roots of (24) satisfying the condition (22). Then the general solution of the equation (15) is:

RðyÞ ¼ c1

A1
B1

!

eip1 y þ c2

A2
B2

!
eip2 y ;

ð28Þ

where Ak ; Bk ðk ¼ 1; 2Þ are given by (27) in which p is replaced by pk , c1 ; c2 are non-zero constants to be determined from the
boundary condition (11) that reads:

Rð0Þ ¼ 0:

ð29Þ

Making use of (28) into (29) yields two equations for c1 , c2 :

"

g 1 p1 þ iaðD À p21 Þ g 1 p2 þ iaðD À p22 Þ
ðX þ

P0 Þp21

þ h0

ðX þ

P 0 Þp22

þ h0

#

!

c1
¼0
c2

ð30Þ

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 is:

g 1 /ðX þ P0 Þp1 p2 þ ia½h0 þ D/ðX þ P0 ފðp1 þ p2 Þ À g 1 h0 ¼ 0:

ð31Þ

Suppose p1 ; p2 are the two roots of (24) satisfying the condition (22). We shall show that:

P > 0;

pffiffiffi
2 P À S > 0;

pffiffiffi
p1 p2 ¼ À P ;

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

ð32Þ

where S; P are defined by (25).
Indeed, if the discriminant D of the quadratic Eq. (24) for p2 is non-negative, then its two roots must be negative in order
that (22) is to be satisfied. In this case, P ¼ p21 p22 > 0 and the pair p1 ; p2 are of the form: p1 ¼ Àir 1 ; p2 ¼ Àir 2 where r 1 ; r 2 are
positive. If D < 0, the quadratic Eq. (24) for p2 has two conjugate complex roots, again P ¼ p21 p22 > 0, and in order to ensure
the condition (22): p1 ¼ t À ir; p2 ¼ Àt À 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 negative imaginary part, hence ðp1 þ p2 Þ2 is a negative
number. Therefore, with the help of (26), it follows that the relations (32) are true.
Taking into account (32) Eq. (31) becomes:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i
pffiffiffi
pffiffiffi
g 1 /ðX þ P0 Þ P þ h0 À a½h0 þ D/ðX þ P0 ފ 2 P À S ¼ 0:

ð33Þ

Eq. (33) is the (exact) secular equation of Rayleigh waves in orthotropic elastic media under the gravity and the initial compression. Since P; S, g 1 , h0 , /, d, D are explicitly expressed in terms of c; k; . . ., it is clear that the secular Eq. (33) is fully explicit.
As a depends on the wave number k, so does the Rayleigh wave velocity defined by Eq. (33). Thus, the Rayleigh wave in
orthotropic elastic media under the gravity and the initial compression is dispersive.
When the pre-stress is absent, i.e. P 0 ¼ 0, the Eq. (33) coincides with equation (2.15) in [21] with a ¼ 0. However, it
should be observed, as above, that the expressions in the square roots of equation (2.15) in [21] have positive values.

4. Special cases
4.1. Rayleigh waves in isotropic elastic half-spaces under gravity and initial stresses
The problem was considered by Dey and Mahto [12], and the authors have been derived the secular equation in the implicit form. In their notations, the explicit secular equation for this problem is:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
i
pffiffiffiffiffi
pffiffiffiffiffi ffi
g 1 /ðX þ PÞ Pà þ h0 À a½h0 þ D/ðX þ Pފ 2 P à À S ¼ 0;

ð34Þ


P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

399

where:

1
1
S ¼ 2D þ
ðX À d À PÞ þ
ðX þ PÞ;
/Q 2
B22



B11
X
XþP
a2
PÃ ¼
1À
À
À
;
/Q 2
B22 B22

/Q 2 B22
P
B12 À P
ðB12 À PÞ2
; D¼
; d ¼ B11 À P À
;
2Q 2
B22
B22
g 1 ¼ ðd þ P À XÞ À DðX þ PÞ;


XþP
a2
;
h0 ¼ ðX À d À PÞ / À
þ
Q2
Q2

/¼1þ

ð35Þ

B11 , B12 , B22 , Q 2 are given by the formula (8a) in [12] (or by (13) in [22]). It is noted that Eq. (34) is derived from Eq. (33) in
which c11 , c33 , c13 , c44 , P 0 , P are replaced by B11 À P, B22 , B12 À P, Q 2 , P, P Ã respectively.
4.2. Rayleigh waves in orthotropic elastic half-spaces under the gravity
When the initial stress is absent, i.e. P0 ¼ 0, the Eq. (33) becomes:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h pffiffiffi
i
pffiffiffi
g 1 X P þ h0 À a½h0 þ DX Š 2 P À S ¼ 0

ð36Þ

in which:

S ¼ 2D þ ðX À dÞ=c44 þ X=c33 ;

P¼




c11
X
X
a2
À
;
1À
À
c44
c33 c33
c33 c44

g 1 ¼ Àð1 þ DÞX þ d; h0 ¼ ðX À dÞð1 À X=c44 Þ þ a2 =c44 :

ð37Þ
ð38Þ

Eq. (36) is the (exact) explicit secular equation of Rayleigh waves in orthotropic elastic media under the effect of gravity. In
this case we can show that the Rayleigh wave velocity is limited by:

0 < X ¼ qc2 < minðc44 ; c11 Þ:

ð39Þ

Indeed, first we rewrite (37)1 as follows:

h
i
S ¼ c33 ðX À c11 Þ þ c44 ðX À c44 Þ þ ðc13 þ c44 Þ2 =ðc33 c44 Þ:

ð40Þ

It follows from (32)1 and (37)2 that ðc11 À XÞ and ðc44 À XÞ must have the same sign. This yields:

0 < X < minðc11 c44 Þ or X > maxðc11 ; c44 Þ:

ð41Þ

2

On use of (40) we see that the discriminant D ¼ S À 4P of Eq. (24) is given by:

o

n
4a2
D ¼ ðc13 þ c44 Þ4 þ 2ðc13 þ c44 Þ2 ½c33 ðX À c11 Þ þ c44 ðX À c44 ފ þ ½c33 ðX À c11 Þ À c44 ðX À c44 ފ2 =ðc33 c44 Þ2 þ
:
c33 c44

ð42Þ

Now, if the (41)2 exists, then it follows from (42) that D P 0, so Eq. (24) for this case has two real roots p21 ; p22 with the same sign,
according to (32)1. On the other hand, it is clear from (40) and (41)2 that S ¼ p21 þ p22 > 0. Thus, both p21 and p22 are positive. This
leads to the contradiction to the requirement that p1 ; p2 must have negative imaginary part. The inequalities (39) are proved.
4.3. Rayleigh waves in transversely isotropic elastic media under the effect of gravity
This problem was considered by Dey and Sengupta [8], but only the implicit form of the secular equation has been derived
in their work. In their notations, the explicit secular equation for this problem is:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h pffiffiffi
i
pffiffiffi
g 1 X P þ h0 À a½h0 þ DX Š 2 P À S ¼ 0

ð43Þ

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;


P¼

A X
À
C C


1À


2X
2a2
À
;
L
CL

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

ð44Þ
ð45Þ

here A; C; F; L are the material constants (see also [23]). It is noted that Eq. (43) is Eq. (36) in which c11 , c33 , c13 , c44 are replaced
by A, C, F, L=2, respectively. The Rayleigh wave velocity is also subjected to the limitation (39) in which c11 , c44 are replaced by
A, L=2, respectively.


400

P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

4.4. Rayleigh waves in isotropic elastic half-spaces under the gravity
When the material is isotropic we have:

c11 ¼ c33 ¼ k þ 2l;
where k,

c44 ¼ l;

c13 ¼ k;

ð46Þ

l are Lame’s constants. On view of (46) the limitation (39) becomes:

0 < x < 1;

ð47Þ

pffiffiffiffiffiffiffiffiffi
where x ¼ c2 =c22 (dimensionless Rayleigh wave speed), c2 ¼ l=q (the shear wave velocity), and the expression (37)1 of S is
simplified to:

S ¼ ð1 þ cÞx À 2;

0 < c ¼ c22 =c21 < 1;

ð48Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where c1 ¼ ðk þ 2lÞ=q is the longitudinal wave velocity. Now, making use of (46) along with (37)2, (38), (48) into (36) we
have:

!
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
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Â
Ã
À  ðx þ 4c À 4Þð1 À xÞ þ ð1 À 2cÞx þ 2 2 ð1 À xÞð1 À cxÞ À c2 2 þ 2 À ð1 þ cÞx ¼ 0;

ð49Þ

2

where  ¼ g=ðkc2 Þ. Eq. (49) is the (exact) secular equation, in the explicit form, of Rayleigh waves in isotropic elastic halfspaces under the influence of gravity. It is noted that this problem was considered by De and Sengupta [7] and Datta [9],
but only the implicit dispersion equation of the wave have been derived.
2
Now suppose that  ¼ g=ðkc2 Þ is much small comparison with the unit. Then by omitting the powers of order bigger than
one in terms of , from the exact secular Eq. (49) we have immediately:

pffiffiffiffiffiffiffiffiffiffiffih
pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffii
2ðc À 1Þð2 À xÞ 1 À x ð2 À xÞ2 À 4 1 À x 1 À cx þ x½ðx þ 4c À 4Þð1 À xÞ þ ð1 À 2cÞxŠ ¼ 0:

ð50Þ

Eq. (50) is an approximate dispersion equation of Rayleigh waves in isotropic elastic media under the effect of gravity, in the
2
case 0 <  ¼ g=ðkc2 Þ ( 1.
Remark 1. (i) As noted in Remark 2.3i (Section 5), when the material is isotropic and the stress is absent ðP0 ¼ 0Þ we can
express the displacement components u1 , u3 in terms of the potentilas u, w by (71), in which u, w satisfy (81). It is not
difficult to see that corresponding to a surface wave travelling with velocity c and wave number k in the x1 -direction and
decaying in the x3 -direction, the potentials u, w are given by:

Â

Ã

u ¼ AÃ1 eikp1 x3 þ AÃ2 eikp2 x3 eikðx1 ÀctÞ ;

Â
Ã
w ¼ ik AÃ1 m1 eikpx3 þ AÃ2 m2 eikp2 x3 eikðx1 ÀctÞ ;

ð51Þ
AÃ1 ,

c21 ðp2j

AÃ2

where mj ¼
À 1Þ; j ¼ 1; 2, p1 , p2 are roots of Eq. (24) with negative imaginary parts,
are non-zero constants. On use
of (51), (71) into (2)2,3 in which c13 ¼ k, c33 ¼ k þ 2l, c44 ¼ l, and taking into account (12), we have:

"

RðyÞ ¼ c1

A1
B1

#

e

"

ip1 y

2
þc

A2
B2

#

eip2 y ;

ð52Þ

where:

Aj ¼ l½2pj þ ikmj ð1 À p2j ފ;
Bj ¼ kð1 þ p2j Þ þ 2lðp2j þ ikmj pj Þ;

j ¼ 1; 2;

ð53Þ

c1 , c2 , are non-zero constants. Substituting (52) into (29) leads to a homogeneous linear system for c1 , c2 . Vanishing the
determinant of this system provides:



 A1

 B1




A2 
 ¼ 0:
B2 

ð54Þ

With the help of (32), it is not difficult to verify that the Eq. (54) is equivalent to the equation:

ffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ð1 À xÞð1 À cxÞ À c2 þ 2 À ð1 þ cÞx ðx À 2Þ2 À 4 ð1 À xÞð1 À cxÞ À c2 À 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
þ 4 c À ð1 À xÞð1 À cxÞ À c2 ¼ 0

ð55Þ

pffiffiffiffiffiffiffiffiffi
in the interval x 2 ð0; 1Þ. Note that, since the limiting velocity (see, for instance, [24,25]) in this case is c2 ¼ l=q, the imaginary parts p1 , p2 are negative for the values of x 2 ð0; 1Þ. Consequently, the relations (32) hold for values of x belonging to this
interval.


P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

401

(2i) Eqs. (49) and (55) are different in form, but they are equivalent to each other in the interval (0, 1). This is proved as
follows. First, we recall that Eq. (54) is equivalent to Eq. (55) in the interval (0, 1). It is clear from (27)–(29) that Eq. (49) is

equivalent to the equation:


e 2 
A
¼0
e2 
B


e
 A1

e1
B

ð56Þ

in the interval (0, 1), where:



e j ¼ g p þ ia D À p2 ;
A
1 j
j

e j ¼ Xp2 þ h0 ;
B
j

j ¼ 1; 2;

ð57Þ

g 1 , h0 , D correspond to the isotropic elastic solids without pre-stress ðP0 ¼ 0Þ.
Now, suppose that x is a root of (55) and 0 < x < 1, then x is a root of (54) and RðyÞ given by (52) is a solution of Eq. (16),
i.e.:

a^ R00 À ib^R0 À c^R  0;

ð58Þ

^ c
^ , b,
^ are correspond to the isotropic elastic solids without pre-stress ðP 0 ¼ 0Þ. Since functions eip1 y , eip2 y are linearly
where a
independent of each other (noting that p1 – p2 ), from (58) it follows:



^ p2j

a

" #
 A
j
^
^

¼ 0;
À bpj þ c
Bj

j ¼ 1; 2:

ð59Þ

From (59) it deduces that:

Bj

¼

Aj

ej
B
ej
A

or

Bj Aj
¼ ¼ Lj – 0;
ej A
ej
B

j ¼ 1; 2:

ð60Þ

On view of (60) it is clear that Eqs. (54) and (56) are equivalent to each other, therefore, x is a root of Eq. (56). Since (56) is
equivalent to (49), x is a root of (49) also. Thus, it has been observed that if x is a root of Eq. (55) then it is a root of Eq. (49).
Now let x be a root of (49) and 0 < x < 1, then it is a solution of (56), and the corresponding potentials u, w given by (51)
satisfy (81). Therefore RðyÞ defined by (52) is a solution of (16). This again leads to the relations (60), and as its consequence,
x is a root of (54), thus x is a root of (55), because (54) is equivalent to (55). The proof is finished.
(3i) In the case that 0 <  ( 1, by neglecting the powers of order bigger than one in terms of , and taking into account the
equalities:

ffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ð1 À xÞð1 À cxÞ þ 2 À ð1 þ cÞx ¼ 1 À x þ 1 À cx;
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À x À 1 À cx
1
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼
:
xðc À 1Þ
1 À x þ 1 À cx

ð61Þ
ð62Þ

Eq. (55) becomes:

h

pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffii
ðx À 2Þ2 À 4 1 À x 1 À cx þ

Replacing x by c2 =c22 , c by c22 =c21 ,

À

2 À c2 =c22

À

Á2

pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffii
4 h
ð1 þ c À cxÞ 1 À x À ð1 þ c À xÞ 1 À cx ¼ 0:
xðc À 1Þ

ð63Þ

 by g=ðkc22 Þ, from (63) we have:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
À 4 1 À c2 =c21 1 À c2 =c22

!
À 2
Áqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 À 2
Áqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

4g
2
2
2 =c À c þ c 2 À c 2 c 2 =c 2
2 =c
Á
1
À
c
1
À
c
þ
c
À
c
c
1
2
1
2
1
2
2
1 ¼ 0:
c2 c21 À c22 k
À

ð64Þ

Eq. (64) is an approximate secular equation of Rayleigh waves in isotropic elastic media under the effect of gravity, in the
 
2
case 0 <  ¼ g= kc2 ( 1, and it was first derived by Love [4] in a different way.
Interestingly that, the (original) exact Eqs. (49) and (55) give the same roots, while the corresponding approximate Eqs.
(50) and (63) give different solutions (see Fig. 1).
4.5. Rayleigh waves in orthotropic elastic media without the effect of gravity and initial stress
When both initial stress and gravity are absent, i.e. P 0 ¼ a ¼ 0, the Eq. (33) simplifies to (see also [26,27]):

Â
à pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðc44 À XÞ c213 À c33 ðc11 À XÞ þ c33 c44 X ðc11 À XÞðc44 À XÞ ¼ 0:

ð65Þ


402

P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

0.8
0.7

2

γ=0.5

0.5

2

x=c /c2

0.6

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

ε
2

Fig. 1. Dependence on  ¼ g=kc2 of x ¼ c2 =c22 defined by the exact secular Eqs. (49) and (55) (solid line), approximate Eq. (50) (dashed line) and approximate
one (63) (dash-dot line), with c ¼ 0:5.

In this case we can obtain the explicit formula for the Rayleigh wave velocity (see [27]), namely:

pffiffiffi

pffiffiffi

qc2 =c44 ¼ b1 b2 b3 = ð b1 =3Þðb2 b3 þ 2Þ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
pffiffiffiffi 3
3
Rþ Dþ RÀ D ;

ð66Þ

where b1 ¼ c33 =c11 , b2 ¼ d=c11 , b3 ¼ c11 =c44 , 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

ð67Þ

in which

hðb1 ; b2 ; b3 Þ ¼

i
pffiffiffi h
b1 2b1 ð1 À b2 b3 Þ3 þ 9ð3b2 À b2 b3 À 2Þ

ð68Þ

and the roots in (65) taking their principal values. It is clear that the speed of Rayleigh waves in orthotropic elastic solids is a
continuous function of three dimensionless parameters
5. On Abd-Alla’s representation of solution
We recall briefly Abd-Alla’s representation of solution, and then show that it is uncorrect. First, substituting (2) into (3)
and taking into account the assumption: c44 ¼ ðc11 À c13 Þ=2, he obtained:

€1 ;
ðc11 þ P0 Þð2u1;11 þ u1;33 þ u3;13 Þ þ c13 ðu3;13 À u1;33 Þ À 2qgu3;1 ¼ 2qu

ð69Þ

€3 :

c11 ðu1;13 þ u3;11 Þ þ ðc13 þ P0 Þðu1;13 À u3;11 Þ þ 2c33 u3;33 þ 2qgu1;1 ¼ 2qu

ð70Þ

According to his argument, by expressing the displacement components u1 , u3 in terms of the displacement potentials
uðx1 ; x3 ; tÞ and wðx1 ; x3 ; tÞ as:

u1 ¼ u;1 À w;3 ;

u3 ¼ u;3 þ w;1 :

ð71Þ

Eqs. (69) and (70) reduce, respectively, to:

€;
ðc11 þ P0 ÞO2 u À qgw;1 ¼ qu

ð72Þ

€
ðc13 À c11 À P0 ÞO w þ 2qg u;1 ¼ 2qw

ð73Þ

€;
c11 u;11 þ c33 u;33 À qgw;1 ¼ qu

ð74Þ

€
c11 ðw;11 À w;33 Þ À ðc13 þ P0 ÞO w þ 2c33 w;33 þ 2qg u;1 ¼ 2qw;

ð75Þ

2

and
2

where O2 f ¼ f;11 þ f;33 .


403

P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

Also by his argument, Eqs. (72) and (74) represent the compressive wave along the x1 and x3 directions, respectively, and
Eqs. (73) and (75) represent the shear wave along those directions. According to him, since he considered the propagation of
Rayleigh waves in the direction of x1 only, thus he restricted his attention only to Eqs. (72) and (75).
Now we show that u1 , u3 expressed by (71) in which u and w is a solution of the system (72) and (75) does not satisfy the
system (69), (70), in general. Indeed, substituting (71) into (69) and (70) leads, respectively, to:

h
i
Â
Ã
€ ¼0
€ ;1 þ ðc13 À c11 À P0 ÞO2 w À 2qg u;1 þ 2qw
2 ðc11 þ P0 ÞO2 u À qgw;1 À qu

ð76Þ

;3

and

h

h
i
i
€ þ 2 c11 u þ c33 u À qgw À qu
€ ¼ 0:
c11 ðw;11 À w;33 Þ À ðc13 þ P0 ÞO2 w þ 2c33 w;33 þ 2qg u;1 À 2qw
;1
;11
;33
;1

;3

ð77Þ

Since u, w is a solution of the system (72) and (75), the first terms of the left-hand sides of (76) and (77) vanish, thus they
become:

h

i

€ ¼0
ðc13 À c11 À P0 ÞO2 w À 2qg u;1 þ 2qw

ð78Þ

;3

and

h

€
c11 u;11 þ c33 u;33 À qgw;1 À qu

i
;3

¼ 0;

ð79Þ

respectively.
It is clear that a pair u and w which is a solution of the system (72) and (75) does not necessarily satisfy the Eqs. (78) and
(79). The observation is demonstrated.
Remark 2
(i) It is not difficult to verify that if c33 – c11 þ P 0 (being valid in general) and u, w given by:

w  0;

u ¼ Aeipx3 eixt ; p2 ¼ qx2 =ðc11 þ P 0 Þ; x ¼ const – 0; A ¼ const – 0;

ð80Þ

then u, w is a solution of the system (72) and (75), however, they do not satisfy the Eqs. (78) and (79).
(ii) We can say, from (76) and (77), that Eq. (69) [Eq. (70)] is satisfied if u1 , u3 is defined by (71) in which u and w is a
solution of the system (72) and (78) [system (75) and (79)].From these, it is clear that the system (69), (70) is satisfied
if u1 , u3 is given by (71) in which u and w is a solution of a system of four equations, namely: (72), (75), (78) and (79).
(3i) When the initial stress is absent (P 0 ¼ 0) and the material is isotrpic, Eqs. (78) and (79) are satisfied if u and w is a
solution of the system (72) and (75), which now is:

O2 u À

g
1
€ ¼ 0;
w À u
c21 ;1 c21

O2 w þ

g
1
u À w€ ¼ 0:
c22 ;1 c22

ð81Þ

That means, the representation of solution (71), (72) and (75) is valid for this case.
(4i) If unknown functions u and w are sought in the form (as in [13]):

u ¼ AeÀikpx3 eikðx1 ÀctÞ ; w ¼ BeÀikpx3 eikðx1 ÀctÞ ;
2

ð82Þ
2

2

where A; B are constants satisfying A þ B – 0, then p has to satisfy a system of 3 quadratic equations that has no
solution in general.
(5i) The assumption: c44 ¼ ðc11 À c13 Þ=2 is not taken in the present paper.

6. Conclusions
In this paper the propagation of Rayleigh waves in homogeneous orthotropic elastic media under the influence of gravity
and initial stress is investigated. We have found the exact secular equation in the explicit form, and it is a new result. By
considering its special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity,
corresponding to some previous studies in which only implicitdispersion
equations have been found. In the case that the

2
material is isotropic, the initial stress is absent, and 0 <  ¼ g= kc2 ( 1, we have derived directly, from the exact secular
equations, approximate dispersion equations, and one of them coincides with the one obtained by Love.
Acknowledgement
The author wish to thank an anonymous reviewer for recommending him some references useful with the research.


404

P.C. Vinh / Applied Mathematics and Computation 215 (2009) 395–404

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