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

doi:10.1016/j.wavemoti.2009.04.003


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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



!
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

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

M¼

M1

M2

Q

M3

!
;

M1 ¼

!
kðX À dÞ ia
Q¼
;
Àia
kX

Àiðm=kÞ

À1

ÀD

Àiðm=kÞ


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


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


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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


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


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



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

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


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


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

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


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


 :  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

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

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.
References
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[2] D. Samuel et al, Rayleigh waves guided by topography, Proc. Roy. Soc. A 463 (2007) 531–550.
[3] T.J. I’A. Bromwhich, On the influence of gravity on elastic waves, and, in particular, on the vibrations of an elastic globe, Proc. Lond. Math. Soc. 30 (1898)
98–120.
[4] A.E. Love, Some Problems of Geodynamics, Dover, New York, 1957.
[5] M.A. Biot, Mechanics of Incremental Deformation, Wiley, New York, 1965.
[6] F. Gilbert, Gravitationally perturbed elastic waves, Bull. Seism. Soc. Am. 57 (1967) 783–794.
[7] S.N. De, P.R. Sengupta, Surface waves under the influence of gravity, Gerlands Beitr. Geophys. 85 (1976) 311–318.
[8] S.K. Dey, P.R. Sengupta, Effects of anisotropy on surface waves under the influence of gravity, Acta Geophys. Polonica XXVI (1978) 291–298.
[9] B.K. Datta, Some observation on interaction of Rayleigh waves in an elastic solid medium with the gravity field, Rev. Roumaine Sci. Tech. Ser. Mec. Appl.