DSpace at VNU: Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

Wave Motion 51 (2014) 496–504

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

Rayleigh waves in an isotropic elastic half-space coated by a
thin isotropic elastic layer with smooth contact
Pham Chi Vinh ∗ , Vu Thi Ngoc Anh, Vu Phuong Thanh
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

highlights
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The propagation of Rayleigh waves in an elastic half-space coated by a thin elastic layer is considered.
The half-space and the layer are both isotropic and the contact between them is smooth.
By using the effective boundary condition method an approximate secular equation of fourth-order has been derived.
From it, an explicit third-order approximate formula for the Rayleigh wave velocity has been established.
The approximate secular equation and the formula for the velocity will be useful in practical applications.

article

info

Article history:
Received 3 May 2013

Received in revised form 9 October 2013
Accepted 24 November 2013
Available online 2 December 2013
Keywords:
Rayleigh waves
An elastic half-space coated with a thin
elastic layer. Approximate secular
equations
Approximate formulas for the velocity

abstract
In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic
elastic half-space coated with a thin isotropic elastic layer. The contact between the layer
and the half space is assumed to be smooth. The main purpose of the paper is to establish
an approximate secular equation of the wave. By using the effective boundary condition
method, an approximate, yet highly accurate secular equation of fourth-order in terms of
the dimensionless thickness of the layer is derived. From the secular equation obtained, an
approximate formula of third-order for the velocity of Rayleigh waves is established. The
approximate secular equation and the formula for the velocity obtained in this paper are
potentially useful in many practical applications.
© 2013 Elsevier B.V. All rights reserved.

1. Introduction
The structures of a thin film attached to solids, modeled as half-spaces coated with a thin layer, are widely applied in
modern technology. Measurement of mechanical properties of thin supported films is therefore very significant [1]. Among
various measurement methods, the surface/guided wave method [2] is used most extensively in which the Rayleigh wave
is a most convenient tool. For the Rayleigh-wave approach, the explicit dispersion relations of Rayleigh waves supported
by thin-film/substrate interactions are employed as theoretical bases for extracting the mechanical properties of the thin
films from experimental data. They are therefore the key factor of the investigations of Rayleigh waves propagating in halfspaces covered by a thin layer. Taking the assumption of a thin layer, explicit secular equations can be derived by replacing
approximately the entire effect of the thin layer on the half-space by the so-called effective boundary conditions which relate the

displacements with the stresses of the half-space at its surface. For obtaining the effective boundary conditions Achenbach [3]
and Tiersten [4] replaced the thin layer by a plate modeled by different theories: Mindlin’s plate theory and the plate theory
of low-frequency extension and flexure (classical plate theory), while Bovik [5] expanded the stresses at the top surface of

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Corresponding author. Tel.: +84 4 35532164; fax: +84 4 38588817.
E-mail addresses: , (P.C. Vinh).

0165-2125/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
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P.C. Vinh et al. / Wave Motion 51 (2014) 496–504

497

the layer into Taylor series in its thickness. The Taylor expansion approach was then employed by Niklasson [6], Rokhlin [7,8],
Benveniste [9], Steigmann and Ogden [10], Steigmann [11], Ting [12], Vinh and Linh [13,14], Kaplunov and Prikazchikov [15]
to establish the effective boundary conditions.
Achenbach [3], Tiersten [4], Bovik [5], Tuan [16] assumed that the layer and the substrate are both isotropic and derived
approximate secular equations of second-order (these equations do not coincide totally with each other). In [10] Steigmann
and Ogden considered a transversely isotropic layer with residual stress overlying an isotropic half-space and the authors
obtained an approximate second-order dispersion relation. In [17] Wang et al. considered an isotropic half-space covered
by a thin electrode layer and the authors obtained an approximate secular equation of first-order. In [13] the layer and the
half-space were both assumed to be orthotropic and an approximate secular equation of third-order was obtained. In [14]
the layer and the half-space were both subjected to homogeneous pre-strains and an approximate secular equation of thirdorder was established which is valid for any pre-strain and for a general strain energy function.
In all investigations mentioned above, the contact between the layer and the half-space is assumed to be welded. For
the case of smooth contact, there exists only one approximate secular equation of third-order in the literature established
by Achenbach and Keshava [3]. This approximate secular equation includes the shear coefficient, originating from Mindlin’s
plate theory [18], whose usage should be avoided as noted by Muller and Touratier [19], Touratier [20]. This remark was

also mentioned in [21].
It should be noted that for the case of smooth contact, one could not arrive at the effective boundary conditions from the
relations between the displacements and the stresses at the bottom surface of the layer which were derived by Tiersten [4]
and Bovik [5]. In contrast, for the case of welded contact, the effective boundary conditions were immediately obtained.
The main purpose of the paper is to establish an approximate secular equation of Rayleigh waves propagating in
an isotropic elastic half-space coated with a thin isotropic elastic layer for the case of smooth contact. By using the
effective boundary condition method, an approximate effective boundary condition of fourth-order which relates the
normal displacement with the normal stress at the surface of the half space is derived. Using this condition along with
the vanishing of the shear stress at the surface of the half-space, an approximate secular equation of fourth-order in terms
of the dimensionless thickness of the layer is derived. We will show that the approximate secular equation obtained is a very
good approximation. Based on it, an approximate formula of third-order for the velocity of Rayleigh waves is established.
2. Effective boundary condition of fourth-order
Consider an elastic half-space x3 ≥ 0 coated by a thin elastic layer −h ≤ x3 ≤ 0. Both the layer and half-space are
homogeneous, isotropic and linearly elastic. The layer is assumed to be thin and has a smooth contact with the half-space.
In particular, the normal component of the particle displacement vector and the normal component of the stress tensor are
continuous, while the shearing stress vanishes across the interface x3 = 0, see Achenbach [3] and Murty [22]. Note that the
same quantities related to the half-space and the layer have the same symbol but are systematically distinguished by a bar
if pertaining to the layer.
If it is assumed that a state of plane strain exists, whereby the x2 component of displacement vanishes and the x1 and x3
components are functions of x1 , x3 and t only, i.e.
ui = ui (x1 , x3 , t ),

u¯ i = u¯ i (x1 , x3 , t ),

i = 1, 3,

u2 = u¯ 2 ≡ 0

(1)

where t is the time. Since the layer is made of isotropic elastic materials, the strain–stress relations take the form

¯ u3 , 3 ,
σ¯ 11 = (λ¯ + 2µ)¯
¯ u1,1 + λ¯
¯ u1,1 + (λ¯ + 2µ)¯
σ¯ 33 = λ¯
¯ u3,3 ,

(2)

σ¯ 13 = µ(¯
¯ u1,3 + u¯ 3,1 )
¯ and µ
where σ¯ ij is the stress of the layer, commas indicate differentiation with respect to spatial variables xk , λ
¯ are Lame
constants. In the absent of body forces, the equations of motion for the layer is
σ¯ 11,1 + σ¯ 13,3 = ρ¯ u¨¯ 1 ,
σ¯ 13,1 + σ¯ 33,3 = ρ¯ u¯¨ 3

(3)

where a dot signifies differentiation with respect to t. From Eqs. (2), (3) we have

 ′ 
M1
U¯
=
M3
T¯ ′

M2
M4

 
U¯
T¯

(4)

where
U¯ = u¯ 1



u¯ 3

T

,

T¯ = σ¯ 13



σ¯ 33

T

498

P.C. Vinh et al. / Wave Motion 51 (2014) 496–504

the symbol ‘‘T ’’ indicate the transpose of a matrix, the prime signifies differentiation with respect to x3 and

M1 = 

−∂1

0



−λ¯
∂1
¯λ + 2µ
¯

0





1

µ
¯

,

M2 = 


(λ¯ + 2µ)
¯ 2 − λ¯ 2 2
∂1 + ρ∂
¯ t2
−
M3 = 
λ¯ + 2µ
¯


,


1

0





0

λ¯ + 2µ
¯

(5)


0 

ρ∂
¯ t2

0

,

M4 = M1 T

here we use the notations ∂1 = ∂/∂ x1 , ∂12 = ∂ 2 /∂ x1 2 , ∂t2 = ∂ 2 /∂ t 2 . From (4) it follows
U¯ (n)
T¯ (n)





 
U¯
=M ¯ ,
T



M1
M =
M3

n



M2
,
M4

n = 1, 2, 3, . . . , x3 ∈ [−h, 0].

(6)

Let h be small (i.e. the layer is thin), then expanding into Taylor series T¯ (−h) at x3 = 0 up to the fourth-order of h we have
T¯ (−h) = T¯ (0) + T¯ ′ (0)(−h) +

1 ′′
1
1
T¯ (0)h2 − T¯ ′′′ (0)h3 + h4 T¯ ′′′′ (0).
3!
4!

(7)

2!

Suppose that surface x3 = −h is free of traction, i.e. T¯ (−h) = 0. Introducing (6) with n = 1, 2, 3, 4 at x3 = 0 into (7) yields


I − hM4 +

1
2

2

h M6 −

1
6

3

h M8 +

1
24


h M10 T¯ (0) =
4


hM3 −

1

2

2

h M5 +

1
6

3

h M7 −

1
24


h M9 U¯ (0)
4

(8)

where I is the identity matrix of order 2, M3 , M4 are defined by (5) and
M5 = M3 M1 + M4 M3 ,
M6 = M3 M2 + M42 ,
M7 = M3 M12 + M4 M3 M1 + M3 M2 M3 + M42 M3 ,
M8 = M3 M1 M2 + M4 M3 M2 + M3 M2 M4 + M43 ,

(9)

M9 = M3 M13 + M4 M3 M12 + M3 M2 M3 M1 + M42 M3 M1 + M3 M1 M2 M3 + M4 M3 M2 M3 + M3 M2 M4 M3 + M43 M3 ,
M10 = M3 M12 M2 + M4 M3 M1 M2 + M3 M2 M3 M2 + M42 M3 M2 + M3 M1 M2 M4 + M4 M3 M2 M4 + M3 M2 M42 + M44 .
Taking into account (5) and (9), the relation (8) in component form is of the form



σ¯ 13 + h (1 − 2γ¯ )σ¯ 33,1 + 4ρ¯ c¯22 (1 − γ¯ )¯u1,11 − ρ¯ u¨¯ 1


1
h2
2
¨
¨
+
(2γ¯ − 3)σ¯ 13,11 + 2 σ¯ 13 + 4ρ¯ c¯2 (1 − γ¯ )¯u3,111 − 2ρ(
¯ 1 − γ¯ )u¯ 3,1
2
c¯2



2
1
h3
(4γ¯ − 3)σ¯ 33,111 + 2 (1 − γ¯ ) + 2 (1 − 2γ¯ ) σ¯¨ 33,1 − 8ρ¯ c¯22 (1 − γ¯ )¯u1,1111
+
6
c¯1
c¯2




ρ¯ ¨
h4
1
2 ¨
(5 − 4γ¯ )σ¯ 13,1111 + 2 2 (1 − γ¯ )
+ ρ(
¯ 5 − 4γ¯ )u¯ 1,11 − 2 u¯ 1,tt +
24
c¯2
c¯1

1
1
2
+ 2 (2γ¯ − γ¯ − 2) σ¨¯ 13,11 + 4 σ¨¯ 13,tt + 4ρ(
¯ 2 − γ¯ − γ¯ 2 )u¨¯ 3,111 − 8ρ¯ c¯22 (1 − γ¯ )¯u3,11111
c¯2
c¯2



1
1
¨
− 2ρ(
¯ 1 − γ¯ ) 2 + 2 u¯ 3,1tt = 0 at x3 = 0,
c¯1

c¯2

h2
1
¨
σ¯ 33 + h(σ¯ 13,1 − ρ¯ u¯ 3 ) +
(1 − 2γ¯ )σ¯ 33,11 + 2 σ¨¯ 33 + 4ρ¯ c¯22 (1 − γ¯ )¯u1,111
2
c¯1




h3
2
1
− 2ρ(
¯ 1 − γ¯ )u¯¨ 1,1 +
(2γ¯ − 3)σ¯ 13,111 + 2 (1 − γ¯ ) + 2 σ¯¨ 13,1
6
c¯2
c¯1

(10)


P.C. Vinh et al. / Wave Motion 51 (2014) 496–504



499



ρ¯
h4
+ 4ρ¯ c¯22 (1 − γ¯ )¯u3,1111 − ρ(
¯ 3 − 4γ¯ )u¨¯ 3,11 − 2 u¨¯ 3,tt +
(4γ¯ − 3)σ¯ 33,1111
24
c¯1


1
1
1
2
+ 2 (4 − 6γ¯ ) + 2 (2 − 6γ¯ + 4γ¯ ) σ¯¨ 33,11 + 4 σ¨¯ 33,tt + 4ρ(
¯ 2 − γ¯ − γ¯ 2 )u¨¯ 1,111
c¯1
c¯2
c¯1



1
1
2
− 8ρ¯ c¯2 (1 − γ¯ )¯u1,11111 − 2ρ(
¯ 1 − γ¯ ) 2 + 2 u¯¨ 1,1tt = 0 at x3 = 0

c¯1
c¯2

(11)

where


c¯1 =



λ¯ + 2µ
¯
,
ρ¯

c¯2 =

µ
¯
,
ρ¯

γ¯ =

c¯22
c¯12

.

(12)

Remark 1. If the contact between the layer and the half-space is welded, i.e. the displacements and the stresses are
continuous through the interface of the layer and the half-space, we immediately obtain the effective boundary conditions
from Eqs. (10) and (11) by replaced u¯ 1 , u¯ 3 , σ¯ 13 and σ¯ 33 by u1 , u3 , σ13 and σ33 , respectively. These effective boundary
conditions are valid not only for the displacements and the stresses of Rayleigh waves but also for those of any dynamic
problem. However, for the case of smooth contact the situation is rather different. The horizontal displacement is not
required to be continuous through the interface, the effective boundary conditions are therefore not immediately obtained
from Eqs. (10) and (11). As shown below, the effective boundary conditions obtained for the case of smooth contact are valid
for only the displacements and the stresses of Rayleigh waves.
Now we consider the propagation of a Rayleigh wave, travelling (in the coated half-space) with velocity c (>0) and wave
number k (>0) in the x1 -direction and decaying in the x3 -direction. The displacements and the stresses of the wave are
sought in the form
u¯ 1 = U¯ 1 (y)eik(x1 −ct ) ,
ik(x1 −ct )

σ¯ 13 = −ikT¯1 (y)e

u¯ 3 = U¯ 3 (y)eik(x1 −ct ) ,

(13)

σ¯ 33 = −ikT¯3 (y)eik(x1 −ct )

,

for the layer, and
u1 = U1 (y)eik(x1 −ct ) ,

u3 = U3 (y)eik(x1 −ct ) ,

σ13 = −ikT1 (y)eik(x1 −ct ) ,

(14)

σ33 = −ikT3 (y)eik(x1 −ct )

for the half-space, where y = kx3 . Substituting (13) into (10) and (11) yields





1 c2
1
1 c2
1 c2
4
2 1
¯
(2γ¯ − 3) +
+
ε
(
4
γ
¯
−
5

)
−
(
1
−
γ
¯
)
−
(2γ¯ 2
iT1 (0) −1 + ε
2
2 c¯22
24
12 c¯12
12 c¯22



1 c4
1
1 c2
3
− γ¯ − 2) −
+ T¯3 (0) ε(1 − 2γ¯ ) + ε
(3 − 4γ¯ ) −
(1 − γ¯ )
4
24 c¯2
6

3 c¯12

 


1 c2
4 2
2
2
3
− 2 (1 − 2γ¯ )
+ U¯ 1 (0) ε 4ρ¯ c¯2 (γ¯ − 1) + ρ¯ c + ε
ρ¯ c¯2 (γ¯ − 1)
6 c¯2
3

 

1 2
1 c4
2
2
2
2
¯
+ iU3 (0) ε 2ρ¯ c¯2 (γ¯ − 1) + ρ¯ c (1 − γ¯ )
+ ρ¯ c (5 − 4γ¯ ) − ρ¯ 2
6
6 c¯2




+ ε4

1

6





T¯1 (0) ε + ε

+

1 c2
2 c¯12

1

1

3

12

ρ¯ c 2 (2 − γ¯ − γ¯ 2 ) − ρ¯ c¯22 (1 − γ¯ ) −
3


+

1
6

ε4
24

(3 − 2γ¯ ) −

1 c2
3 c¯22


(3 − 4γ¯ ) −

c2
c¯12

(1 − γ¯ ) −

1 c2
6 c¯12

(4 − 6γ¯ ) −

c2
c¯22

1

ρ¯ c 4 (1 − γ¯ )



c¯12

+

1





+ iT¯3 (0) −1 + ε

(2 − 6γ¯ + 4γ¯ ) −
2

= 0,

c¯22

c4
c¯14

2



1
2

(1 − 2γ¯ )

(15)


500

P.C. Vinh et al. / Wave Motion 51 (2014) 496–504





 ε4
 2
1
2
2
2
2
4
+ iU¯ 1 (0) ε ρ(
¯ 1 − γ¯ ) c − 2c¯2 + ρ¯ 2c (2 − γ¯ − γ¯ ) − 4c¯2 (1 − γ¯ ) − c
12
c¯12






1
1 2
1 c4
2
3 2
2
¯
+ 2 (1 − γ¯ )
+ U3 (0) ε ρ¯ c + ε
=0
ρ¯ c¯2 (1 − γ¯ ) − ρ¯ c (3 − 4γ¯ ) − ρ¯ 2
3
6
6 c¯1
c¯2
2

where ε = kh is the dimensionless thickness of the layer.
Let the contact between the layer and the half-space is smooth, i.e.

σ13 = 0,

σ¯ 13 = 0,

u3 = u¯ 3 ,

σ33 = σ¯ 33 at x3 = 0

(16)

or equivalently
T1 (0) = 0,

T¯1 (0) = 0,

U3 (0) = U¯ 3 (0),

T3 (0) = T¯3 (0)

(17)

according to (13) and (14). Introducing (17)2 into (15) yields
T¯3 (0)(a1 + a2 ε 2 ) + U¯ 1 (0)(a3 + a4 ε 2 ) + iU¯ 3 (0)(a5 ε + a6 ε 3 ) = 0,
iT¯3 (0)(−1 + a7 ε 2 + a8 ε 4 ) + iU¯ 1 (0)(a5 ε 2 + a6 ε 4 ) + U¯ 3 (0)(a9 ε + a10 ε 3 ) = 0

(18)

in which
a1 = 1 − 2γ¯ ,
a2 =

1
6

1



(3 − 4γ¯ ) − c

2

2
c¯12

6

(1 − γ¯ ) +



1
c¯22

(1 − 2γ¯ ) ,

a3 = −4ρ¯ c¯22 (1 − γ¯ ) + ρ¯ c 2 ,
1 c4
4
1
a4 = − ρ¯ c¯22 (1 − γ¯ ) + ρ¯ c 2 (5 − 4γ¯ 2 ) − ρ¯ 2 ,
3
6
6 c¯2
a5 = −2ρ¯ c¯22 (1 − γ¯ ) + ρ¯ c 2 (1 − γ¯ ),

a6 =

a7 =

a8 =

1
6
1
2

1

ρ¯ c (2 − γ¯ − γ¯ ) − ρ¯ ¯ (1 − γ¯ ) −
2

2

3

(1 − 2γ¯ ) +

1
24

1 c2
2 c¯12

(3 − 4γ¯ ) −

24



1
12

ρ(
¯ 1 − γ¯ )c

4

1
c¯12

+

1
c¯22


,

(19)

,


1

c22

c

2

1

1

c¯1

c¯2

(4 − 6γ¯ ) +
2



(2 − 6γ¯ + 4γ¯ ) −
2
2

1 c4
24 c¯14

,

a9 = ρ¯ c 2 ,
a10 =

2
3

1

1 c4

6

6 c¯12

ρ¯ c¯22 (1 − γ¯ ) − ρ¯ c 2 (3 − 4γ¯ ) − ρ¯

.

Eliminating U¯ 1 from (18) we have
iT¯3 (0)(−a3 + a11 ε 2 + a12 ε 4 ) = −U¯ 3 (0)(a3 a9 ε + a13 ε 3 )

(20)

where
a11 = −a4 + a3 a7 − a1 a5 ,
a12 = a4 a7 + a3 a8 − a2 a5 − a1 a6 ,
a13 =

a25

(21)

+ a4 a9 + a3 a10 .

From the last two equations of (17) and Eq. (20) it follows
T3 (0)(−a3 + a11 ε 2 + a12 ε 4 ) = iU3 (0)(a3 a9 ε + a13 ε 3 ).

(22)

From the first of (17) and (22) we see that the surface x3 = 0 of the half-space is subjected to the following conditions
T1 (0) = 0,
T3 (0)(−a3 + a11 ε 2 + a12 ε 4 ) = iU3 (0)(a3 a9 ε + a13 ε 3 ).

(23)

The second of (23) is the approximate effective boundary condition (of fourth-order). The total effect of the layer on the
half-space is replaced approximately by this condition.


P.C. Vinh et al. / Wave Motion 51 (2014) 496–504

501

3. An approximate secular equation of fourth-order
Now we can ignore the layer and consider the propagation of Rayleigh waves in the isotropic elastic half-space x3 ≥ 0
whose surface x3 = 0 is subjected to the boundary conditions (23). According to Achenbach [23], the displacement
components of a Rayleigh wave travelling with velocity c and wave number k in the x1 -direction and decaying in the
x3 -direction are determined by (14)1,2 in which U1 (y) and U3 (y) are given by
U1 (y) = A1 e−b1 y + A2 e−b2 y ,

(24)

U3 (y) = α1 A1 e−b1 y + α2 A2 e−b2 y
where A1 and A2 are constant to be determined and
b1 =

γ =



√

1 − γ x,



c22

,
2

c1 =

c1

1 − x,

b2 =

λ + 2µ
,
ρ

b1

α1 = −

c2 =

i

µ
,
ρ

,

α2 =

x=

c2
c22

,

i
b2

,

0 < x < 1.

(25)

Substituting (14)1,2 and (24) into the stress–strain relations (2) without the bar yields that the stresses σ13 and σ33 are given
by (14)3,4 in which


T1 (y) =

ic22

ρ −2b1 A1 e

−b 1 y


T3 (y) = −c22 ρ

c2
c22

+

1
b2



c2
c22




−b 2 y

− 2 A2 e



,
(26)



− 2 A1 e−b1 y − 2A2 e−b2 y .

Introducing (24), (26) into (23) provides a homogeneous system of two linear equations for A1 , A2 namely



f1 A1 + f2 A2 = 0
F1 A1 + F2 A2 = 0

(27)

where
f1 = −2b1 ,
f2 = (x − 2),
F1 =

F2 =

−a3 + a11 ε 2 + a12 ε 4
b1
(x − 2) − 4 2 (a3 a9 ε + a13 ε 3 ),
2
c2 ρ
c2 ρ
2b2
c22 ρ

(a3 − a11 ε 2 − a12 ε 4 ) −

1
c24 ρ 2

(28)

(a3 a9 ε + a13 ε 3 ).

For a non-trivial solution, the determinant of the matrix of the system (27) must vanish


 f1

F1



f2 

= 0.
F2 

Expanding this determinant and using (28) lead to the dispersion equation of the wave, namely
A0 + A1 ε + A2 ε 2 + A3 ε 3 + A4 ε 4 = 0

(29)

where
A0 = rν2 x − 4(1 − γ¯ ) (x − 2)2 − 4b1 b2 ,







A1 = rµ rν2 x2 b1 rν2 x − 4(1 − γ¯ ) ,



A2 = −
A3 =

1
6

1
6




8(1 − γ¯ ) + 4rν x(γ¯ − 2) + rν x (1 + 3γ¯ )
2



2

4 2





(x − 2)2 − 4b1 b2 ,

rµ xb1 8(1 − γ¯ )2 + rν2 x 8(−2 + 3γ¯ − γ¯ 2 ) + 2rν2 x(4 − 2γ¯ − γ¯ 2 ) − rν4 x2 (1 + γ¯ )

A4 = −



1 
24

(30)



,



(x − 2)2 − 4b1 b2 4(1 − γ¯ ) + rν2 x(4γ¯ 2 − 7) + 2rν4 x2 (1 + 4γ¯ − 2γ¯ 2 ) − rν6 x3 γ¯ (2 + γ¯ )

where rµ = µ/µ,
¯
rv = c2 /¯c2 . Eq. (29) is the desired approximate secular equation.


502

P.C. Vinh et al. / Wave Motion 51 (2014) 496–504

√

Fig. 1. Dependence on ε = k · h ∈ [0 1] of the dimensionless Rayleigh wave velocity x = c /c2 that is calculated by the exact secular equation and by
the approximate secular equations of fourth-order (29). Two corresponding curves almost totally coincide with each other. Here we take rµ = 0.5, rv =
5, γ = 1/4 and γ¯ = 2/3.
Table 1

√
Some values of x, corresponding to Fig. 1 (rµ = 0.5, rv = 5, γ = 1/4 and
√
γ¯ = 2/3), that are calculated by the exact secular equation ( xext ), by the
√
approximate secular equation (29) ( xapp ).

ε
√
x
√ ext

xapp

0

0.2

0.4

0.6

0.8

1.0

0.9325
0.9325

0.6146
0.6152

0.4535
0.4555

0.3647
0.3674

0.3102
0.3125

0.2767
0.2781

√

√

From (29) and the first of (30) it follows that, when ε = 0 either (x − 2)2 − 4 1 − x 1 − γ x = 0 or x = 4(1 − γ¯ )¯c22 /c22 .
That means, in the limit ε → 0 two modes are possible, one of which approaches the classical Rayleigh√
wave in the isotropic
half-space and the other approaches the longitudinal wave of the layer with the velocity c = 2c¯2 1 − γ¯ , as noted by
Achenbach and Keshava [3].
√
Fig. 1 presents the dependence on ε = k · h ∈ [0 1] of the dimensionless Rayleigh wave velocity x = c /c2 that
is calculated by the exact secular equation and by the approximate
secular equations of fourth-order (29). Here we take
√
rµ = 0.5, rv = 5, γ = 1/4 and γ¯ = 2/3. Some values of x are listed in Table 1. Note that the exact secular equation is
similar in form to Eq. (30) in Ref. [3], and is not reproduced here. It is seen from Fig. 1 that the exact velocity curve and the
approximate velocity curve of fourth-order almost totally coincide with each other for the values of ε ∈ [0 1]. This shows
that the approximate secular equation (29) is a very good approximation.
4. An approximate formula of third-order for the velocity
In this section we establish an approximate formula of third-order for the squared dimensionless Rayleigh wave velocity
x(ε) that is of the form
x(ε) = x(0) + x′ (0) ε +

x′′ (0)
2

ε2 +

x′′′ (0)
6

ε3 + O(ε4 )

(31)

where x(0) is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic half-space that is
given by (see [24])



x(0) = 4(1 − γ ) 2 −

4
3

γ+

√


3

R+

D+

√


3

R−

 −1
D

(32)

in which
R = 2(27 − 90γ + 99γ 2 − 32γ 3 )/27,
D = 4(1 − γ )2 (11 − 62γ + 107γ 2 − 64γ 3 )/27

(33)

and the roots in the formula (32) taking their principal values. Note that x(0) can be calculated by another formula derived
by Malischewsky [25].


P.C. Vinh et al. / Wave Motion 51 (2014) 496–504

503

√

Fig. 2. Plots of the dimensionless Rayleigh wave velocity x(ε) in the interval [0 1] that is calculated by the exact secular equation and by the formula
(31). Here we take rµ = 4, rv = 0.5, γ = 1/4 and γ¯ = 2/3.
Table 2

√

Some values of x, corresponding to Fig. 2 (rµ = 4, rv = 0.5, γ = 1/4 and
√
γ¯ = 2/3), that are calculated by the exact secular equation ( xext ), by the
√
approximate formula (31) ( xapp ).

ε
√
x
√ ext

xapp

0

0.2

0.4

0.6

0.8

1.0

0.9325
0.9325

0.9063
0.9062

0.8815
0.8806

0.8635
0.8606

0.8557
0.8515

0.8586
0.8584

From (29) it follows that



2A2 A20x − 2A0x A1 A1x + A0xx A21 
′′


,
x

(
0
)
=
−
,

A0x x=x(0)
A30x
x=x(0)
 


′′′
′
′2
′′
′
′′
′3
x (0) = − 6A3 + 6A2x x (0) + 3A1xx x (0) + 3A1x x (0) + 3A0xx x (0)x (0) + A0xxx x (0) /A0x 
A1 

x′ (0) = −

(34)
x=x(0)

where A1 , A2 and A3 are given by (30) and

√











A0x = rν2 (x − 2)2 − 4 1 − x 1 − γ x + 2 4(γ¯ − 1) + rν2 x
2

1 + γ − 2γ x



A0xx = 4rν

x−2+ √


2

A0xxx = 6rν



√

1 − x 1 − γx

(γ − 1)2
1+ 

2 (1 − x)3 (1 − γ x)3



A1x =
A1xx =

+ 2 4(γ¯ − 1) + rν x


2

√

rµ rν2
4 (1 − γ x)3





1 + γ − 2γ x
x−2+ √

√
1 − x 1 − γx



,


2 (1 − x)5





,





−



(γ − 1)2
1+ 

2 (1 − x)3 (1 − γ x)3

3(1 − γ )2 (1 + γ − 2γ x) 4(1 − γ¯ ) − rν2 x

rµ rν2 x 4(1 − γ¯ )(5γ x − 4) + rν2 x(6 − 7γ x)
2 1 − γx



(1 − γ x)5

,


,




4(1 − γ¯ )(−8 + 24γ x − 15γ 2 x2 ) + rν2 x(24 − 60γ x + 35γ 2 x2 ) ,




√
1 
A2x = − rν2 2(γ¯ 2 − 2) + rν2 x(1 + 3γ¯ ) (x − 2)2 − 4 1 − x 1 − γ x
3


1 2
1 + γ − 2γ x
2

4 2
− 4rν x(γ¯ − 2) + 8(1 − γ¯ ) + rν x (1 + 3γ¯ ) x − 2 + √
.
√
3
1 − x 1 − γx
√

Fig. 2 shows the plots of the dimensionless Rayleigh wave velocity x(ε) in the interval [0 1] that is calculated by the
√
exact secular equation and by the formula (31). Here we take rµ = 4, rv = 0.5, γ = 1/4 and γ¯ = 2/3. Some values of x
are listed in Table 2. It is shown that the approximate velocity curve is close to the exact velocity curve in the interval [0 1].


504

P.C. Vinh et al. / Wave Motion 51 (2014) 496–504

5. Conclusions
In this paper the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer
is considered. The contact between the layer and half space is assumed to be smooth. An approximate secular equation of
fourth-order in terms of the dimensionless thickness of the layer is derived using the effective boundary condition method.
We have shown that the approximate secular equation obtained has high accuracy. An approximate formula of third-order
for the velocity of Rayleigh waves is established using the obtained approximate secular equation. The approximate secular
equation and the formula for the velocity are potentially useful in many practical applications.
Acknowledgment
The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED).
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