DSpace at VNU: Formulas for the speed and slowness of Stoneley waves in bonded isotropic elastic half-spaces with the same bulk wave velocities

International Journal of Engineering Science 60 (2012) 53–58

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International Journal of Engineering Science
journal homepage: www.elsevier.com/locate/ijengsci

Formulas for the speed and slowness of Stoneley waves in bonded
isotropic elastic half-spaces with the same bulk wave velocities
Pham Chi Vinh a,⇑, Peter. G. Malischewsky b, Pham Thi Ha Giang a
a
b

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam
Institute for Geosciences, Friedrich-Schiller University Jena, Burgweg 11, 07749 Jena, Germany

a r t i c l e

i n f o

Article history:
Received 5 August 2011
Received in revised form 20 March 2012
Accepted 12 May 2012
Available online 21 June 2012
Keywords:
Stoneley waves
The wave velocity
The wave slowness
Two bonded isotropic elastic half-spaces
Holomorphic function

a b s t r a c t
This paper is concerned with the propagation of Stoneley waves in two bonded isotropic
elastic half-spaces with the same bulk wave velocities. Our main purpose is to find formulas for the wave velocity and the wave slowness. By applying the complex function method,
the exact formulas for the wave velocity and the wave slowness have been derived. The
derivation of these formulas also shows that there always exists a unique Stoneley wave
for the case under consideration.
Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction
Interfacial waves traveling along the welded plane boundary of two different isotropic elastic half-spaces were first investigated by Stoneley (1924). He derived the secular equation of the wave, and showed by means of examples that such interfacial waves do not always exist. Subsequent studies by Sezawa and Kanai (1939) and Scholte (1942, 1947) focused on the
range of existence of Stoneley waves. Scholte (1947) found the equations expressing the boundaries of the existence domain
and they are in complete agreement with the corresponding curves numerically obtained by Sezawa and Kanai (1939) for the
case of Poisson solids (for which the corresponding Lame constants of the half-spaces are the same). Their studies showed
that the restriction on material constants that permit the existence of Stoneley waves are rather severe. However, Sezawa
and Kanai (1939) and Scholte did not prove the uniqueness of Stoneley waves. This question was settled by Barnett, Lothe,
Gavazza, and Musgrave (1985) for general anisotropic half-spaces with bonded interface. The propagation of Stoneley waves
in anisotropic media was also studied by Stroh (1962) and Lim and Musgrave (1970). Much of the early attentions to Stoneley waves was directed toward geophysical applications. Latter studies have indicated that interfacial waves may prove to be
useful probes for the non-destructive evaluations (see Lee & Corbly, 1977, Rokhlin, Hefet, & Rosen, 1980).
The propagation of Stoneley waves in two isotropic elastic half-spaces with the loosely bonded interface was studied by
Murty (1975b, 1975a). The author has derived the secular equation of the wave and obtained a lot of numerical values of the
Stoneley wave velocity by directly solving that equation for the case of Poisson solids. The existence and uniqueness of
Stoneley waves in two half-spaces in sliding contact was investigated by Barnett, Gavazza, and Lothe (1988). The authors
showed that for the isotropic elastic half-spaces, if a Stoneley exists, then it is unique, while for the anisotropic half-spaces,
the possibility of a new slip-wave mode, called the second slip-wave mode, arises.
⇑ Corresponding author. Tel.: +84 4 5532164; fax: +84 4 8588817.
E-mail address: (P.C. Vinh).
0020-7225/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
/>

54

P.C. Vinh et al. / International Journal of Engineering Science 60 (2012) 53–58

For the Stoneley wave, its velocity is of great interest to researchers in various fields of science. The formulas for the
Stoneley wave velocity are powerful tools 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. Recently, the exact formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces have been derived by Vinh and Giang (2011).
The main aim of this paper is to find exact formulas for the velocity and the slowness of Stoneley waves propagating in
two bonded isotropic elastic half-spaces which have the same bulk wave velocities. By employing the complex function
method that is based on the properties of Cauchy integrals and the generalized Liouville theorem, the authors derive the exact formulas for the velocity and for the slowness. The derivation of these formulas shows that: if a Stoneley wave exists then it
is unique; and there always exits a Stoneley wave propagating along the bonded interface of two bonded isotropic elastic halfspaces which have the same bulk wave velocities. The former was proved by Barnett et al. (1985) by another method, and
the latter was shown by Stoneley (1924) expanding the corresponding secular equation into Taylor series.
2. Exact formula for the velocity
Let us consider two isotropic elastic solids X and X⁄ occupying the half-space x2 P 0 and x2 6 0, respectively. Suppose
that these two elastic half-spaces are in welded contact with each other at the plane x2 = 0. Then, the component of the particle displacement vector and the component of the stress tensor are continuous across the interface x2 = 0. Note that same
quantities related to X and X⁄ have the same symbol but are systematically distinguished by an asterisk if pertaining to X⁄.
Suppose that the two half-spaces have the same bulk wave velocities, i.e. ck ¼ cÃk ðk ¼ 1; 2Þ, where c1, cÃ1 are the longitudinal
wave velocities and c2, cÃ2 are the transverse wave velocities of the half-spaces. These conditions appear to be satisfied at the
Wiechert surface of discontinuity within the Earth, as indicated by Stoneley (1924). Since these two half-spaces become the
same if q = q⁄, we therefore assume that the mass densities of the half-spaces are different from each other, i.e. q – q⁄.
According to Stoneley (1924) the secular equation of Stoneley waves for the case under consideration is (see Eq. (3) in
Stoneley, 1924):

h
h
i
pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffii
x2 ðq À qà Þ2 À ðq þ qà Þ2 1 À x 1 À cx þ 4ðq À qà Þ2 ð1 À xÞð2 À cxÞ þ 1 À x 1 À cxðx À 2Þ ;

ð1Þ

where c ¼ c22 =c21 ð0 < c < 1Þ and x ¼ c2 =c22 , c is the velocity of Stoneley waves. Since 0 < c < c2, it follows that x 2 (0, 1). The
secular Eq. (1) can be rewritten as follows:

pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f ðxÞ  q1 ðxÞ À q2 ðxÞ 1 À x 1 À cx ¼ 0;

0 < x < 1;

ð2Þ

where

Ã
2Â
q1 ðxÞ ¼ b ð1 þ 4cÞx2 À 4ð2 þ cÞx þ 8 ;

2

2

q2 ðxÞ ¼ a2 x2 À 4b x þ 8b ;

ð3Þ

in which

a ¼ 1 þ r;

b ¼ 1 À r;
2

r ¼ q=qà ð–1Þ:

ð4Þ

2

Note that, since b < a it follows:

q2 ðxÞ ¼ a2


2

 !
2
2
2
x À 2b =a2 þ 4b 2 À b =a2 =a2 > 0 8 x;

ð5Þ

and according to Stoneley (1924) we have:

lim
x!0

f ðxÞ
2
¼ Àa2 þ b c < 0:
x2

ð6Þ

Now, in the complex plane C we consider the equation:

pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f ðzÞ  q1 ðzÞ þ q2 ðzÞ z À 1 cz À 1 ¼ 0;

ð7Þ

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where qk(z) are given by (3), and z À 1; cz À 1 are chosen as the principal branches of the corresponding square roots.
With the fact 0 < c < 1 it is easy to see that for z 2 (0, 1), Eq. (7) becomes Eq. (2). We will prove the following theorem:
Theorem 1. If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs ¼ c2 =c22 is defined by

xs ¼ 1 À


2
p
ffiffiffi 2
b
1
4
c
þ I0 ;

À
2
ffiffiffi
p
a2 4 c

ð8Þ

where

I0 ¼

1

p

Z

(

1=c

hðtÞdt;
1

hðtÞ ¼ atan

)
q1 ðtÞ
pffiffiffiffiffiffiffiffiffiffi

ffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
q2 ðtÞ t À 1 1 À ct

ð9Þ


P.C. Vinh et al. / International Journal of Engineering Science 60 (2012) 53–58

55

Proof. Denote L = [1,1/c], S = {z 2 C, z R L}, N(z0) = {z 2 S: 0 < jz À z0j < e}, e is a sufficient small positive number, z0 is some
point of the complex plane C. If a function /(z) is holomorphic in X & C we write /(z) 2 H(X). In order to solve Eq. (2) in
the interval (0, 1) we will find zeros of f(z) in the domain S ('(0, 1)).
Theorem 1 is proved as follows:
Step 1: To express f(z) by product of two functions, the first function is non-zero in S, and the second one is a polynomial of
third-order (see Eq. (16)).
Sine the first factor is non-zero in S, the zeros of f(z) are identical with those of the third-order polynomial in S.
Due to (6), z1 = 0, z2 = 0 are zeros of the third-order polynomial in S.
Step 2: Finding the coefficients of z3 and z2 of the third-order polynomial (see (26) and (27)) by expanding f(z) and the
inverse of the first factor into Laurent series at infinity. Knowing these coefficients we obtain immediately the
expression of the third zero, given by (8), of the third-order polynomial.
Step 3: Showing that: (i) if a Stoneley wave exists then it is unique. (ii) if a Stoneley wave exists then its squared dimensionless velocity is the third zero of the third-order polynomial.
Step 1
From (3) and (7) it is not difficult to show that the function f(z) has the properties:
(f1) f(z) 2 H(S).
(f2) f(z) is bounded in N(1) and N(1/c).
(f3) f(z) = O(z3) as jzj ? 1.
(f4) f(z) is continuous on L from the left and from the right (see Muskhelishvili, 1953) with the boundary values f+(t) (the
right boundary value of f(z)), fÀ(t) (the left boundary value of f(z)) defined as follows:

pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f þ ðtÞ ¼ q1 ðtÞ þ iq2 ðtÞ t À 1 1 À ct;

f À ðtÞ ¼ f þ ðtÞ;

ð10Þ

where the bar indicates the complex conjugate.
Now we define the function g(t) (t 2 L) as follows:

gðtÞ ¼

f þ ðtÞ
;
f À ðtÞ

ð11Þ

then it is obvious that:

f þ ðtÞ ¼ gðtÞf À ðtÞ;

t 2 L:

ð12Þ

Consider the function C(z) defined as:

CðzÞ ¼

1
2pi

Z
L

log gðtÞ
dt:
tÀz

ð13Þ

The function C(z) is an integral of the Cauchy type whose properties are examined in detail in Muskhelishvili (1953) (chaps.
2–4). It is not difficult to verify that:
(c1) C(z) 2 H(S).
(c2) C(1) = 0.
(c3) For c – 1/2: C(z) = X0(z), z 2 N(1), C(z) = X1(z), z 2 N(1/c), where X0(z) (X1(z)) bounded in N(1) (N(1/c)) and takes a
defined value at z = 1 (z = 1/c).
(c4) For c = 1/2: C(z) = X2(z), z 2 N(1), C(z) = (1/2) log (z À 2) + X3(z), z 2 N(2), where X2(z) (X3(z)) bounded in N(1) (N(2))
and takes a defined value at z = 1 (z = 2).
It is noted that (c3) and (c4) come respectively from the facts (see Muskhelishvili, 1953, chap. 4, Section 29): log g(1) = log
g(1/c) = 0 for c – 1/2 and logg(1) = 0, log g(2) = À1 for c = 1/2.
We now consider the function Y(z) defined by

YðzÞ ¼ f ðzÞeÀCðzÞ :

ð14Þ

From (f1) À (f3), (c1) À (c4), (12), (14) and the Plemelj formula (see Muskhelishvili, 1953, Chapter 2, Section 17), it is not difficult to assert that (see also Vinh & Giang, 2011, 2012):
(y1)

(y2)
(y3)
(y4)

Y(z) 2 H(S).
Y(z) = O(z3) as jzj ? 1.
Y(z) is bounded in N(1) and N(1/c).
Y+(t) = YÀ(t),t 2 L.


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P.C. Vinh et al. / International Journal of Engineering Science 60 (2012) 53–58

Properties (y1) and (y4) of the function Y(z) show that Y(z) is holomorphic in entire complex plane C, with the possible
exception of points: z = 1 and z = 1/c. By (y3) these points are removable singularity points and it may be assumed that the
function Y(z) is holomorphic in the entire complex plane C (see Muskhelishvili, 1963). Thus, by the generalized Liouville
theorem (Muskhelishvili, 1963) and taking account into (y2) we have:

YðzÞ ¼ PðzÞ;

ð15Þ

where P(z) is a third-order polynomial. From (14) and (15) we have:

f ðzÞ ¼ eCðzÞ PðzÞ:
C(z)

Since e

ð16Þ

– 0 "z 2 S (by (c1)), from (16) it follows that:

f ðzÞ ¼ 0 $ PðzÞ ¼ 0 in S:

ð17Þ

From (6), (16) and eC(z) – 0 "z 2 S, it follows that z = 0 is a double root of Eq. P(z) = 0.
Step 2
Also from (16) we have:

PðzÞ ¼ f ðzÞeÀCðzÞ :

ð18Þ

From (5), (10) and (11) it implies:

log gðtÞ ¼ i/ðtÞ;

/ðtÞ ¼ p À 2hðtÞ;

ð19Þ

where h(t) is given by (9)2. From (13) and (19) it follows (see also Nkemzi, 1997):

ÀCðzÞ ¼

1
X

In
;
nþ1
z
n¼0

ð20Þ

in which:

In ¼

1
2p

Z

1=c

t n /ðtÞdt;

n ¼ 0; 1; 2; 3; . . . :

ð21Þ

1

On use of (20) one can express eÀC(z) as follows:

eÀCðzÞ ¼ 1 þ

a1 a2 a3
þ 2 þ 3 þ OðzÀ4 Þ;
z
z
z

ð22Þ

where a1, a2, a3 are constants to be determined. Employing the identity:

À ÀCðzÞ Á0
e
¼ ðÀCðzÞÞ0 eÀCðzÞ ;

ð23Þ

and substituting (20), (22) into (23) yield:

a1 ¼ I0 ; a2 ¼
By expanding

I20
þ I1 ;
2

a3 ¼

I30
þ I1 I0 þ I 2 :

6

ð24Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À 1=z; 1 À 1=ðczÞ into Laurent series at infinity, it is not difficult to see that:

À Á
f ðzÞ ¼ A3 z3 þ A2 z2 þ A1 z þ A0 þ O zÀ1 ;

ð25Þ

where A2, A3 are given by



pffiffiffi 2
pffiffiffi
1 a2
2
A2 ¼ b ð1 À 2 cÞ À
c þ pffiffiffi
;
c 2

A3 ¼

pffiffiffi

ca2 :

ð26Þ

Substituting (22) and (25) into (18) yields:

PðzÞ ¼ A3 z3 þ z2 ðA2 þ A3 a1 Þ þ zðA1 þ A3 a2 þ A2 a1 Þ þ A3 a3 þ A2 a2 þ A1 a1 þ A0 :

ð27Þ

Since z = 0 is a double root of Eq. P(z) = 0, as mentioned above., from (19), (21), (24)1 and (27), the third root of Eq. P(z) = 0,
denoted by xs, is given by

xs ¼ À



A2 1 1
À
À 1 þ I0 ;
A3 2 c

ð28Þ

where A2, A3 are given by (26) and I0 is calculated by (9).
Step 3
Now we suppose that there exist two different Stoneley waves with the corresponding velocities x(1), x(2) (x(1) – x(2)). Then
x(1), x(2) are two different roots of Eq. f(z) = 0, and 0 < x(1), x(2) < 1. From (17) it follows P(x(1)) = P(x(2)) = P(0) = 0. But this is
impossible because P(z) is a third-order polynomial and z = 0 is a double root of Eq. P(z) = 0. Thus, if a Stoneley wave exists, it

57

P.C. Vinh et al. / International Journal of Engineering Science 60 (2012) 53–58

c

0.98
0.96
0.94

c2

10
8
0.1

6 r
0.2

γ

4

0.3

Fig. 1. Dependence of

0.4

0.5 2

pffiffiffi
x ¼ c=c2 on c and r.

Table 1
Values of the squared dimensionless velocity of Stoneley waves calculated by directly solving the secular Eq. (2) (x⁄), by the exact formulas (8) (xs) and (32) (ys)
for some given values of r. Here c = 1/3.
r

2

3

4

5

6

7

8

9

x⁄
xs
ys
1/ys

0.9901
0.9901
1.0100
0.9901

0.9686
0.9686
1.0324
0.9686

0.9501
0.9501
1.0525
0.9501

0.9357
0.9357
1.0687
0.9357

0.9246
0.9246
1.0815
0.9246

0.9158
0.9158
1.0919
0.9158

0.9087
0.9087
1.10043
0.9087

0.9029
0.9029
1.1075
0.9029

must be unique. This fact was also proved by Barnett et al. (1985) for the generally anisotropic case using the surface
impedance method.
Suppose that there exists a (unique) Stoneley wave. Then by above arguments, its squared dimensionless velocity x is a
zero of the third-order polynomial P(z), and x – 0. This implies that x must be xs given by (28), i.e. by (8). The proof of
Theorem 1 is finished h.
Fig. 1 demonstrates the dependence of the Stoneley-wave velocity on c and r as a 3D-picture.
3. Exact formula for the slowness
In this section we derive the formula for the squared dimensionless slowness y = 1/x of Stoneley waves that satisfies the
equation:

pffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi
FðyÞ  p1 ðyÞ À p2 ðyÞ y À 1 y À c ¼ 0;

y > 1;

Ã
2Â
p1 ðyÞ ¼ b 8y2 À 4ð2 þ cÞy þ 1 þ 4c y;

p2 ðyÞ ¼ 8b y2 À 4b y þ a2 :

ð29Þ

where
2

2

ð30Þ

Note that p2(y) > 0 "y. Eq. (29) is derived from Eq. (2) replacing x by 1/y. In the complex plane C, Eq. (29) takes the form:

pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi
FðzÞ  p1 ðzÞ À p2 ðzÞ z À 1 z À c ¼ 0;

ð31Þ

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
where pk(z) are given by (30), and z À 1; z À c are chosen as the principal branches of the corresponding square roots. For
z 2 (1 + 1) Eq. (31) becomes Eq. (29). Following the same procedure carried out in the previous section we have:
Theorem 2. If a Stoneley wave exists, then it is unique, and its squared dimensionless slowness ys ¼ c22 =c2 is given by
2

ys ¼

2a2 þ b cð1 À cÞð1 À 2cÞ


À I0 ;
2

2 a2 À b c2

ð32Þ

where

I0 ¼

1

p

Z

(

1

hðtÞdt;
c

hðtÞ ¼ atan

)
p1 ðtÞ
p
ffiffiffiffiffiffiffiffiffiffi
ffi
:
pffiffiffiffiffiffiffiffiffiffiffi

p2 ðtÞ t À c 1 À t

ð33Þ

Table 1 shows the numerical values of the squared dimensionless velocity of Stoneley waves which are calculated by directly
solving the secular Eq. (2) (for x⁄), by using the exact formulas (8) (for xs) and (32) (for ys) for some given values of r with
c = 1/3. It is seen from Table 1 that the corresponding values of x⁄, xs and 1/ys totally coincide with each other.


58

P.C. Vinh et al. / International Journal of Engineering Science 60 (2012) 53–58

4. On the existence of Stoneley waves
We will prove the following theorem:
Theorem 3. There always exists a (unique) Stoneley wave propagating along the bonded interface of two isotropic elastic halfspaces with the same bulk wave velocities.
Proof. It is clear that in order a Stoneley wave to exist it is sufficient that ys > 1, where ys is given by (32). Since Eq. F(z) = 0
has no solution in the interval (c, 1] due to the discontinuity of F(z) in (0, 1), it follows ys R (c, 1]. Therefore if ys > c then ys > 1.
This yields if ys > c, then a Stoneley wave can be propagate along the bonded interface of two isotropic elastic half-spaces
with the same bulk wave velocities. Now we prove that ys > c. Indeed, from (33) it implies that ÀI0 P À(1 À c)/2. Taking into
account this fact, from (32) we have:

h
i
2
a2 ð1 À cÞ þ b c ð1 À cÞ2 þ 2c2


ys À c P
> 0;

2
2 a2 À b c2

ð34Þ

due to c 6 3/4 < 1 and a2 > b2. The proof of Theorem 3 is finished. Note that the existence of Stoneley waves for the case under
consideration was also asserted by Stoneley (1924) by another technique. h

5. Conclusions
In this paper, the exact formulas for the velocity and the slowness of Sroneley waves propagating along the bonded interface of two isotropic elastic half-spaces having the same bulk wave velocities are derived using the complex function method. By using the obtained exact formulas, it is shown that there always exists a unique Stoneley wave propagating in two
bonded isotropic elastic half-spaces with the same bulk wave velocities. Since the obtained formulas are explicit they are
useful in practical applications.
Acknowledgments
The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED), and
by the DAAD.
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