DSpace at VNU: New results on Rayleigh waves in incompressible elastic media subjected to gravity

Pham Chi Vinh · Nguyen Thi Khanh LinhNew results on Rayleigh waves in incompressible elastic media subjected to gravity Received: 11 October 2011 / Revised: 20 March 2012 / Published onl

Pham Chi Vinh · Nguyen Thi Khanh Linh

New results on Rayleigh waves in incompressible elastic media subjected to gravity

Received: 11 October 2011 / Revised: 20 March 2012 / Published online: 19 May 2012

© Springer-Verlag 2012

Abstract In this paper, the following new results related to Rayleigh waves in incompressible elastic media

under the influence of gravity are presented: (i) the exact formulas for the velocity of Rayleigh waves prop-agating along the free-surface of an incompressible isotropic elastic half-space under the gravity are derived, and (ii) two approximate formulas for the velocity of the Rayleigh waves are established and it is shown that their accuracy is very high To derive the exact formulas, we use the theory of cubic equation, and to establish the approximate formulas, we employ the best approximate second-order polynomials of the cubic power The obtained formulas are powerful tools for analyzing the effect of gravity on the propagation of Rayleigh waves and for solving the inverse problem

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 It would not be far-fetched to say that Rayleigh’s study of surface waves in an elastic half-space has had fundamental and far-reaching effects upon modern engineering, stretching from mobile phones to the study of earthquakes, as addressed by Adams et al [2]

The problem of propagation of Rayleigh waves under the effect of gravity is an important problem in seis-mology and geophysics, and many investigations on this topic have been carried out, see for example [3 19]

In most of these investigations, the material is assumed to be compressible, and among them, the studies by Vinh and Seriani [18] and Vinh [19] provide the explicit secular equations, and the remaining investigations provide the equations in an implicit form

Bromwich [3], Biot [6] and Kuipers [12] assumed incompressibility in their studies for the sake of simplic-ity and derived the explicit secular equation of Rayleigh waves in an incompressible isotropic elastic half-space under the effect of gravity by employing different approaches

The Rayleigh wave velocity is a physical quantity of great importance It is discussed in almost every text book and monograph on the subject of surface acoustic waves in solids Further, the Rayleigh wave velocity also appears in Green’s function of many elastodynamic problems for a half-space, and explicit formulas for the Rayleigh wave velocity are thus clearly of practical as well as theoretical interest It is worth noting that although the existence of Rayleigh waves has been discovered by Rayleigh [1] more than 120 years ago, the

P C Vinh (B)

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science,

334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

E-mail: pcvinh@vnu.edu.vn

Tel.: +84-4-5532164

Fax: +84-4-8588817

N T K Linh

Department of Engineering Mechanics, Water Resources University of Vietnam, 175 Tay Son Str., Hanoi, Vietnam

exact formulas for the velocity of Rayleigh waves in various elastic media were found only recently, see for example [20,21] and the references herein

In this paper, we obtain the exact formulas for the velocity of Rayleigh waves in an incompressible isotropic elastic half-space by applying the theory of cubic equation Two approximate formulas for the Rayleigh wave velocity are established by employing the best approximate second-order polynomials of the cubic power It is shown that they are very good approximations The necessary and sufficient conditions for the existence and uniqueness of Rayleigh waves are also provided The obtained formulas might be powerful tools for evaluating the effect of gravity on Rayleigh waves and for solving inverse problems

2 Secular equation

In this section, we briefly recall the derivation of the secular equation of Rayleigh waves in an incompressible elastic half-space under gravity We follow Biot’s approach [6] but do not employ the displacement potentials

in the following analysis

An incompressible isotropic elastic half-space subjected to gravity occupies the half-space x3≥ 0 We are concerned with the plane strain

where uk are the displacement components, and t is time The equations of motion are then [6]

(s11+ ρgu3) ,1 + s13,3 = ρ ¨u1, s13,1 + (s33+ ρgu3) ,3 = ρ ¨u3, (2) whereρ is the mass density of the medium, g is the acceleration due to gravity, s i j are the stress components,

commas indicate the differentiation with respect to spatial variables xk, and a superposed dot denotes the differentiation with respect to t The stress–strain relation is of the form [6]

s11− s = 2μu1,1 , s33− s = 2μu3,3 , s13= μ(u1,3 + u3,1 ), (3)

where s = (s11+s33)/2, and μ is the Lame constant In addition to Eqs (2) and (3), the traction-free condition

on the planar surface x3= 0 is

and the decay conditions require that

Introducing the notion of fictitious stresses [6]

s

11= s11+ ρgu3, s

33= s33+ ρgu3, s

Eqs (2)–(5) become

s

11,1 + s

13,3 = ρ ¨ u1, s

13,1 + s

s

11− s = 2μu1,1 , s33 − s= 2μu3,3 , s13 = μ(u1,3 + u3,1 ), (8)

s

where s = (s

11+ s

33)/2 = s + ρgu3 Substituting Eqs (8) into Eqs (7), and taking into account the incompressibility condition

we obtain

μ(u1,11 + u1,33 ) + s

,1 = ρ ¨ u1,

Next, we consider propagation of a Rayleigh wave, traveling with the velocity c and the wave number k (> 0)

in the x1-direction and decaying in the x3-direction Then, u1, u3, sare sought in the form

u1= A1e−λkx3ei k (x1−ct), u3= A3e−λkx3ei k (x1−ct), s= Ae−λkx3ei k (x1−ct), (13)

where A1, A3and Aare nonzero constants, andλ must have a positive real part to satisfy the decay

condi-tion (10) Substituting expressions (13) into Eqs (11) and (12), one obtains a homogeneous linear system of

equations for A1, A3and A The vanishing of the determinant of the system provides an equation forλ

where x = c2/c2

2, c22 = μ/ρ The quadratic equation (14) forλ2has two real rootsλ2

1= 1 and λ2

2 = 1 − x.

Since Re(λ2) > 0, it follows that

Suppose that inequalities (15) hold, thenλ1= 1 and λ2=√1− x, and the general solution of the system of

equations (11)–(12) that satisfies the decay condition (10) is then

u1= −(ikγ1e−kx3 + ikλ2γ2e−λ2 kx3)e i k (x1−ct),

u3= (kγ1e−kx3 + kγ2e−λ2 kx3)e i k (x1−ct),

s = γ1ρk2c2e−kx3ei k (x1−ct),

(16)

whereγ1, γ2are constants to be determined Substituting the relations (16) into the boundary conditions (9),

we obtain

2γ1+ (2 − x)γ2= 0,

The vanishing of the determinant of the system of Eq (17) yields the secular equation

where  = ρg/(kμ) ≥ 0 Equation (18) is the secular equation of Rayleigh waves in an incompressible isotropic elastic half-space under gravity that was derived by Bromwich [3], Biot [5,6] and Kuipers [12] by applying different approaches

3 Exact formulas for the velocity of Rayleigh waves

Let x ∈ (0, 1) and define the function φ(x) as follows:

φ(x) = (2 − x)2− 4

√

1− x

Then, Eq (18) can be written as

It is readily to see that

x2√

1− x φ(x) = (2 − x)[2 −√1− x(2 + x)] > 0 ∀ x ∈ (0, 1). (21) Therefore,φ(x) > 0 ∀ x ∈ (0, 1), that is, φ(x) is strictly monotonously increasing in the interval (0, 1).

Recalling Eq (20), usingφ(+0) = −2 and φ(1) = 1 and observing that the function φ(x) is strictly

monoto-nously increasing in the interval(0, 1) (Fig.1), we formulate Proposition1

Proposition 1 Let  ≥ 0, then:

(i) If 0 ≤  < 1, then Eq ( 20 ) [also Eq ( 18 )] has a unique real solution in the interval (0, 1), denoted

by x r ().

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

−1.5

−1

−0.5 0 0.5 1

x0

x

Fig 1 Plot of the functionφ(x)given in Eq (19 )

(ii) If  ≥ 1, Eq ( 20 ) [also Eq ( 18 )] has no real solution in the interval (0, 1).

(iii) The function xr () is strictly increasingly monotonous in the interval [0, 1) from x0= x r (0) to 1 (but not equal to 1) (Fig.1), where x r (0) is the velocity of Rayleigh waves in incompressible isotropic elastic half-spaces without the effect of gravity.

Note that the approximate value of x0is 0.9126 (see, e.g., [22]) and its exact value is given by (see [20,23])

x0= 1 − η2

where

η0=

 26

27 +2 3

 11 3

1/3

−8 9

 26

27+2 3

 11 3

−1/3

−1

It is not difficult to verify that for ∈ [0, 1), the function ϕ(x) = (2 − x)2− x is strictly monotonously

decreasing in the interval(0, 1), consequently

Using the relations (24), one can show that, in the interval(0, 1), the secular equation (18) is equivalent to

where

Therefore, from Proposition1follows Proposition2

Proposition 2 Let  ≥ 0, then:

(i) If 0 ≤  < 1, Eq (25) has a unique real solution in the interval(0, 1), namely x r

(ii) For  ≥ 1, Eq (25) has no real root in the interval(0, 1).

Introducing the new variable z defined by

z = x +1

Equation (25) becomes

where

q2= (a2

2− 3a1)/9, r =2a23− 9a1a2+ 27a0



Using the theory of cubic equation, three roots of Eq (28) are given by Cardan’s formula [24]:

z1= S + T,

z2= −1

2(S + T ) +1

2i

√

3(S − T ),

z3= −1

2(S + T ) −1

2i

√

3(S − T ),

(30)

where i2= −1 and

S=3

R+√D , T =3

R−√D ,

D = R2+ Q3, R = −1

Remark 1 The nature of the three roots of Eq (28) depends on the sign of its discriminant D, in particular:

If D > 0, then Eq (28) has one real root and two complex conjugate roots; if D = 0, Eq (28) has three real

roots, at least two of which are equal; if D < 0, then Eq (28) has three real distinct roots

From Eqs (29) and (31), it follows that

R=9a1a2− 27a0− 2a3

2



/54,

D=4a0a32− a2

1a22− 18a0a1a2+ 27a2

0+ 4a3 1



Using the relations (26), the first of Eqs (29) and (32) can be written as

q2 = (2+ 8 − 8)/9,

D = 16( + 11)(2+ 4)/27.

It is clear from the third of Eqs (33) that D> 0∀ ∈ [0, 1) Thus, recalling Remark1, Eq (28) has only one

real root, denoted by zr, and with the aid of Eqs (30), it is given by

z r = 3



R+√D+ 3



From Eqs (27) and (34), it follows that

x r = 2(4 + )

3 −3

16( + 11)(2+ 4)/27 + (3+ 122+ 12 + 136)/27

93

16( + 11)(2+ 4)/27 + (3+ 122+ 12 + 136)/27 ,  ∈ [0, 1).

(35)

The formula (35) is the exact expression for the velocity of Rayleigh waves in incompressible isotropic elas-tic half-spaces under the effect of gravity Figure2shows its dependence on the gravity parameter = g/(kc2

2).

One can see again from Fig.2that the Rayleigh wave velocity is a strictly monotonously increasing function

of ∈ [0, 1).

From the formula (35), we may derive an alternative exact expression for x0

x0=8

3− 2

 17

27 +1 3

 11 3

1/3

+4 9

 17

27+1 3

 11 3

−1/3

Using the transformation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.91

0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01

ε

2 /c 2

Fig 2 Plot of the dimensionless Rayleigh wave velocity x r() versus the gravity parameter  ∈ (0, 1),  = g/(kc2)

Equation (18) takes the form

It is readily to see that, in the interval(0, 1), Eq (38) is equivalent to

We can thus formulate Proposition3

Proposition 3 Suppose  ≥ 0, then:

(i) Equation (39) has a unique real solution in the interval (0, 1), denoted by t r , if and only if  < 1.

(ii) For  ∈ [0, 1), t r is given by

t r= 3

26 − 9

( + 11)(2+ 4)

93

26− 9



( + 11)(2+ 4)

27

−1

thus, the squared dimensionless Rayleigh wave velocity is

x r = 1 −

⎛

⎜

⎜

⎝ 3

26 − 9

( + 11)(2+ 4)

93

26− 9



( + 11)(2+ 4)

27

−1 3

⎞

⎟

⎟

⎠

2

(41)

We summarize the obtained results in Theorem1

Theorem 1 Let  ≥ 0, then:

(i) A Rayleigh wave exists if and only if 0 ≤  < 1.

(ii) If a Rayleigh wave exists, then it is unique, and its squared dimensionless velocity xr () is given by Eqs (35) or (41)

(iii) The squared dimensionless Rayleigh wave velocity xr () is a strictly monotonously increasing function

in the interval [0, 1), from x0, given in Eqs (22) or (36), to 1 (but not equal to 1)

4 Approximate formulas for the velocity of Rayleigh waves

In this section, we set down approximate formulas for the velocity of Rayleigh waves by applying the best approximate second-order polynomials of the cubic power Since these approximate formulas are simpler than the exact ones and have a very high accuracy, they are useful in applications

Following Vinh and Malischewsky [25], one can see that in the interval [x0, 1], the best approximate

second-order polynomial of x3in the sense of least squares is

Introducing expression (42) into Eq (25), we obtain a quadratic equation

whose solution corresponding to the Rayleigh wave is

x1= B−

√

B2− 4AC

where

A = −(5.1311 + 2), B = −(21.2576 + 8 + 2), C = −(15.1266 + 8). (45)

Similarly, the best approximate second-order polynomial of t3in the interval[0, t0] in the sense of least squares is

Introducing the expression (46) into Eq (39), we obtain a quadratic equation for t

whose solution corresponding to the Rayleigh wave is

t r= −(2.9475724 + ) +

√

2+ 0.1215448 + 14.4543266

The squared dimensionless velocity is given by x2= 1 − t2

r Figure3shows the dependence on the gravity parameter of the exact velocity x r () and the approximate

velocities x1() and x2() From Fig.3, one can see that the plots of xr(), x1() and x2() totally coincide

with each other This means that the obtained approximate formulas have a very high accuracy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.91

0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01

ε

Fig 3 The plots of the exact velocity x r () and the approximate velocities x1() and x2() They totally coincide with each other

5 Conclusions

In this paper, the exact and highly accurate approximate formulas for the velocity of Rayleigh waves in an incompressible isotropic elastic half-space under gravity are derived These formulas are useful tools for evaluating the effect of gravity on propagation of Rayleigh waves and for solving the inverse problem as well

Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development

(NAFOSTED).

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5 Conclusions

In this paper, the exact and highly accurate approximate formulas for the velocity of Rayleigh waves in an incompressible isotropic elastic. ..

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