DSpace at VNU: Rayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity

DOI 10.1007/s11012-013-9723-xRayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity Pham Chi Vinh · Nguyen Thi Khanh Linh Received

DOI 10.1007/s11012-013-9723-x

Rayleigh waves in an incompressible elastic half-space

overlaid with a water layer under the effect of gravity

Pham Chi Vinh · Nguyen Thi Khanh Linh

Received: 13 May 2012 / Accepted: 28 February 2013

© Springer Science+Business Media Dordrecht 2013

Abstract This paper is concerned with the

propaga-tion of Rayleigh waves in an incompressible isotropic

elastic half-space overlaid with a layer of non-viscous

incompressible water under the effect of gravity The

authors have derived the exact secular equation of the

wave which did not appear in the literature Based on

it the existence of Rayleigh waves is considered It is

shown that a Rayleigh wave can be possible or not,

and when a Rayleigh wave exists it is not necessary

unique From the exact secular equation the authors

arrive immediately at the first-order approximate

secu-lar equation derived by Bromwich [Proc Lond Math

Soc 30:98–120, 1898] When the layer is assumed to

be thin, a fourth-order approximate secular equation

is derived and of which the first-order approximate

secular equation obtained by Bromwich is a special

case Some approximate formulas for the velocity of

Rayleigh waves are established In particular, when the

layer being thin and the effect of gravity being small,

a second-order approximate formula for the velocity is

created which recovers the first-order approximate

for-mula obtained by Bromwich [Proc Lond Math Soc

P.C Vinh ()

Faculty of Mathematics, Mechanics and Informatics,

Hanoi University of Science, 334, Nguyen Trai Str., Thanh

Xuan, Hanoi, Vietnam

e-mail: pcvinh@vnu.edu.vn

N.T.K Linh

Department of Engineering Mechanics, Water Resources

University of Vietnam, 175 Tay Son Str., Hanoi, Vietnam

30:98–120, 1898] For the case of thin layer, a second-order approximate formula for the velocity is provided and an approximation, called global approximation, for it is derived by using the best approximate second-order polynomials of the third- and fourth-powers

Keywords Rayleigh waves· An incompressible elastic half-space· A layer of non-viscous water · Gravity· Secular equations · Formulas for the velocity

1 Introduction

Elastic surface waves in isotropic elastic solids, dis-covered by Lord 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 materi-als 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 mo-bile phones through to the study of earthquakes, as ad-dressed by Adams et al [2]

The problem on the propagation of Rayleigh waves under the effect of gravity is a significant problem in Seismology and Geophysics, and many investigations

on this topic have been carried out, see for examples [3 21]

The propagation of Rayleigh waves in an

incom-pressible isotropic elastic half-space underlying a

non-viscous incompressible fluid layer under the effect of

gravity was studied also by Bromwich [3] In his study

Bromwich assumed that the fluid layer is thin and

the effect of gravity is small With these assumption

the author derived the first-order approximate

disper-sion equation of the wave by approximating directly

the boundary conditions However, as illustrated

be-low in Sect.2.1, that approximate secular equation is

not valid for all possible values of the Rayleigh wave

velocity (lying between zero and the velocity of the

bulk transverse wave in the elastic substrate) Based

on the obtained first-order approximate secular

equa-tion, Bromwich derived a first-order approximate

for-mula for the Rayleigh wave velocity Bromwich did

not consider the general problem when the depth of

the layer and the effect of gravity being arbitrary This

problem is significant in practical applications

The main aim of this paper is to investigate the

gen-eral problem and to improve on Bromwich’s results In

particular: (i) We first derive the exact secular equation

of Rayleigh waves for the general problem From this

we arrive immediately at the first-order approximate

secular equation derived by Bromwich [3] and

indi-cate that it is not valid for all possible values of the

Rayleigh wave velocity (ii) Based on the exact

sec-ular equation the study of the existence of Rayleigh

waves is carried out It is shown that a Rayleigh wave

can be possible or not, and when a Rayleigh wave

exists it is not necessary unique Note that from the

first-order approximate dispersion equation derived by

Bromwich it is implied that if a Rayleigh wave

ex-ists it must be unique (iii) When the fluid layer being

thin we establish a fourth-order approximate secular

equation and of which the first-order approximate

sec-ular equation obtained by Bromwich is a special case

(iv) For the case of thin layer and small effect of

grav-ity, a second-order approximate formula for the

veloc-ity is created which recovers the first-order

approxi-mate formula obtained by Bromwich [3] (v) When

only the layer being thin, a second-order approximate

formula for the velocity is provided and an

approx-imation, called global approxapprox-imation, of the velocity

is derived by using the best approximate second-order

polynomials of the third- and fourth-powers

We note that, for the Rayleigh wave its speed is a

fundamental quantity which is of great interest to

re-searchers in various fields of science It is discussed

Fig 1 Elastic half-space overlaid with a water layer

in almost every survey and monograph on the subject

of surface acoustic waves in solids Further, it also involves Green’s function for many elastodynamic problems for a half-space, explicit formulas for the Rayleigh wave speed are clearly of practical as well

as theoretical interest

2 Secular equation

2.1 Exact secular equation Consider an incompressible isotropic elastic

half-space x3<0 that is overlaid with a layer of incom-pressible non-viscous water occupying the domain

0 < x3≤ h (see Fig. 1) The elastic half-space and

the water layer is separated by the plane x3= 0 Both the elastic half-space and the water layer are assumed

to be under the gravity We are concerned with a plane strain such that:

u k = u k (x1, x3, t ), k = 1, 3, u2≡ 0

p = p(x1, x3, t ), φ = φ(x1, x3, t ) (1)

where u k and p are respectively the displacement

components and the hydrostatic pressure

correspond-ing to the elastic half-space, φ is the velocity-potential

of the water layer with ∂φ/∂s as the velocity in the direction ds (see [3]), t is the time According to

Bromwich [3], the equations governing the motion of the elastic half-space and the water layer are:

p ,1+ μ 2u1= ρ ¨u1, p ,3+ μ 2u3= ρ ¨u3

where commas indicate differentiation with respect to

spatial variables x k, a superposed dot denotes

differen-tiation with respect to t ,2f = f ,11+ f ,33, ρ and μ

are the mass density and the Lame constant of the

elas-tic solid Addition to Eqs (2) are required the

bound-ary condition at x3= h [3]:

the continuity conditions at x3= 0 [3]:

μ(u 1,3 + u 3,1 ) = 0 at x3= 0 (5)

p + 2μu 3,3 + gρ − ρu3− ρ˙φ = 0 at x3= 0 (6)

and the decay condition at x3= −∞:

u k = 0 (k = 1, 3), p = 0, at x3= −∞ (7)

where ρ is the mass density of the water, g is the

acceleration due to the gravity Now we consider the

propagation of a Rayleigh wave, travelling with the

ve-locity c (> 0) and the wave number k (> 0) in the x1

-direction, and decaying in the x3-direction According

to Bromwich [3], the solution of Eqs (2) satisfying the

decay condition (7) is:

p

μk22 = Qe kx3exp(ikx1+ iωt) (8)

u1= −p ,1

μk22+ Ae sx3exp(ikx1+ iωt) (9)

u3= −p ,3

μk22+ Be sx3

exp(ikx1+ iωt) (10)

φ=C cosh(kx3) + D sinh(kx3)

exp(ikx1+ iωt)

(11)

where ω = kc is the circular frequency, k2= ω/c2<

k , c2=√μ/ρ , s=k2− k2

2 (> 0), Q, A, B, C, D

are constants to be determined from the conditions (3),

(4) and the relation ikA + sB = 0 Using Eqs (8)–

(10) into Eq (5) and taking into account ikA +sB = 0

yield:

where ˆB = B/k, x = c2/c22called the squared

dimen-sionless velocity of Rayleigh waves and 0 < x < 1 in

order to satisfy the decay condition (7) From Eqs (6),

(8), (10) and (11) we have:

μk22Q + 2μsB − k2Q

+ gρ − ρ(B − kQ) − iρωC= 0 (13)

It follows from Eqs (4), (8), (10) and (11):

On use of (11) in (3) and taking into account (14) yield:

(x − ε tanh δ)C = iω( ˆB − Q)(ε − x tanh δ) (15)

where ε = g/(kc2

2) (> 0) and δ = kh (> 0).

Since 0 < tanh δ < 1, it follows from (15) that:

x = ε tanh δ, because otherwise either ˆB = Q or ε −

x tanh δ = 0 If ˆB = Q then ˆB = Q = D = C = A = 0

by (12)–(14) and ikA + sB = 0 It is impossible be-cause this leads to a trivial solution If ε −x tanh δ = 0, from x = ε tanh δ we have immediately tanh δ = 1.

From (15) and x = ε tanh δ we have:

where:

f (x, ε, δ)=ε − x tanh δ

With the help of (16), Eq (13) becomes:



(x − 2) − ε(1 − r) − rf xQ

+2√

1− x + ε(1 − r) + rf x ˆB= 0 (18)

where r = ρ/ρ (> 0) Equations (12) and (18) estab-lish a homogeneous system of two linear equations for

Q and ˆB Vanishing the determinant of this system gives:

(2− x)2− 4√1− x − εx + rεx

− rf (x, ε, δ)x2= 0, 0 < x < 1 (19) Equation (19) is the exact secular equation of Rayleigh waves propagating in an incompressible isotropic elas-tic half-space overlaid with a layer of incompressible

non-viscous water of the finite depth h under the grav-ity The dimensionless parameters ε and δ characterize

the effect on the Rayleigh waves of the gravity and the water layer, respectively

Remark 1

(i) To the best knowledge of the authors the exact sec-ular equation (19) did not appear in the literature

(ii) With the assumption that ε and δ are both

suf-ficiently small, Bromwich [3] derived the first-order approximate secular equation of the wave, namely:

(2− x)2− 4√1− x − εx + rδx2= 0 (20)

by using approximations: sinh δ = δ, cosh δ = 1

(equivalently, tanh δ = δ) and neglecting the

quan-tity εδ/x (see [3], lines 12–14, p 107)

Unfor-tunately, for x ∈ (0, ε tanh δ) this quantity is not

small at all, therefore, in the interval (0, ε tanh δ)

Eq (20) is not an approximate equation of the

exact equation (19), i.e the approximate secular

equation (20) holds for only the values of x ∈

(ε tanh δ, 1) It will be shown later that Eq (20)

can be derived from the exact equation (19) by

approximating its left-hand side in the domain

ε tanh δ < x < 1.

When ρ→ 0, then r → 0, from (19) we have:

(2− x)2− 4√1− x − x = 0 (21)

that is the secular equation of Rayleigh waves

propagating in an incompressible isotropic

elas-tic half-space under the gravity (see also [3])

When h → 0, then δ → 0, it follows from (17)

that f (x, ε, δ) → ε/x This fact yields rεx −

rf (x, ε, δ)x2→ 0 and we again arrive at the

sec-ular equation (21)

When ε → 0, f → − tanh δ by (17), then Eq (19)

simplifies to:

(2− x)2− 4√1− x + rx2

This is the exact secular equation of Rayleigh waves

propagating in an incompressible isotropic elastic

half-space underlying a layer of non-viscous

incom-pressible fluid (without effect of gravity)

Now, suppose that δ and ε are both sufficiently

small By approximating tanh δ by δ and ε tanh δ by

zero, from Eq (19) we arrive immediately at Eq (20)

Since ε tanh δ ≈ 0, it follows that x −ε tanh δ ≈ 0 ∀x ∈

( 0, ε tanh δ) Therefore, the function f does not define

in the interval (0, ε tanh δ) This fact says that Eq (20)

is the first-order approximate equation of the exact

secular equation (19) only in the domain (ε tanh δ, 1),

not in the interval (0, ε tanh δ) at all.

2.2 On existence of Rayleigh waves

Since the existence of Rayleigh waves depends on the

existence of solution of Eq (19) in the interval (0, 1),

we first prove the proposition:

Proposition 1

(i) If 0 < ε < 1, then Eq (19) has a unique real root

belong to ( 0, 1) for r ≥ 1 + 2/ε and for 0 < r <

1+ 2/ε it has exactly two real roots x ( 1) , x ( 2)

in the interval (0, 1): x ( 1) ∈ (0, ε tanh δ), x ( 2)∈

(ε tanh δ, 1).

(ii) If ε ≥ 1 and 0 < ε tanh δ ≤ 1, then Eq (19) has

no real roots in the interval (0, 1) for r ≥ 1 + 2/ε

and it has a unique real solution belong to (0, 1) for 0 < r < 1 + 2/ε.

(iii) If ε ≥ 1 and ε tanh δ > 1, then Eq (19) has no

real roots in the interval (0, 1) for r ∈ (0, m] ∪ [1 + 2/ε, +∞) and it has a unique real solution

belong to (0, 1) for m < r < 1 + 2/ε, where m =

(ε tanh δ − 1)/((1 + ε) tanh δ).

Proof Equation (19) is equivalent to:

φ2(x) ≡ φ(x) + φ1(x) + ε(r − 1) = 0

x ∈ (0, 1), x = ε tanh δ (23) where:

φ (x)=(2− x)2− 4

√

1− x

and:

φ1(x) = −rxf (x, ε, δ), x ∈ (0, 1), x = ε tanh δ

(25)

It is not difficult to see that:

x2√

1− xφ(x)

= (2 − x)2−√1− x(2 + x)>0 ∀x ∈ (0, 1)

(26)

Therefore, φ(x) >0∀x ∈ (0, 1), i.e φ(x) is strictly increasingly monotonous in the interval (0, 1) Since (noting that 0 < tanh δ < 1):

φ

1(x)=r tanh δ[(x − ε)2+ 2εx(1 − tanh δ)]

(x − ε tanh δ)2 >0

∀x ∈ (0, 1), x = ε tanh δ, ∀ε > 0 (27)

the function φ2(x)is strictly increasingly monotonous

in the intervals (0, ε tanh δ) and (ε tanh δ, 1) ∀δ, r, ε >

0 It follows from (23)–(25) that:

φ2( 1) = 1 − ε + r tanh δ(1 − ε2)

(i) Suppose 0 < ε < 1, it follows that 0 < ε tanh δ <

1 (due to 0 < tanh δ < 1) and φ2( 1) > 0 (according to

(31)) From φ2( 1) > 0 and (30) it implies that Eq (19)

has alway a unique real root in (ε tanh δ, 1) From (28)

and (29), if−2 + ε(r − 1) ≥ 0 ↔ r ≥ 1 + 2/ε Eq (19)

has no real roots in (0, ε tanh δ) and it has exactly one

real root belong to (0, ε tanh δ) if 0 < r < 1 + 2/ε The

observation (i) is proved

(ii) (+) Let ε ≥ 1 and 0 < ε tanh δ < 1 Then

φ2( 1)≤ 0 by (31), therefore Eq (19) has no real root

in the interval (ε tanh δ, 1) due to (30) By (29), if

φ2( 0)≥ 0, Eq (19) thus has no real root in the

in-terval (0, ε tanh δ) and it has exactly one real root in

( 0, ε tanh δ) if φ2( 0) < 0 With the help of these facts

and (28) the observation (ii) for 0 < ε tanh δ < 1 is

proved

(+) Suppose ε ≥ 1 and ε tanh δ = 1 One can

see that for this case φ2( +1) = +∞ Since φ2(x)

is strictly increasingly monotonous in the intervals

( 0, 1), Eq (19) has no real root in the interval (0, 1)

if φ2( 0) ≥ 0 and it has exactly one real root in (0, 1)

if φ2( 0) < 0 These facts along with (28) leads to the

observation (ii) for ε tanh δ= 1

(iii) Let ε ≥ 1 and ε tanh δ > 1 Since φ2(x)is

con-tinuous and strictly increasingly monotonous in the

in-terval (0, 1), (⊂ (0, ε tanh δ)), Eq (19) has a unique

real root in the interval (0, 1) if φ2( 0) < 0 and φ2( 1) >

0, and it has no real root in the interval (0, 1) if either

φ2( 0) ≥ 0 or φ2( 1)≤ 0 With these facts we arrive

im-mediately at the observation (iii) 

Remark 2 When ε → 0: x ( 1) → 0 and x ( 2) → x r (δ),

where x r (δ)is the unique real root of Eq (22) (see

Re-mark3) The wave corresponding to x ( 2)is therefore

originates from the classical Rayleigh wave and the

wave corresponding to x ( 1)exists only when the

grav-ity is present To distinguish between these waves the

former is called “classical Rayleigh wave (CRW)” and

the latter is called “gravity-Rayleigh wave (GRW)”

From Proposition1and its proof we have the

follow-ing theorem sayfollow-ing about the existence of Rayleigh

waves

Theorem 1

(i) A Rayleigh wave is impossible if either {ε ≥

1, 0 < ε tanh δ ≤ 1, r ≥ 1 + 2/ε} or {ε ≥ 1,

ε tanh δ > 1, r ∈ (0, m] ∪ [1 + 2/ε, +∞)}.

(ii) There exists a unique Rayleigh wave, namely

CRW, if {0 < ε < 1, r ≥ 1 + 2/ε}.

(iii) There exists a unique Rayleigh wave, namely

GRW, if either {ε ≥ 1, 0 < ε tanh δ ≤ 1, 0 <

r <1+ 2/ε} or {ε ≥ 1, ε tanh δ > 1, m < r <

1+ 2/ε}.

(iv) There exist exactly two Rayleigh waves, one CRW

and one GRW, if {0 < ε < 1, 0 < r < 1 + 2/ε}.

Remark 3 By the same argument used for

Proposi-tion1, one can prove that:

(i) Equation (21) has a (unique) real solution in the

interval (0, 1) if and only if 0 ≤ ε < 1.

(ii) Equation (22) has always exactly one real root in

the interval (0, 1).

While the exact secular equation (19) has either no

root or one root, or two roots in the interval (0, 1), the

approximate secular equation (20) has at most one root

in the interval (0, 1) as shown below.

Proposition 2

(i) If Eq (20) has a real solution in the interval (0, 1),

then it is unique.

(ii) Equation (20) has a real solution in the interval

( 0, 1) if and only if 0 ≤ ε < 1 + rδ.

Proof By the same argument used for Proposition1.

We note that Bromwich [3] did not consider the ex-istence and uniqueness of solution of Eq (20) 2.3 Approximate secular equations

Let 0 < ε < 1, then according to Theorem1, a (unique) CRW exists and its squared dimensionless

veloc-ity x ( 2) is determined by Eq (19) in the domain

ε tanh δ < x < 1 Since 0 < ε∗tanh δ < 1, ε∗= ε/x, for x ∈ (ε tanh δ, 1), the following expansion holds for

x : ε tanh δ < x < 1:

(1− ε∗tanh δ)−1

= 1 + ε∗tanh δ + ε2

∗tanh2δ + ε3

∗tanh3δ

+ ε4

∗tanh4δ + Oδ5

(32)

here δ is assumed to be sufficiently small From (17) and (32) we have:

f (x, ε, δ) = (ε∗ − tanh δ)(1 − ε∗ tanh δ)−1

= ε∗+ε∗2− 1tanh δ + ε∗tanh2δ

+ ε2

∗tanh3δ + ε3

∗tanh4δ

+ Oδ5

(33)

Using the expansion tanh δ = δ − δ3/3+ O(δ5)into

(33) leads to:

f (x, ε, δ) = ε∗+ε∗2− 1δ + ε∗ δ2+ε2∗− 1/3δ3

+ε3∗− 2ε∗/3

δ4

+ Oδ5

(34) Substituting (34) into Eq (19) yields the fourth-order

approximate secular equation of the exact secular

equation (19) in the domain ε tanh δ < x < 1, namely:

F (x, ε, δ) ≡ (2 − x)2− 4√1− x − εx − rε2− x2

δ

− r



ε3

x − εx



δ2

− r



ε4

x2−4ε2

3 +x2 3



δ3

− r



ε5

x3−5ε3

3x +2εx 3



δ4= 0

In the first-order approximation, Eq (35) takes the

form:

F (x, ε, δ) ≡ (2 − x)2− 4√1− x − εx

− rε2− x2

δ = 0, ε tanh δ < x < 1

(36)

If ε and δ are both sufficiently small, from (36) we

immediately arrive at Eq (20) by neglecting−rε2δ

Remark 4

(i) Equation (35) determines the approximation of

x ( 2) , not of x ( 1)

(ii) To obtain approximate equations for x ( 1)(i.e

ap-proximate secular equations for GRWs) we can

start from:

(x − ε tanh δ)(2− x)2− 4√1− x − εx + rεx

− rx2(ε − x tanh δ) = 0

0 < x < 1, x = ε tanh δ (37)

that is equivalent to Eq (19)

3 Approximate formulas for the velocity

3.1 Both ε and δ being small

In [3], with the assumption that both ε and δ being

sufficiently small, Bromwich derived a first-order

ap-proximate formula for the dimensionless velocity√

x

of Rayleigh wave, namely:

x

x0 = 1 + 0.109ε − 0.099rδ (38)

where x0 is the squared dimensionless velocity of Rayleigh waves propagating in an incompressible

isotropic elastic half-space (i.e x0 = x(0, 0)) It is well-known that x0is approximately 0.9126 (see [6]), and its exact value is given by [22]:

x0=2 3



4+3

−17 + 3√33−3

17+ 3√33

(39)

or [23,24]:

x0= 1 − 26

27+2 3

11 3

1/3

−8 9

 26

27+2 3

11 3

−1/3

−1 3

2

(40) Now we extend the expression (38) to the one of

second-order Let x(ε, δ) is the solution of Eq (35), then we have:

ϕ(ε, δ) = Fx(ε, δ), ε, δ

From (41) it follows:

ϕ ε = 0, ϕ δ = 0, ϕ εε= 0

here we use the notations f ε = ∂f/∂ε, f δ = ∂f/∂δ,

f εε = ∂2f/∂ε2, f εδ = ∂2f/∂ε∂δ , f δδ = ∂2f/∂δ2,

f = f (ε, δ) Using (41) and (42) provides:

x ε = −F ε /F x , x δ = −F δ /F x

x εε= −F xx x ε2+ 2F xε x ε + F εε



/F x

x εδ = −(F xx x ε x δ + F xδ x ε + F xε x δ + F εδ )/F x

x δδ= −F xx x2δ + 2F xδ x δ + F δδ



/F x

(43)

here F = F [x(ε, δ), ε, δ] On the other hand, by ex-panding x(ε, δ) into Taylor series about the point

( 0, 0) up to the second order we have:

x(ε, δ) = x(0, 0) + x0

ε ε + x0

δ δ

+x εε0ε2+ 2x0

εδ εδ + x0

δδ δ2

where f0= f (0, 0) One can see that in order to get a second-order approximation for x(ε, δ) we can neglect

the terms of order bigger than two in the expression (35), i.e it is sufficient to take the function F as:

F (x, ε, δ) = (2 − x)2− 4√1− x − εx + rδx2

(45)

From (43) and (45), after some manipulations we

have:

x ε0= 0.1988, x δ0= −0.1814r

x εε0 = −0.2638, x εδ0 = 0.2012r

x δδ0 = −0.1475r2

(46)

Substituting these results into (44) yields:

x(ε, δ) = 0.9126 + 0.1988ε − 0.1814rδ − 0.1319ε2

+ 0.2012rεδ − 0.0737r2δ2 (47)

This is the second-order approximation of the squared

dimensionless velocity of Rayleigh waves From (47)

it is not difficult to get the second-order

approxima-tion of the dimensionless velocity of Rayleigh waves,

namely:

x

x0 = 1 + 0.1089ε − 0.0994rδ − 0.0782ε2

+ 0.1211rεδ − 0.0453r2δ2 (48)

that recovers the first-order approximation (38) Note that following the same procedure one can obtain the

higher-orders approximations of x and√

x

3.2 Only δ being small Now suppose that δ is sufficiently small and 0 < ε < 1 Let x(δ) is solution of (35) for a fixed given value of

ε , then: F [x(δ), δ] ≡ 0, where F is given by (35) By

expanding x(δ) into Taylor series about δ= 0 we have:

x(δ) = x0+ x( 0)δ + Oδ2

(49)

where x0= x(0) is the velocity of Rayleigh waves

propagating in an incompressible isotropic elastic half-space under the gravity and:

x( 0) = −F0

here f0= f [x(0), 0], f = f [x(δ), δ] Note that x0is determined by the following exact formula (see [25]):

x0=2(4 + ε)

3 − 3

16(ε + 11)ε2+ 4/27+ε3+ 12ε2+ 12ε + 136/27

93

16(ε + 11)(ε2+ 4)/27 + (ε3+ 12ε2+ 12ε + 136)/27

or it is calculated by a very highly accurate

approxi-mation, namely (see [25]):

x0=B−

√

B2− 4AC

where:

A = −(5.1311 + 2ε)

B= −21.2576 + 8ε + ε2

C = −(15.1266 + 8ε)

(53)

For obtaining the first-order approximation of x(δ) we

can ignore three last terms of F in (35), i.e the

func-tion F is taken as:

F (x, ε, δ) = (2 − x)2− 4√1− x − εx

− rε2− x2

Using (54) in (50) gives:

2

0)

2(x0− 2) + 2(1 − x0) −1/2 − ε (55)

Therefore, at the first-order approximation x(δ) is

given by:

x(δ) = x0+ r(ε2− x02)δ

2(x0− 2) + 2(1 − x0) −1/2 − ε (56)

Following the same procedure one can see that the

second-order approximation x(δ) is:

x(δ) = x0+r(ε2− x02)δ

where:

a1= 2(x0− 2) + 2(1 − x0) −1/2 − ε

a2= −[2 + (1 − x0) −3/2 ]r

2(ε2− x2

0)2

a13

−4r2x0(ε2− x2

0)

a21 −2rε(x0−

ε2

x0)

a1

(58)

and x0given by (51) or (52)

3.3 Global approximations

Suppose 0 < ε < 1 and δ is sufficiently small Then,

according to Theorem1, a (unique) CRW exists and

its squared dimensionless velocity x ( 2) is determined

approximately by Eq (36) By dividing its two sides

by x (> 0), Eq (36) is equivalent to:

Φ

x, ε, δ∗

≡ φ3



x, δ∗

− δ∗ε2/x − ε = 0

ε tanh δ < x < 1, δ∗= rδ > 0 (59)

where φ3(x, δ∗)is given by:

φ3(x)=(2− x)2− 4

√

1− x

Following the same procedure presented in

Proposi-tion1, one can prove that:

Proposition 3 Let 0 < ε < 1 and δ is sufficiently

small Then Eq (36) has a unique real solution in the

interval (ε tanh δ, 1).

Now we want to have approximate expressions of

the solution of Eq (36) by using the best approximate

second-order polynomials of the powers x3 and x4

(see [26]) in the sense of least-square We call them the

global approximations After squaring and rearranging

Eq (36) is converted to:

a4x4+ a3x3+ a2x2+ a1x + a0= 0 (61)

where:

a0= δ∗ε2

δ∗ε2− 8,

a1= −16 + 8δ∗ε2+ 2ε3δ∗− 8ε

a2= 8ε − 2δ∗ε2+ 8δ∗+ ε2− 2δ∗2ε2+ 24

a3= −2(4 + ε)1+ δ∗

, a4=1+ δ∗2

(62)

After replacing x4 and x3 by the best approximate

second-order polynomials Eq (62) becomes a

quadra-tic equation of which one solution corresponding to

the Rayleigh waves

Let x(ε, δ∗) is the (unique) solution of Eq (36),

then it is the (unique) solution of Eq (59) Thus we

have: Φ [x(ε, δ∗), ε, δ∗] ≡ 0, where Φ[x(ε, δ∗), ε, δ∗]

is defined by (59) From (59) it follows:

Φ x=∂φ3

∂x +δ∗ε2

x2 >0

Φ ε= −



1+2δ∗ε

x



<0

∀ x ∈ (0, 1), ε > 0, δ∗>0

(63)

because as shown in Sect 2.1: ∂φ3/∂x >0 ∀x ∈

( 0, 1), δ∗> 0 As x

ε = −Φ ε /Φ x, from (63) we

con-clude that: x ε >0∀ε > 0, δ∗>0 That means:

x

ε, δ∗

> x

0, δ∗

∀ε > 0, δ∗>0 (64)

where x(0, δ∗)is the (unique) solution of the equation:

φ3



x, δ∗

=(2− x)2− 4

√

1− x

On use of (65) it is not difficult to verify that dx(0, δ∗)/

dδ∗<0, ∀δ∗> 0, therefore: x(0, δ∗) > x( 0, δ∗

0) if

0 < δ∗< δ∗

0 From this fact and (64) we conclude that:

x

ε, δ∗

> x

0, δ∗

0



∀ε > 0, ∀δ : 0 < δ∗< δ∗

0 (66) Inequality (66) says that the best interval on which

we determine the best approximate second-order

poly-nomials of the powers x3 and x4 is the interval

[x(0, δ∗

0),1] Note that for a given value of δ∗

0 it is

easy to calculate x(0, δ∗

0)by solving directly Eq (65)

As an example, let r = 0.5, δ = 0.1, then δ∗

0= 0.05.

By solving directly Eq (65) we have: x(0, 0.05)=

0.9034 Following Vinh and Malischewsky [26], the best approximate second-order polynomials of the

powers x3 and x4 in the interval [0.9034, 1] in the

sense of least-square are:

x4= 5.4364x2− 6.8944x + 2.4578 (67)

x3= 2.8551x2− 2.7158x + 0.8607 (68)

Replacing x4 and x3 in Eq (61) by (67) and (68),

respectively, we obtain a quadratic equation for x,

namely:

whose solution corresponding to Rayleigh waves is:

x=−B +

√

B2− 4AC

where 0 < ε < 1, 0 < δ∗< 0.05, and:

A = 5.43641+ δ∗2

+ 8ε − 2δ∗ε2+ 8δ∗+ ε2

− 2δ∗2ε2+ 24 − 2.8551(8 + 2ε)1+ δ∗

B = −6.89441+ δ∗2

− 16 + 8δ∗ε2+ 2ε3δ∗

− 8ε + 2.7158(8 + 2ε)1+ δ∗

C = 2.45781+ δ∗2

− 8δ∗ε2

− 0.8607(8 + 2ε)1+ δ∗

+ δ∗2ε4

(71)

Fig 2 Plots of x(ε, 0.04) calculated by the approximate

for-mula (56), by the global approximation (70), (71) and by solving

directly Eq (36) They most totally coincide with each other

Figure2 shows the dependence on ε ∈ [0, 0.9] of

x(ε, 0.04) which is calculated by the approximate

for-mula (56), by the globally approximate formulae (70),

(71) and by solving directly Eq (36) They most

to-tally coincide with each other This says that the

ap-proximation (70) has a very high accuracy

Remark 5

(i) Since 0 < x < 1, we can take the interval [0, 1] for

determining the best approximate second-order

polynomials of the powers x3and x4 According

to Vinh and Malischewsky [26], the best

approx-imate second-order polynomials of the powers x3

and x4in[0, 1] in the sense of least-square are:

x4=12

7 x

2−32

35x+ 3

x3= 1.5x2− 0.6x + 0.05 (73)

With the approximations (72), (73), x is given by

(70) in which A, B, C are calculated by:

A=12

7



1+ δ∗2+ 8ε − 2δ∗ε2+ 8δ∗+ ε2

− 2δ∗2ε2+ 24 −3

2(8+ 2ε)1+ δ∗

B= −32

35



1+ δ∗2

− 16 + 8δ∗ε2+ 2ε3δ∗− 8ε

+3

5(8+ 2ε)1+ δ∗

35



1+ δ∗2− 8δ∗ε2

− 1

20(8+ 2ε)1+ δ∗+ δ∗2ε4

(74)

Fig 3 Plots of x(ε, 0.04) calculated by the globally

approxi-mate formulae (70), (74) (dashed line), (70), (71) (solid line) and by solving directly Eq (36) (solid line)

in which 0 < ε < 1 and δ∗ >0 However, this

approximation of x is less accurate than the one

given by (70)–(71), as shown in Fig.3, because

the polynomials given by (72) and (73) are not the

best approximate second-order polynomials of x4 and x3, respectively, in the interval [x(0, δ∗

0),1] (⊂ [0, 1])

(ii) While the accuracy of the global approximation (70) is the same as that of the approximation (56),

as shown in Fig.2, the global approximation (70)

is more simple, it is therefore more useful in prac-tical applications

4 Conclusions

In this paper, the propagation of Rayleigh waves in

an incompressible isotropic elastic half-space overlaid with a layer of non-viscous water under the effect of gravity is investigated The exact secular equation of the wave is derived and based on it the existence of Rayleigh waves is examined When the layer being thin, a fourth-order approximate secular equation is established and using it some approximate formulas for the velocity are established The obtained secular equations and formulas for the Rayleigh wave velocity are powerful tools for analyzing the effect of the water layer and the gravity on the propagation of Rayleigh waves, especially for solving the inverse problems

Acknowledgements The work was supported by the Vietnam

National Foundation For Science and Technology Development

(NAFOSTED) under Grant no 107.02-2012.12.

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an incompressible isotropic elastic half-space overlaid with a layer of non-viscous water under the effect of gravity is investigated The exact secular equation of. ..

18 Abd-Alla AM, Hammad HAH (2004) Rayleigh waves in< /small>

a magnetoelastic half-space of orthotropic material under the influence of initial stress and gravity field Appl Math Comput... approximate formulas for the velocity are established The obtained secular equations and formulas for the Rayleigh wave velocity are powerful tools for analyzing the effect of the water layer and the