DSpace at VNU: Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial stress

DSpace at VNU: Explicit secular equations of Rayleigh waves in elastic media under the influence of gravity and initial...

Explicit secular equations of Rayleigh waves in elastic media under

the influence of gravity and initial stress

Pham Chi Vinh

Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

a r t i c l e i n f o

Keywords:

Rayleigh waves

Rayleigh wave velocity

Gravity

Initial stress

Orthotropic

Secular equation

a b s t r a c t

The problem of Rayleigh waves in an orthotropic elastic medium under the influence of gravity and initial stress was investigated by Abd-Alla [A M Abd-Alla, Propagation of Ray-leigh waves in an elastic half-space of orthotropic material, Appl Math Comput 99 (1999) 61–69], and the secular equation of the wave in the implicit form was derived However, due to the uncorrect representation of the solution, the secular equation is not right The main aim of the present paper is to reconsider this problem We find the secular equation

of the wave in explicit form By considering some special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity of some previous studies, in which only implicit secular equations were derived

Ó 2009 Elsevier Inc All rights reserved

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, 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 mobile phones through to the study of earthquakes, as addressed by Samuel[2]

For the Rayleigh wave, its dispersion equations in the explicit form are very significant in practical applications They can

be used for solving the forward (direct) problems, and especially for the inverse problems Thus, the secular equations in the explicit form are always the main purpose of investigations related to the Rayleigh wave

The problem on the propagation of Rayleigh waves under the effect of gravity is a significant problem in seismology and geophysics, and it has attracted attention of many researchers such as[3–11]

Bromwhich[3]and Love[4]treated the force of gravity as a type of body force while Biot[5,6]and the other authors, following him, assumed that the force of gravity to create a type of initial stress of hydrostatic nature Bromwhich[3] as-sumed that the material is incompressible for the sake of simplicity Love[4]finished Bromwhich’s investigation by consid-ering the compressible case Biot[5,6], Kuipers [10]also took the assumption of incompressibility in their studies The material is assumed to be isotropic in the investigations[3–7,9,10], transversely isotropic in[8], and orthotropic in[11] Following Biot’s approach, Dey and Mahto[12]investigated the influence of gravity on the propagation of the Rayleigh wave in an isotropic elastic medium, taking into account the effect of initial stress The authors have derived the implicit secular equation of the wave Recently, Abd-Alla[13]extended this problem to the orthotropic case He employed two dis-placement potentials for representing the solution, and has also derived the dispesion equation of Rayleigh waves in the im-plicit form However, as will be shown, his represention of solution is uncorrect, the secular equation is, thus, not true

0096-3003/$ - see front matter Ó 2009 Elsevier Inc All rights reserved.

E-mail address: pcvinh@vnu.edu.vn

Contents lists available atScienceDirect

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c

The main purpose of the present paper is to re-investigate the problem on the propagation of Rayleigh waves in an ortho-tropic elastic medium under the effect of gravity and initial stress Unlike Abd-Alla, we seek the solution directly, do not use the displacement potentials Interestingly that, we have found the dispersion equations of the wave in the explicit form From this we obtain the explicit secular equation for Dey and Mahto’s investigation[12] When the initial stress is absent,

by considering its special cases, we derive the (exact) explicit secular equations of Rayleigh waves under the effect of gravity

of the previous studies[7–9], in which only the implicit secular equations have been found

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 parameters characterizing the material and external effects (see for example,[14–16]) Otherwise we call it an implicit secular equation

2 Basic equations

Consider a homogeneous orthotropic elastic body occupying the half-space x360 subject to the gravity and an initial compression P0along the x1-direction (see[13]) We are interested in a plane motion in ðx1;x3Þ-plane with displacement components u1;u2;u3such that:

Then the components of the stress tensorrij;i; j ¼ 1; 3 are related to the displacement gradients by the following equations

[13]:

r11¼ ðc11þ P0Þu1;1þ ðc13þ P0Þu3;3;

r33¼ c13u1;1þ c33u3;3;

r13¼ c44ðu1;3þ u3;1Þ;

ð2Þ

where cijare the material constants

Equations of motion are[13]:

r11;1þr13;3þ ðP0=2Þðu1;3 u3;1Þ;3qgu3;1¼qu€1;

r13;1þr33;3þ ðP0=2Þðu1;3 u3;1Þ;1þqgu1;1¼qu€3

ð3Þ

in whichqis the mass density of the medium, and g is the acceleration due to gravity, a superposed dot signifies differen-tiation with respect to the time t, commas indicate differendifferen-tiation with respect to the spatial variables xi From(2)3 it follows:

u1;3¼ 1

c44

Analogously, from(2)2we have:

u3;3¼ 1

c33

r33c13

c33

Employing(3)2and using(4)yield:

where / ¼ 1 þ P0=ð2c44Þ From(3)1and taking into account(2)1,(4), (5)we have:

r13;3¼ ðq=/Þ€u1 ½ðd þ P0Þ=/u1;11þ ðqg=/Þu3;1 ðD=/Þr33;1; ð7Þ

where d ¼ c11 c2

13=c33,D¼ c13=c33 In matrix (operator) form, following the Stroh formalism (see[17,18]), the Eqs.(4)–(7)

are written as:

u0

r0

 

¼ N u

r

 

where: u ¼ u1;u

3

;u

3¼ u3=/;r¼ ½r13;r33T, the symbol T indicates the transpose of matrices, the prime indicates the derivative with respect to x3and:

N ¼ N1 N2

K N3

; N1¼ 0 /@1

ðD=/Þ@1 0

; N2¼ 1=c44 0

0 1=ð/c33Þ

;

K ¼ ðq=/Þ@2

t ½ðd þ P0Þ=/@2 qg@1

qg@1 q@2tþ P0@2

; N3¼ NT1:

ð9Þ

Here we use the notations: @1¼ @=ð@x1Þ; @2¼ @2=ð@x2Þ; @2t ¼ @2=ð@t2Þ

In addition to Eq.(8), the displacement vector u and the traction vectorrare required to satisfy the decay condition:

uð1Þ ¼ 0; rð1Þ ¼ 0 ð10Þ

and the free-traction condition at the plane x3¼ 0:

3 Secular equation

Now we consider the propagation of a Rayleigh wave, travelling with velocity c and wave number k in the x1-direction The components u1;u

3of the displacement vector andr13;r33of the traction vector at the planes x3¼ const are found in the form:

u1;u

3;rj3

ðx1;x3;tÞ ¼ fU1ðx3Þ; U3ðx3Þ; iRjðx3Þgeikðx 1 ctÞ; j ¼ 1; 3: ð12Þ

Substituting(12)into(8)yields:

U0

R0

 

¼ iM U

R

 

where: U ¼ ½U1 U3T;R¼ ½R1 R3T, and:

M ¼ M1 M2

Q M3

; M1¼ 0 /

D=/ 0

; M2¼ ð1=kÞ 1=c44 0

0 1=ð/c33Þ

;

Q ¼ k ðX  d  P0Þ=/ ia

ia /ðX þ P0Þ

; M3¼ MT1;

ð14Þ

here a ¼qg=k, X ¼qc2, the prime indicates the derivative with respect to y ¼ kx3

Following the approach employed in[16,19,20], by eliminating U from(13), we obtain the equation for the traction vector

RðyÞ, namely:

^

where the matrices ^a; ^b; ^care given by:

^

a¼ Q1¼ 1

kd

/ðX þ P0Þ ia

ia ðX  d  P0Þ=/

; d ¼ ðX þ P0ÞðX  d  P0Þ  a2; ð16Þ

^

b¼ M1Q1

þ Q1M3¼ 1

kd

0 g1

g1 0

where

^

c¼ M1Q1M3 M2¼ 1

kd

h0 iDa

iDa h1

ð19Þ

in which

h0¼ /ðX  d  P0Þ  d=c44;

Now we seek the solution of the Eq.(15)in the form:

whereR0is a non-zero constant vector, p is a complex number which must satisfy the condition:

in order to ensure the decay condition(10) Substituting(21)into(15)leads to:

AsR0is a non-zero vector, the determinant of the system(23)must vanish This provides the equation for determining p, namely:

where

S ¼ 2Dþ 1

/c44

ðX  d  P0Þ þ 1

c33

ðX þ P0Þ;

P ¼ c11

c33X  P0

c33

1 X þ P0 /c44

 a

2

/c33c44

:

ð25Þ

It follows from(24)that:

p2

þ p2¼ S; p2p2

where p2;p2are two roots of the quadratic equation(24)for p2 It is not difficult to demonstrate that vectorR0¼ ½A BT, the solution of(23), is given by:

A ¼ g1p þ iaðD p2Þ;

Let p1, p2be the two roots of(24)satisfying the condition(22) Then the general solution of the equation(15)is:

RðyÞ ¼c1 A1

B1

 

eip1yþc2 A2

B2

 

where Ak;Bkðk ¼ 1; 2Þ are given by(27)in which p is replaced by pk,c1;c2are non-zero constants to be determined from the boundary condition(11)that reads:

Making use of(28)into(29)yields two equations forc1,c2:

g1p1þ iaðD p2Þ g1p2þ iaðD p2Þ

ðX þ P0Þp2þ h0 ðX þ P0Þp2þ h0

c1

c2

 

and vanishing the determinant of the system leads to the secular equation that defines the Rayleigh wave velocity After some algebraic manipulations and removing the factor ðp2 p1Þ, the secular equation is:

g1/ðX þ P0Þp1p2þ ia½h0þD/ðX þ P0Þðp1þ p2Þ  g1h0¼ 0: ð31Þ

Suppose p1;p2are the two roots of(24)satisfying the condition(22) We shall show that:

P > 0; 2 ffiffiffi

P

p

 S > 0; p1p2¼  ffiffiffi

P

p

; p1þ p2¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

 S

q

where S; P are defined by(25)

Indeed, if the discriminant D of the quadratic Eq.(24)for p2is non-negative, then its two roots must be negative in order that(22)is to be satisfied In this case, P ¼ p2p2>0 and the pair p1;p2are of the form: p1¼ ir1;p2¼ ir2where r1;r2are positive If D < 0, the quadratic Eq.(24)for p2has two conjugate complex roots, again P ¼ p2p2>0, and in order to ensure the condition(22): p1¼ t  ir; p2¼ t  ir where r is positive and t is a real number In both cases, P ¼ p2p2>0, p1p2is a negative real number and p1þ p2is a purely imaginary number with negative imaginary part, hence ðp1þ p2Þ2is a negative number Therefore, with the help of(26), it follows that the relations(32)are true

Taking into account(32)Eq.(31)becomes:

g1 /ðX þ P0Þ ffiffiffi

P

p

þ h0

 a h½ 0þD/ðX þ P0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

 S

q

Eq.(33)is the (exact) secular equation of Rayleigh waves in orthotropic elastic media under the gravity and the initial com-pression Since P; S, g1, h0, /, d,Dare explicitly expressed in terms of c; k; , it is clear that the secular Eq.(33)is fully explicit

As a depends on the wave number k, so does the Rayleigh wave velocity defined by Eq.(33) Thus, the Rayleigh wave in orthotropic elastic media under the gravity and the initial compression is dispersive

When the pre-stress is absent, i.e P0¼ 0, the Eq.(33)coincides with equation (2.15) in[21]witha¼ 0 However, it should be observed, as above, that the expressions in the square roots of equation (2.15) in[21]have positive values

4 Special cases

4.1 Rayleigh waves in isotropic elastic half-spaces under gravity and initial stresses

The problem was considered by Dey and Mahto[12], and the authors have been derived the secular equation in the im-plicit form In their notations, the exim-plicit secular equation for this problem is:

g1 /ðX þ PÞ ffiffiffiffiffi

P

p

þ h0

 a h½ 0þD/ðX þ PÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffiffiffi

P

p

 S q

S ¼ 2Dþ 1

/Q2

ðX  d  PÞ þ 1

B22ðX þ PÞ;

P¼ B11

B22

 X

B22

1 X þ P /Q2

 a

2

/Q2B22

;

/¼ 1 þ P

2Q2

; D¼B12 P

B22

; d¼ B11 P ðB12 PÞ

2

B22

;

g1¼ ðd þ P  XÞ DðX þ PÞ;

h0¼ ðX  d  PÞ / X þ P

Q2

þa

2

Q2

;

ð35Þ

B11, B12, B22, Q2are given by the formula (8a) in[12](or by (13) in[22]) It is noted that Eq.(34)is derived from Eq.(33)in which c11, c33, c13, c44, P0, P are replaced by B11 P, B22, B12 P, Q2, P, Prespectively

4.2 Rayleigh waves in orthotropic elastic half-spaces under the gravity

When the initial stress is absent, i.e P0¼ 0, the Eq.(33)becomes:

g1 X ffiffiffi

P

p

þ h0

 a h½ 0þDX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

 S

q

in which:

S ¼ 2Dþ ðX  dÞ=c44þ X=c33; P ¼ c11

c33 X

c33

1  X

c44

 a

2

c33c44

g1¼ ð1 þDÞX þ d; h0¼ ðX  dÞð1  X=c44Þ þ a2=c44: ð38Þ

Eq.(36)is the (exact) explicit secular equation of Rayleigh waves in orthotropic elastic media under the effect of gravity In this case we can show that the Rayleigh wave velocity is limited by:

Indeed, first we rewrite(37)1as follows:

S ¼ ch 33ðX  c11Þ þ c44ðX  c44Þ þ ðc13þ c44Þ2i

It follows from(32)1and(37)2that ðc11 XÞ and ðc44 XÞ must have the same sign This yields:

0 < X < minðc11c44Þ or X > maxðc11;c44Þ: ð41Þ

On use of(40)we see that the discriminant D ¼ S2 4P of Eq.(24)is given by:

D ¼ ðcn 13þ c44Þ4þ 2ðc13þ c44Þ2½c33ðX  c11Þ þ c44ðX  c44Þ

þ c½ 33ðX  c11Þ  c44ðX  c44Þ2o

=ðc33c44Þ2þ 4a

2

c33c44

: ð42Þ

Now, if the(41)2exists, then it follows from(42)that D P 0, so Eq.(24)for this case has two real roots p2;p2with the same sign, according to(32)1 On the other hand, it is clear from(40)and(41)2that S ¼ p2þ p2>0 Thus, both p2and p2are positive This leads to the contradiction to the requirement that p1;p2must have negative imaginary part The inequalities(39)are proved

4.3 Rayleigh waves in transversely isotropic elastic media under the effect of gravity

This problem was considered by Dey and Sengupta[8], but only the implicit form of the secular equation has been derived

in their work In their notations, the explicit secular equation for this problem is:

g1 X ffiffiffi

P

p

þ h0

 a h½ 0þDX

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

 S

q

in which:

S ¼2F

C þ

2F2

CL 

2A

L þ

2

Lþ

1 C

X; P ¼ A

C

X C

1 2X L

2a

2

g1¼ ð1 þ F=CÞX þ A  F2=C; h0¼ ðX  A þ F2=CÞð1  2X=LÞ þ 2a2=L; ð45Þ

here A; C; F; L are the material constants (see also[23]) It is noted that Eq.(43)is Eq.(36)in which c11, c33, c13, c44are replaced

by A, C, F, L=2, respectively The Rayleigh wave velocity is also subjected to the limitation(39)in which c11, c44are replaced by

A, L=2, respectively

4.4 Rayleigh waves in isotropic elastic half-spaces under the gravity

When the material is isotropic we have:

where k,lare Lame’s constants On view of(46)the limitation(39)becomes:

where x ¼ c2=c2(dimensionless Rayleigh wave speed), c2¼ ffiffiffiffiffiffiffiffiffi

l=q

p (the shear wave velocity), and the expression(37)1of S is simplified to:

where c1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðk þ 2lÞ=q

p

is the longitudinal wave velocity Now, making use of(46)along with(37)2, (38), (48)into(36)we have:

2ð1 cÞð2  xÞ x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  xÞð1 cxÞ c22

q

þ ðx þ 4c 4Þð1  xÞ þ2

ðx þ 4c 4Þð1  xÞ þ ð1  2cÞx þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  xÞð1 cxÞ c22

q

þ 2  ð1 þcÞx

r

where¼ g=ðkc2Þ Eq.(49)is the (exact) secular equation, in the explicit form, of Rayleigh waves in isotropic elastic half-spaces under the influence of gravity It is noted that this problem was considered by De and Sengupta[7]and Datta[9], but only the implicit dispersion equation of the wave have been derived

Now suppose that¼ g=ðkc2Þ is much small comparison with the unit Then by omitting the powers of order bigger than one in terms of, from the exact secular Eq.(49)we have immediately:

2ðc 1Þð2  xÞ ffiffiffiffiffiffiffiffiffiffiffi

1  x

p ð2  xÞ2 4 ffiffiffiffiffiffiffiffiffiffiffi

1  x

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx p

þx½ðx þ 4c 4Þð1  xÞ þ ð1  2cÞx ¼ 0: ð50Þ

Eq.(50)is an approximate dispersion equation of Rayleigh waves in isotropic elastic media under the effect of gravity, in the case 0 <¼ g=ðkc22Þ  1

Remark 1 (i) As noted in Remark 2.3i (Section5), when the material is isotropic and the stress is absent ðP0¼ 0Þ we can express the displacement components u1, u3 in terms of the potentilasu, w by(71), in whichu, w satisfy(81) It is not difficult to see that corresponding to a surface wave travelling with velocity c and wave number k in the x1-direction and decaying in the x3-direction, the potentialsu, w are given by:

u¼ A 1eikp 1 x 3þ A2eikp 2 x 3

eikðx 1 ctÞ;

w¼ ik A 1m1eikpx 3þ A2m2eikp 2 x 3

where mj¼ c2ðp2

j  1Þ; j ¼ 1; 2, p1, p2are roots of Eq.(24)with negative imaginary parts, A

1, A

2are non-zero constants On use

of(51),(71)into(2)2,3in which c13¼ k, c33¼ k þ 2l, c44¼l, and taking into account(12), we have:

RðyÞ ¼ c1 A1

B1

" #

eip1y

þ c2 A2

B2

" #

where:

Aj¼l½2pjþ ikmjð1  p2

jÞ;

Bj¼ kð1 þ p2jÞ þ 2lðp2j þ ikmjpjÞ; j ¼ 1; 2; ð53Þ



c1, c2, are non-zero constants Substituting(52)into(29)leads to a homogeneous linear system for c1, c2 Vanishing the determinant of this system provides:

A1 A2

B1 B2

With the help of(32), it is not difficult to verify that the Eq.(54)is equivalent to the equation:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1  xÞð1 cxÞ c 2

q

þ 2  ð1 þcÞx

r

ðx  2Þ2 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1  xÞð1 cxÞ c 2

q

2

þ 4 c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1  xÞð1 cxÞ c 2

q

in the interval x 2 ð0; 1Þ Note that, since the limiting velocity (see, for instance,[24,25]) in this case is c2¼ ffiffiffiffiffiffiffiffiffi

l=q

p , the imag-inary parts p1, p2are negative for the values of x 2 ð0; 1Þ Consequently, the relations(32)hold for values of x belonging to this interval

(2i) Eqs.(49) and (55)are different in form, but they are equivalent to each other in the interval (0, 1) This is proved as follows First, we recall that Eq.(54)is equivalent to Eq.(55)in the interval (0, 1) It is clear from(27)–(29)that Eq.(49)is equivalent to the equation:

eA1 eA2

eB1 eB2

in the interval (0, 1), where:

eAj¼ g1pjþ ia D p2

j

g1, h0,Dcorrespond to the isotropic elastic solids without pre-stress ðP0¼ 0Þ

Now, suppose that x is a root of(55)and 0 < x < 1, then x is a root of(54)andRðyÞ given by(52)is a solution of Eq.(16), i.e.:

^

a R00

 i^bR0

where ^a, ^b, ^care correspond to the isotropic elastic solids without pre-stress ðP0¼ 0Þ Since functions eip1y, eip2yare linearly independent of each other (noting that p1–p2), from(58)it follows:

^

ap2

j  ^bpjþ ^c

Bj

" #

From(59)it deduces that:

Bj

Aj

¼eBj

eAj or

Bj

eBj

¼Aj

On view of(60)it is clear that Eqs.(54) and (56)are equivalent to each other, therefore, x is a root of Eq.(56) Since(56)is equivalent to(49), x is a root of(49)also Thus, it has been observed that if x is a root of Eq.(55)then it is a root of Eq.(49) Now let x be a root of(49)and 0 < x < 1, then it is a solution of(56), and the corresponding potentialsu, w given by(51)

satisfy(81) ThereforeRðyÞ defined by(52)is a solution of(16) This again leads to the relations(60), and as its consequence,

x is a root of(54), thus x is a root of(55), because(54)is equivalent to(55) The proof is finished

(3i) In the case that 0 < 1, by neglecting the powers of order bigger than one in terms of, and taking into account the equalities:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1  xÞð1 cxÞ

p

þ 2  ð1 þcÞx

q

¼ ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx

p

1

ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx

ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx p

Eq.(55)becomes:

ðx  2Þ2 4 ffiffiffiffiffiffiffiffiffiffiffi

1  x

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx p

þ 4

xðc 1Þ ð1 þccxÞ ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

 ð1 þc xÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx p

Replacing x by c2=c2,cby c2=c2,by g=ðkc22Þ, from(63)we have:

2  c2=c2

 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  c2=c2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  c2=c2

q

 4g

c2c2 c2

k c

2þ c2 c2

  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  c2=c2

q

 c2þ c2 c2c2=c2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  c2=c2

q

Eq.(64)is an approximate secular equation of Rayleigh waves in isotropic elastic media under the effect of gravity, in the case 0 <¼ g= kc 2

 1, and it was first derived by Love[4]in a different way

Interestingly that, the (original) exact Eqs.(49) and (55)give the same roots, while the corresponding approximate Eqs

(50) and (63)give different solutions (seeFig 1)

4.5 Rayleigh waves in orthotropic elastic media without the effect of gravity and initial stress

When both initial stress and gravity are absent, i.e P0¼ a ¼ 0, the Eq.(33)simplifies to (see also[26,27]):

ðc44 XÞ c 213 c33ðc11 XÞ

þ ffiffiffiffiffiffiffiffiffiffiffiffiffic

33c44

p

X ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðc11 XÞðc44 XÞ p

In this case we can obtain the explicit formula for the Rayleigh wave velocity (see[27]), namely:

qc2=c44¼ ffiffiffi

b p

1b2b3= ð ffiffiffi

b p

1=3Þðb2b3þ 2Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

q

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R  ffiffiffiffi D p

3

q

where b1¼ c33=c11, b2¼ d=c11, b3¼ c11=c44, R and D are given by:

R ¼  1

54hðb1;b2;b3Þ;

D ¼  1

108 2

ffiffiffiffiffi

b1

p ð1  b2Þhðb1;b2;b3Þ þ 27b1ð1  b2Þ2þ b1ð1  b2b3Þ2þ 4

ð67Þ

in which

hðb1;b2;b3Þ ¼ ffiffiffi

b p

1h2b1ð1  b2b3Þ3þ 9ð3b2 b2b3 2Þi

ð68Þ

and the roots in(65)taking their principal values It is clear that the speed of Rayleigh waves in orthotropic elastic solids is a continuous function of three dimensionless parameters

5 On Abd-Alla’s representation of solution

We recall briefly Abd-Alla’s representation of solution, and then show that it is uncorrect First, substituting(2)into(3)

and taking into account the assumption: c44¼ ðc11 c13Þ=2, he obtained:

ðc11þ P0Þð2u1;11þ u1;33þ u3;13Þ þ c13ðu3;13 u1;33Þ  2qgu3;1¼ 2q€u1; ð69Þ

c11ðu1;13þ u3;11Þ þ ðc13þ P0Þðu1;13 u3;11Þ þ 2c33u3;33þ 2qgu1;1¼ 2q€u3: ð70Þ

According to his argument, by expressing the displacement components u1, u3 in terms of the displacement potentials

uðx1;x3;tÞ and wðx1;x3;tÞ as:

Eqs.(69) and (70)reduce, respectively, to:

and

c11ðw;11 w;33Þ  ðc13þ P0ÞO2

wþ 2c33w;33þ 2qgu;1¼ 2qw;€ ð75Þ

where O2f ¼ f þ f

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

γ=0.5

2 /c 2

ε Fig 1 Dependence on¼ g=kc2of x ¼ c 2 =c 2 defined by the exact secular Eqs (49) and (55) (solid line), approximate Eq (50) (dashed line) and approximate one (63) (dash-dot line), withc¼ 0:5.

Also by his argument, Eqs.(72) and (74)represent the compressive wave along the x1and x3directions, respectively, and Eqs.(73) and (75)represent the shear wave along those directions According to him, since he considered the propagation of Rayleigh waves in the direction of x1only, thus he restricted his attention only to Eqs.(72) and (75)

Now we show that u1, u3expressed by(71)in whichuand w is a solution of the system(72)and(75)does not satisfy the system(69), (70), in general Indeed, substituting(71)into(69) and (70)leads, respectively, to:

2 ðc 11þ P0ÞO2uqgw;1q u€

;1þ ðch 13 c11 P0ÞO2w 2qgu;1þ 2qw€i

and

c11ðw;11 w;33Þ  ðc13þ P0ÞO2

wþ 2c33w;33þ 2qgu;1 2qw€

;1þ 2 ch11u;11þ c33u;33qgw;1q u€i

Sinceu, w is a solution of the system(72) and (75), the first terms of the left-hand sides of(76) and (77)vanish, thus they become:

ðc13 c11 P0ÞO2

w 2qgu;1þ 2qw€

and

c11u;11þ c33u;33qgw;1q u€

respectively

It is clear that a pairuand w which is a solution of the system(72) and (75)does not necessarily satisfy the Eqs.(78) and (79) The observation is demonstrated

Remark 2

(i) It is not difficult to verify that if c33–c11þ P0(being valid in general) andu, w given by:

w 0; u¼ Aeipx 3ei x t; p2¼qx2=ðc11þ P0Þ; x¼ const – 0; A ¼ const – 0; ð80Þ

thenu, w is a solution of the system(72) and (75), however, they do not satisfy the Eqs.(78) and (79)

(ii) We can say, from(76) and (77), that Eq.(69)[Eq.(70)] is satisfied if u1, u3is defined by(71)in whichuand w is a solution of the system(72) and (78)[system(75) and (79)].From these, it is clear that the system(69), (70)is satisfied

if u1, u3is given by(71)in whichuand w is a solution of a system of four equations, namely:(72), (75), (78) and (79) (3i) When the initial stress is absent (P0¼ 0) and the material is isotrpic, Eqs.(78) and (79)are satisfied ifuand w is a solution of the system(72) and (75), which now is:

O2ug

c2w;11

c2u€¼ 0; O2wþg

c2u;11

That means, the representation of solution(71), (72) and (75)is valid for this case

(4i) If unknown functionsuand w are sought in the form (as in[13]):

where A; B are constants satisfying A2þ B2

–0, then p2has to satisfy a system of 3 quadratic equations that has no solution in general

(5i) The assumption: c44¼ ðc11 c13Þ=2 is not taken in the present paper

6 Conclusions

In this paper the propagation of Rayleigh waves in homogeneous orthotropic elastic media under the influence of gravity and initial stress is investigated We have found the exact secular equation in the explicit form, and it is a new result By considering its special cases, we obtain the exact explicit secular equations of Rayleigh waves under the effect of gravity, corresponding to some previous studies in which only implicit dispersion equations have been found In the case that the material is isotropic, the initial stress is absent, and 0 <¼ g= kc 22

 1, we have derived directly, from the exact secular equations, approximate dispersion equations, and one of them coincides with the one obtained by Love

Acknowledgement

The author wish to thank an anonymous reviewer for recommending him some references useful with the research

[1] L Rayleigh, On waves propagating along the plane surface of an elastic solid, Proc Roy Soc Lond A17 (1885) 4–11.

[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, The influence of initial stress on elastic waves, J Appl Phys 11 (1940) 522–530.

[6] M.A Biot, Mechanics of Incremental Deformation, Wiley, New York, 1965.

[7] S.N De, P.R Sengupta, Surface waves under the influence of gravity, Gerlands Beitr, Geophysik 85 (1976) 311–318.

[8] S.K Dey, P.R Sengupta, Effects of anisotropy on surface waves under the influence of gravity, Acta Geophys Pol 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 Mech Appl 31 (1986) 369–374.

[10] M Kuipers, A.A.F van de Ven, Rayleigh-gravity waves in a heavy elastic medium, Acta Mech 81 (1990) 181–190.

[11] A.M Abd-Alla, S.M Ahmed, Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity, Appl Math Comput 135 (2003) 187–200.

[12] S Dey, P Mahto, Surface waves in a highly pre-stressed medium, Acta Geophys Pol 36 (1988) 89–99.

[13] A.M Abd-Alla, Propagation of Rayleigh waves in an elastic half-space of orthotropic material, Appl Math Comput 99 (1999) 61–69.

[14] T.C.T Ting, An explicit secular equation for surface waves in an elastic material of general anisotropy, Quart J Mech Appl Math 55 (2) (2002) 297– 311.

[15] T.C.T Ting, Explicit secular equations for surface waves in an anisotropic elastic half-space from Rayleigh to today, Surface Waves in Anisotropic and Laminated Bodies and Defects Detection, NATO Sci Ser II Math Phys Chem., vol 163, Kluwer Acad Publ., Dordrecht, 2004 pp 95–116.

[16] M Destrade, The explicit secular equation for surface acoustic waves in monoclinic elastic crystals, J Acoust Soc Am 109 (2001) 1398–1402 [17] A.N Stroh, Dislocations and cracks in anisotropic elasticity, Philos Mag 3 (1958) 625–646.

[18] A.N Stroh, Steady state problems in anisotropic elasticity, J Math Phys 41 (1962) 77–103.

[19] M Destrade, Surface waves in orthotropic incompressible materials, J Acoust Soc Am 110 (2001) 837–840.

[20] M Destrade, Rayleigh waves in symmetry planes of crystals: explicit secular equations and some explicit wave speeds, Mech Mater 35 (2003) 931– 939.

[21] M Destrade, Seismic Rayleigh waves on an exponentially graded, orthotropic elastic half-space, Proc Roy Soc A 463 (2007) 495–502.

[22] I Tolstoy, On elastic waves in prestressed solids, J Geophys Res 87 (1982) 6823–6827.

[23] W.M Ewing, W.S Jardetzky, F Press, Elastic Waves in Layered Media, McGraw-Hill Book Comp., New York–Toronto–London, 1957.

[24] D.M Barnett, J Lothe, Consideration of the existence of surface wave (Rayleigh wave) solutions in anisotropic elastic crystals, J Phys F: Metal Phys 4 (1974) 671–686.

[25] P Chadwick, Interfacial and surface waves in pre-stressed isotropic elastic media, Z Angew Math Phys 46 (1995) S51–S71.

[26] P Chadwick, The existence of pure surface modes in elastic materials with orthorhombic symmetry, J Sound Vib 47 (1) (1976) 39–52.

[27] Pham Chi Vinh, R.W Ogden, Formulas for the Rayleigh wave speed in orthotropic elastic solids, Ach Mech 56 (3) (2004) 247–265.

6 Conclusions

In this paper the propagation of Rayleigh waves in homogeneous orthotropic elastic media under the in? ??uence of gravity and initial stress is investigated... investigated We have found the exact secular equation in the explicit form, and it is a new result By considering its special cases, we obtain the exact explicit secular equations of Rayleigh waves under. .. approximate secular equation of Rayleigh waves in isotropic elastic media under the effect of gravity, in the case <¼ g= kc 2

 1, and it was first derived by Love[4]in