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Explicit secular equations of Rayleigh waves in a non-homogeneousorthotropic elastic medium under the inﬂuence of gravity Pham Chi Vinha,*, Géza Serianib a Faculty of Mathematics, Mechan

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Explicit secular equations of Rayleigh waves in a non-homogeneous

orthotropic elastic medium under the inﬂuence of gravity

Pham Chi Vinha,*, Géza Serianib

a

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

b

Istituto Nazionale di Oceanograﬁa e di Geoﬁsica Sperimentale, Borgo Grotta Gigante 42/C, 34100 Sgonico, Trieste, Italy

a r t i c l e i n f o

Article history:

Received 14 November 2008

Received in revised form 26 March 2009

Accepted 21 April 2009

Available online 3 May 2009

Keywords:

Rayleigh waves

Rayleigh wave velocity

Orthotropic

Secular equation

Non-homogeneous

Gravity

a b s t r a c t

The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the inﬂuence of gravity is investigated Using an appropriate representation of the solution we derive the secular equation of the wave motion in the explicit form Moreover, following the same approach, we obtain the explicit secular equations for a number of pre-viously investigated Rayleigh wave problems whose dispersion equations were obtained only in the implicit form

1 Introduction

Elastic surface waves in isotropic elastic solids, discovered by Lord Rayleigh[1]more than 120 years ago, have been stud-ied 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 waves, their dispersion equations in the explicit form are very signiﬁcant in practical applications They can be used 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 Thus, the secular equations in the explicit form are always the main purpose of investigations related to Rayleigh waves 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 Bromwhich[3], Love[4], Biot[5], Gilbert[6], De and Sengupta[7], Dey and Sengupta[8], Datta[9], Das et al.[10], Abd-Alla and Ahmed[12] Bromwhich[3], Gilbert[6]and Love

[4]treated the force of gravity as a type of body force, while Biot[5]and the other authors, following him, assumed that the force of gravity to create a type of initial stress of hydrostatic nature Bromwhich[3]assumed that the material is incom-pressible for the sake of simplicity Love[4]ﬁnished Bromwhich’s investigation by considering the compressible case Biot

[5] also took the assumption of incompressibility in his study Gilbert [6] used the Bromwich’s secular equation to

* Corresponding author Tel.: +84 4 35532164; fax: +84 4 38588817.

E-mail address: pcvinh@vnu.edu.vn (P.C Vinh).

Contents lists available atScienceDirect

Wave Motion

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 / w a v e m o t i

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investigate the inﬂuence of gravity on the Rayleigh wave The material was assumed to be isotropic in the investigations[3– 7,9,10], transversely isotropic in[8] Most of the investigations supposed that the material is homogeneous However, be-cause any realistic model of the earth must take into account continuous changes in the vertical direction of the elastic prop-erties of the material, the problem was extended to the non-homogeneous case by Das et al.[10] Das et al assumed that the material is isotropic and they obtained the implicit secular equation Recently, Abd-Alla and Ahmed[12]extended the prob-lem to the orthotropic case Abd-Alla and Ahmed[12]employed two displacement potentials for expressing the solution, and they have derived the secular equation of the wave in the implicit form

In the present work we analyze the orthotropic case and using an appropriate representation of the solution we derive an explicit form of the secular equation, which also provides the explicit secular equations for a number of previous investiga-tions related to Rayleigh waves under the gravity, where only the implicit dispersion equainvestiga-tions were obtained

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 the parameters characterizing the material and external effects (see for example[13–15]) Otherwise we call it an im-plicit secular equation

2 Basic equations

Consider a non-homogeneous orthotropic elastic body occupying the half-space x3P0 subject to the gravity 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 equa-tions[12]:

r11¼ c11u1;1þ c13u3;3

r33¼ c13u1;1þ c33u3;3

r13¼ c55ðu1;3þ u3;1Þ

ð2Þ

where cijare the material constants

Equations of motion are[12]:

r11;1þr13;3þqgu3;1¼q€u1

in whichqis the mass density of the medium, and g is the acceleration due to gravity, a superposed dot denotes differen-tiation with respect to t, commas indicate differendifferen-tiation with respect to spatial variables xi

In matrix (operator) form, the Eqs.(2) and (3)are written as follows:

u0

r0

¼ N u

r

ð4Þ

where: u ¼ ½u1;u3T;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

ðc13=c33Þ@1 0

; N2¼ 1=c55 0

0 1=c33

K ¼ q@2t þ ½ðc2

13 c11c33Þ=c33@21 qg@1

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

t ¼ @2=ð@t2Þ In addition to Eq.(4), the displacement vector u and the traction vectorrare required to satisfy the decay condition:

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

3 Secular equation

Assume that the half-space x3P0 is made of a material with an exponential depth proﬁle:

where c0;q0;m are constants

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Now we consider the propagation of a Rayleigh wave, travelling with velocity c and wave number k in the x1-direction The components u1;u3of the displacement vector andr13;r33of the traction vector at the planes x3¼ const are found in the form (see[16]):

fuj;rj3gðx1;x3;tÞ ¼ femx 3Ujðx3Þ; iemx 3Rjðx3Þgeikðx 1 ctÞ; j ¼ 1; 3 ð9Þ

Substituting(9)into(4)yields:

U0

R0

¼ iM U

R

ð10Þ

where: U ¼ ½U1U3T;R¼ ½R1R3T, and:

M ¼ M1 M2

Q M3

; M1¼ iðm=kÞ 1

D iðm=kÞ

; M2¼ ð1=kÞ 1=c

0

55 0

0 1=c0

33

Q ¼ kðX dÞ ia

ia kX

; M3¼ iðm=kÞ D

1 iðm=kÞ

ð11Þ

here d ¼ c0

11 ðc0

13Þ2Þ=c0

33,D¼ c0

13=c0

33, a ¼q0g, X ¼q0c2, the prime indicates the derivative with respect to y ¼ kx3

It is not difﬁcult to verify, by eliminating U from(10), that the traction vectorRðyÞ is the solution of the equation:

where the matricesa;b;care given by:

a¼ Q1¼ 1

kd

X ia=k ia=k ðX dÞ

b¼ M1Q1þ Q1M3¼ 1

kd

0 g1

g1 0

c¼ M1Q1M3 M2¼ 1

kd

h0 img0=k þ iag2=k

img0=k iag2=k h1

ð15Þ

in which

g0¼ d ð1 DÞX; g2¼Dm

2

k2

h0¼m

2X

k2 þ ðX dÞ

d

c0 55

þ2ma

k2

h1¼m

2

k2ðX dÞ þD

2X d

c0 33

2maD

k2

ð16Þ

Now we seek the solution of Eq.(12)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(6) Substituting(17)into(12)leads to:

AsR0is a non-zero vector, the determinant of system(19)must vanish This provides the equation for determining p, namely:

p4

where

S ¼ 2D d

c0

55

þ 1

c0 33

þ 1

c0 55

X 2m 2

k2

P ¼ðc

0

11 XÞðc0

55 XÞ

c0

33c0

55

m 2

k2

1

c0 33

þ 1

c0 55

X 2D d

c0 55

þm 4

k4

a2þ 2amðc0

55 c0

13Þ

k2c0

33c0 55

ð21Þ

It follows from(20)that:

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where p2;p2are two roots of the quadratic Eq.(20)for p2 It is not difﬁcult to demonstrate that vectorR0¼ ½A BT, the solution

of(19), is given by:

A ¼ia

kp

2þ g1p img0

k iag2=k

B ¼ Xp2

Let p1, p2be the two roots of(20)satisfying the condition(18) Then the general solution of Eq.(12)is:

RðyÞ ¼c1 A1

B1

eip1y

þc2 A2

B2

eip2y

ð24Þ

where Ak;Bkðk ¼ 1; 2Þ are given by(23)in which p is replaced by pk,c1;c2(c2þc2–0) are constants to be determined from the boundary condition(7)that reads:

Making use of(24)into(25)yields two equations forc1;c2, namely:

ðia=kÞp2þ g1p1 img0=k iag2=k ðia=kÞp2þ g1p2 img0=k iag2=k

Xp2

þ 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 results in:

g1Xp1p2 ði=kÞ½mg0X þ ag2X þ ah0ðp1þ p2Þ g1h0¼ 0 ð27Þ

Suppose that p1;p2are the two roots of(20)satisfying condition(18), we shall show that the following relations hold:

P > 0; 2 ffiffiffi

P

p

S > 0; p1p2¼ ffiffiffi

P

p

; p1þ p2¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

S

q

ð28Þ

where S; P are deﬁned by(21)

Indeed, from(18)it follows that ImðpiÞ > 0 If the discriminant D of the quadratic Equation(20)for p2is non-negative, then its two roots must be negative in order that ImðpiÞ > 0 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, Eq.(20)for p2has two conjugate complex roots, again P ¼ p2p2>0 and

in order to ensure ImðpiÞ > 0, it must be that p1¼ t þ ir; p2¼ t þ 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 positive imaginary part, hence

ðp1þ p2Þ2is a negative number Therefore, with the help of(22), it follows that the relations(28)are true It is noted that the result(28)3, (28)4were obtained in[16], but without showing that P > 0, 2 ffiffiffi

P

p

S > 0

Taking into account(28), (27)becomes:

g1ðX ffiffiffi

P

p

þ h0Þ ðm=kÞg0X

2 ffiffiffi P

p

S

q

ða=kÞðh0þ g2XÞ

2 ffiffiffi P

p

S

q

Since gi, h0, P; S are explicitly expressed in terms of X ¼q0c2, Eq.(29)is fully explicit in terms of the Rayleigh wave speed

Eq.(29)is the (exact) secular equation, in the explicit form, of Rayleigh waves in non-homogeneous orthotropic elastic media under the inﬂuence of gravity, where g0, g1, g2, h0, S, P are deﬁned by(16)1, (14)2, (16)2, (16)3, (21)1, (21)2, respectively Remark 1

(i) One can obtain the quadratic Equation(20)for p2by another way that has been used by Kulkarni and Achenbach[17] First, by substituting(2)into(3)and taking into account the assumption(8), an equation for u is derived and we call it the equation for the displacement vector Then, substituting u, deﬁned as uj¼ Ajei pyeikðx 1 ctÞðj ¼ 1; 3), into this equa-tion yields a homogeneous system of two linear equaequa-tions for constants Aj The vanishing of the determinant of the system leads to a fully quartic equation for p that is quite complicated as remarked by Kulkarni and Achenbach

[17] By a (linear) transformation that cancels the cubic term of the equation, the authors obtain the quadratic Equa-tion(20)for p2 This has also been pointed out in[16] It should be noted that, the quantitiescj(j ¼ 1; 2) of the paper

[17]are not always real number, they can be complex numbers Thus, they are required to have positive real parts, rather than to be positive numbers, in order that the decay condition is satisﬁed Also note that the results(28)1,2(with

a ¼ 0) ensure that the expressions in the square roots of formula(29)in[17]have positive values

(ii) Substituting u deﬁned by(9), where Uj¼ eAjeipyðeAjbeing constantÞ, into the equation for the displacement vector will also yields immediately the bi-quadratic Equation(20) However, the use of the equation for the traction vector(12)is better than that of the equation for the displacement vector, since the boundary condition is expressed in terms of the traction vector The secular equation is derived more quickly if we use the equation for the traction vector This can be seen by comparing the secular Eq.(27)in which a ¼ 0 with the corresponding Equation(39)in[17]

(iii) The representation of solution(9)indicates clearly the decay behaviours of the displacement vector u and the traction vectorr Unlike the homogeneous case, they are quite different from each other

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(iv) One can arrive at the explicit secular equation of the wave by following the procedure carried out by Kulkarni and Achenbach[17]

4 Explicit secular equations of Rayleigh waves under the effect of gravity in special cases

4.1 Case of a non-homogeneous isotropic elastic half-space under gravity

This problem was consider Das et al.[10], but the authors only have derived the implicit form of the secular equation From(29)and(14)2, (16), (21), and taking into account that the material is isotropic, i.e

c0

11¼ c0

33¼ k0þ 2l0; c0

13¼ k0; c0

55¼l0

here k0;l0are constants, the explicit secular equation for this case is:

2ð1 cÞð2 xÞ x ffiffiffi

P

p

þ ðx þ 4c 4Þð1 xÞ þ m2x=k2þ 2m=k þ2

2mxð2 2 ccxÞ=k þx2þ 6ðx 1Þ þ 4ðc 1Þ þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

S

q

¼ 0

ð30Þ

where x ¼ X=l0¼ c2=c0 ;c¼ c0 =c0 ;¼ g=kc022;c0 ¼ ðk0þ 2l0Þ=q0;c0 ¼l0=q0, and

S ¼ ð1 þcÞx 2 2m2=k2

P ¼ ð1 xÞð1 cxÞ m2½ð1 þcÞx þ 2ð4c 3Þ=k2þ m4=k4þ 2mð1 3cÞ=k c 2 ð31Þ

4.2 Case of a non-homogeneous orthotropic elastic half-space without gravity

When the gravity is absent, i.e a ¼ 0, Eq.(29)becomes:

g1ðX ffiffiffi

P

p

þ h0Þ ðm=kÞg0X

2 ffiffiffi P

p

S

q

in which

g0¼ d ð1 DÞX; g1¼ d ð1 þDÞX

h0¼ ðX dÞð1 X=c055Þ þ m2X=k2

S ¼ 2D d

c0

55

þ 1

c0 33

þ 1

c0 55

X 2m 2

k2

P ¼ðc

0

11 XÞðc0

55 XÞ

c0

33c0

55

m 2

k2

1

c0 33

þ 1

c0 55

X 2D d

c0 55

þm 4

Eq.(32)coincides with the secular equation derived recently by Destrade[16] Note that in[16]it is not shown P > 0 and

2 ffiffiffi

P

p

S > 0

4.3 Case of a non-homogeneous transversely isotropic elastic half-space without gravity

This problem was considered by Pal and Acharya[11], but only the implicit form of the secular equation has been derived

in their work In their notations, the explicit secular equation for the problem is Eq.(32), where m is replaced bym=2, the functions g0;g1;h0;S; P (in terms of X) are given by(33)in which c0

11;c0

13;c0

33;c0

55;q0are replaced by A1;F1;C1;L1;q1, respec-tively Here A1;F1;C1;L1are the material constants (see[18])

4.4 Case of a homogeneous orthotropic elastic half-space under gravity

In this case m ¼ 0, Eq.(29)thus is simpliﬁed to:

g1 X ffiffiffi

P

p

þ h0

ða=kÞ h½ 0þDX

2 ffiffiffi P

p

S

q

in which:

S ¼ 2Dþ ðX dÞ=c55þ X=c33; P ¼ c11

c33

X

c33

1 X

c55

a 2

k2c33c55

ð35Þ

g1¼ d ð1 þDÞX; h0¼ ðX dÞð1 X=c55Þ þ a2=ðk2c55Þ ð36Þ

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Eq.(34)is the explicit secular equation of Rayleigh waves in orthotropic elastic media under the effect of gravity In this case one can show that the Rayleigh wave velocity is limited by:

4.5 Case of a homogeneous transversely isotropic elastic half-space under gravity

This problem was considered by Dey and Sengupta[8], and only the implicit form of the secular equation was derived In their notations, the explicit secular equation for this problem is:

g1 X ffiffiffi

P

p

þ h0

ða=kÞ h½ 0þDX

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 ð40Þ

here A; C; F; L are the material constants It is noted that Eq.(38)is Eq.(34)in which c11;c33;c13;c55are replaced by A; C; F; L=2, respectively The Rayleigh wave velocity is also subjected to the limitation(37)in which c11;c55are, respectively, replaced by A; L=2

4.6 Case of a homogeneous isotropic elastic half-space under gravity

By putting m ¼ 0 in Eq.(30)and replacing c0;q0by cij;qwe obtain the explicit secular equation of Rayleigh waves in homogeneous isotropic elastic half-space under gravity, namely:

2ð1 cÞð2 xÞ x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 xÞð1 cxÞ c22

q

þ ðx þ 4c 4Þð1 xÞ þ2

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

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 xÞð1 cxÞ c22

q

þ 2 ð1 þcÞx

r

¼ 0

ð41Þ

where x ¼ c2=c2;c¼ c2=c2;¼ g=kc22;c2¼ ðk þ 2lÞ=q;c2¼l=qand

here k;lare Lame’s constants The Eq.(41)provides the exact secular equation in the explicit form for the investigations by

De and Sengupta[7]and Datta[9]

4.7 Case of a homogeneous orthotropic elastic half-space without gravity

When the material is homogeneous and the gravity is absent we have: m ¼ a ¼ 0 Then Eq.(29)is simpliﬁed to (see also

[19,20]):

ðc55 XÞ½c213 c33ðc11 XÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffic

33c55 p

X ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðc11 XÞðc55 XÞ

p

In this case we can obtain the explicit formula for the Rayleigh wave velocity (see[20]), namely:

qc2=c55¼ ffiffiffiffiffi

b1

p

b2b3 ð ffiffiffiffiffi

b1

p

=3Þðb2b3þ 2Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p 3 q

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R ffiffiffiffi D p 3 q

ð44Þ

where b1¼ c0

33=c0

11;b2¼ d=c0

11;b3¼ c0

11=c0

55;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

ð45Þ

in which

hðb1;b2;b3Þ ¼ ffiffiffi

b

p

1½2b1ð1 b2b3Þ3þ 9ð3b2 b2b3 2Þ ð46Þ

and the roots in(44)taking their principal values It is clear that the speed of Raleigh waves in homogeneous orthotropic elastic solids is a continuous function of three dimensionless parameters b1;b2;b3

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5 A numerical example

As an example, we consider a non-homogeneous orthotropic elastic half-space whose elastic constants and mass density are deﬁned by(8), in which m 6 0 and (see[16]):

c0

11=q0

¼ 9 ðkm=sÞ2; c0

13=q0

¼ 3:6 ðkm=sÞ2

c0

33=q0

¼ 9:89 ðkm=sÞ2; c0

55=q0

Taking into account(47), it is easy to numerically solve the secular Equation(29), and the dependence of squared dimension-less Rayleigh wave velocity x ¼q0c2=c0

55on m=k and¼q0g=kc055are shown inFigs 1 and 2 It appears that the inﬂuence of the inhomogeneity on the Rayleigh wave velocity is stronger than that of the gravity

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0c

2/c

0 55

−m/k

Fig 1 Dependence of squared dimensionless Rayleigh wave velocity x ¼q0 c 2 =c 0

55 on the parameter m=k with different values of:¼ 0 (solid line),

¼ 0:3 (dashed line),¼ 0:5 (dash-dot line),¼ 0:8 (dotted line).

0.4 0.5 0.6 0.7 0.8 0.9 1

0c

2/c

0 55

ε

Fig 2 Dependence of squared dimensionless Rayleigh wave velocity x ¼q0 c 2 =c 0

55 on the parameterwith different values of m=k : m=k ¼ 0 (solid line),

m=k ¼ 0:1 (dotted line), m=k ¼ 0:2 (dashed line), m=k ¼ 0:4 (dash-dot line).

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6 Conclusions

The problem of the Rayleigh waves in a non-homogeneous orthotropic elastic medium under the inﬂuence of gravity is considered and the secular equation of the wave motion in the explicit form is derived Furthermore, by considering various special cases, the explicit secular equations is obtained for the Rayleigh wave motions under the effect of inhomogeneity and/or gravity, corresponding to a number of previous studies in which only the implicit dispersion equations were given The explicit secular equations derived in this work may be useful in practical applications

Acknowledgements

The authors wish to thank Prof J.D Achenbach for helpful discussions They also would like to give thanks to an anon-ymous reviewer for recommending the paper by F Gilbert The first author undertook this work during his visit to the OGS (Istituto Nazionale di Oceanografia e Geofisica Sperimentale) with the support of the ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy

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