Rayleigh waves with impedance boundary condition formula for the velocity, existence and uniqueness

Rayleigh waves with impedance boundarycondition: Formula for the velocity, Existence and Uniqueness Pham Chi Vinh ∗ and Nguyen Quynh Xuan Faculty of Mathematics, Mechanics and Informatic

Pham Chi Vinh, Nguyen Quynh Xuan

DOI: 10.1016/j.euromechsol.2016.09.011

Reference: EJMSOL 3355

To appear in: European Journal of Mechanics / A Solids

Received Date: 25 December 2015

Revised Date: 4 August 2016

Accepted Date: 14 September 2016

Please cite this article as: Vinh, P.C., Xuan, N.Q., Rayleigh waves with impedance boundary condition:

Formula for the velocity, existence and uniqueness, European Journal of Mechanics / A Solids (2016),

doi: 10.1016/j.euromechsol.2016.09.011.

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Rayleigh waves with impedance boundary

condition: Formula for the velocity, Existence and

Uniqueness

Pham Chi Vinh ∗ and Nguyen Quynh Xuan Faculty of Mathematics, Mechanics and Informatics

Hanoi University of Science

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

AbstractThe propagation of Rayleigh waves in an isotropic elastic half-space withimpedance boundary conditions was investigated recently by Godoy et al.[Wave Motion 49 (2012), 585-594] The authors have proved the existenceand uniqueness of the wave However, they were not successful in obtaining

an analytical exact formula for the wave velocity The main purpose of thispaper is to find such a formula By using the complex function method, ananalytical exact formula for the velocity of Rayleigh waves has been derived.Furthermore, from the obtained formula, the existence and uniqueness of thewave has been established easily

Key words: Rayleigh waves, Impedance boundary conditions, Method of complexfunction, Exact formula for the wave velocity

Although the existence and uniqueness theorems for the secular equations ofRayleigh waves were proved, they remained unsolved for more than 100 years because

of their complicated and transcendent nature, as mentioned in Voloshin (2010)

In 1995, the first formula for the Rayleigh wave speed in compressible isotropicelastic solids has been obtained by Rahman and Barber (1995) As this formula isdefined by two different expressions depending on the sign of the discriminant ofthe cubic Rayleigh equation, it gives a big inconvenience when applying it to inverseproblems

Employing Riemann problem theory, Nkemzi (1997) derived a formula for thevelocity of Rayleigh waves that is expressed as a continuous function of Poisson’sratio It is rather cumbersome (Destrade, 2003), and the final result as printed

in his paper is incorrect (Malischewsky, 2000) Malischewsky (2000) obtained aformula, given by one expression, for the speed of Rayleigh waves by using Cardan’s

Vinh and Ogden (2004a) gave a detailed derivation of this formula togetherwith an alternative formula by using the method of cubic equations Followingthis method, these authors derived the Rayleigh wave velocity formulas for theorthotropic materials (Ogden and Vinh, 2004; Vinh and Ogden, 2004b; Vinh andOgden, 2005), for the pre-stressed materials (Vinh, 2010; Vinh and Giang 2010;Vinh, 2011).

In all works mentioned above, it is assumed that the surface of half-spaces is free

of the traction, and the Rayleigh waves are called ”Rayleigh waves with traction-freecondition” As mentioned in Godoy et al (2012), in many fields of physics such asacoustics and electromagnetism, it is common to use impedance boundary condi-tions, that is, when a linear combination of the unknown function and their deriva-tives is prescribed on the boundary See, for examples, Antipov (2002), Zakharov(2006), Yla-Oijala and Jarvenppa (2006), Mathews and Jeans (2007), Castro andKapanadze (2008), Qin and Colton (2012) for the acoustics case and Senior (1960),Asghar and Zahid (1986), Stupfel and Poget (2011), Hiptmair et al (2014) for theelectromagnetism one, and the references therein The Rayleigh waves propagat-ing in half-spaces subjected to impedance boundary conditions are called ”Rayleigh

It should be noted that there are three kinds of Rayleigh waves, namely, subsonic,transonic and supersonic Rayleigh waves (see Lothe & Barnett, 1985) whose velocity

is smaller than, equal to and bigger than the the limiting velocity ˆv, respectively.For compressible isotropic half-spaces ˆv = c2, where c2 =pµ/ρ is the velocity of thetransverse wave Therefore, the velocity c of subsonic Rayleigh waves propagating

in these half-spaces satisfies 0 < c < c2 The Rayleigh waves mentioned above are

The main purpose of this paper is to find such a formula By using the complexfunction method, an analytical exact formula for the velocity of Rayleigh waves hasbeen derived Furthermore, based on the obtained formula it has been easily shownthat there always exists a unique Rayleigh wave.

In this section, we present briefly the derivation of secular equation of Rayleighwaves propagating in an compressible isotropic half-space subjected to impedanceboundary conditions For more details, the reader is referred to the Godoy et al.(2012), Malischewsky (1987)

Let us consider a compressible isotropic elastic half-space occupying the domain

x2 ≥ 0 We are interested in planar motion in the (x1x2)-plane with the displacement

components u1, u2, u3 such that:

ui = ui(x1, x2, t), i = 1, 2, u3 ≡ 0 (1)

σ12 = µ(u1,2+ u2,1), σ22= λ(u1,1+ u2,2) + 2µu2,2 (3)

Suppose that the surface x2 = 0 is subjected to the impedance boundary conditions

Note that both b1 and b2 are positive real numbers due to the fact: 0 < c < c2 < c1

Using (3) and (6) into the impedance boundary conditions (4) yields a system oftwo homogeneous linear equations for A1, A2/b2, namely:

con-Remark 1: If a Rayleigh wave exists, then Eq (9) has a solution xr so that

0 < xr < 1 and xr is the dimensionless squared velocity of the Rayleigh wave

Inversely, if Eq (9) has a solution xr lying in the interval (0, 1), then a Rayleigh

wave is possible

In the next section we will find an exact analytical formula of xr by employing

the complex function method The existence and uniqueness of Rayleigh waves will

be established in Section 4 by using the obtained formula

3.1 Complex form of secular equation

We introduce the transformation:

z + 1, √

z − 1 and p(2 − γ)z − γ are chosen as the principal branches ofthe corresponding square roots When z ∈ R, |z| > 1 Eq (13) coincides with Eq.(11), therefore Eq (13) is called the complex form of Eq (11) In order to find xr

we find a real solution zr of Eq (13) so that |zr| > 1

3.2 Properties of function F (z)

Denote L = L1 ∪ L2, L1 = [−1, γ/(2 − γ)], L2 = [γ/(2 − γ), 1], S = {z ∈ C, z /∈L} N (z0) = {z ∈ S : |z − z0| < ε}, ε is a sufficiently small positive number, z0 is

some point of the complex plane C If a function ϕ(z) is holomorphic in Ω ⊂ C wewrite ϕ(z) ∈ H(Ω) Note that from 0 < γ < 1 it follows 0 < γ/(2 − γ) < 1 Using(13) it is not difficult to prove that:

(f5) F (z) is continuous on L from the left and from the right (see Muskhelishvili,

1953) with the boundary values F+(t) (the right boundary value of F (z)), F−(t)

bounded in N (−1) [N (1)] and takes a defined value at z = −1 [z = 1].

It is noted that (γ3) comes from the fact (see Muskhelishvili, 1953):

Proposition 5: Y (z) is a second-order polynomial.

Proof: Properties (y1) and (y4) of the function Y (z) show that Y (z) is holomorphic

in the entire complex plane C, with the possible exception of points: z = −1 and z =

1 By (y3) these points are removable singularity points and it may be assumed that

the function Y (z) is holomorphic in the entire complex plane C (see Muskhelishvili,1963) Thus, by the generalized Liouville theorem (Muskhelishvili, 1963), Y (z) is apolynomial, and according to (y2), Y (z) is a second-order polynomial:

Y (z) = P2(z) := A2z2+ A1z + A0, A2 6= 0 (23)

3.5 Equation F (z) = 0 equivalent to a quadratic equation

Proposition 6: Equation F (z) = 0 ⇔ P2(z) = 0 in the domain S ∪ {−1} ∪ {1}

From (φ1) and (φ3) it follows that Φ(z) 6= 0 ∀ z ∈ S ∪ {−1} ∪ {1} Proposition 6 is

proved by this fact and the equality (24)

Remark 2:

(i) Equation F (z) = 0 has no solutions in the interval (−1, 1) due to the tinuity of F (z) in this interval, according to (f5) This means that all solutions of

discon-F (z) = 0 fall in the domain S ∪ {−1} ∪ {1}

(ii) As 0 < |Φ±(t)| < ∞ ∀ t ∈ (−1, 1), therefore by (i) and the equality (24) two

roots of the quadratic equation P2(z) = 0 also fall in the domain S ∪ {−1} ∪ {1}

(iii) According to Proposition 6, instead of finding the analytical solution of thetranscendent equation F (z) = 0 we look for the one of a much simpler equation,namely the quadratic equation P2(z) = 0, in the domain S ∪ {−1} ∪ {1}

Proposition 7: Equation F (z) = 0 has exactly two roots, namely z1 = −1 and

z2 = 1 − A1/A2

Proof:

- By (f4), z1 = −1 is a solution of the equation F (z) = 0

- From Proposition 6 and this fact it follows that z1 = −1 is also a root of the

equation P2(z) = 0 According to Vieta’s formulas, the second root of the quadratic

equation P2(z) = 0 is z2 = 1 − A1/A2 and it lies in the domain S ∪ {−1} ∪ {1} due

to Remark 2 (ii) Again according to Proposition 6, z2 is a solution of the equation

in which the functions θk(t) are determined as:

(i) For γ ∈ (0, 1) and δ > 0:

(31)

I0 is given by (34) in which the functions θk(t) are calculated by (28)-(31).

Remark 3: Since γ and δ are all real numbers, it implies from (37), (34) and (31) that the root z2 = 1 − A1/A2 of Eq F (z) = 0 is a real number

3.7 Formula for the wave velocity

Theorem 1: If a Rayleigh wave exists, its dimensionless velocity xr is given by:

xr= 1 + z2

in which z2 is expressed in terms of γ and δ as follows:

(i) For γ ∈ (0, 1) and δ ∈ R, δ > 0:

z2 = 8 + (δ − 6)

√

2 − γ8(2 − γ) − (9 + δ)√

2 − γ +

8π

2 1/2 -1.6085 0.1892 0.1892

0 1/2 1.8944 0.7639 0.7639-2 1/2 1.1239 0.9449 0.9449

3 2/3 -1.0980 0.0446 0.0446

0 2/3 5.0278 0.5994 0.5994-3 2/3 1.0956 0.9564 0.9564

Table 1: Some values of the Rayleigh wave velocity that are computed by using theformulas (38)-(42) (denoted by x(1)r ) and by directly solving the secular equation (9)

in the domain 0 < x < 1 (denoted by x(2)r ) They are the same

(10) Consequently, the (complex) equation F (z) = 0 has a real root zr: |zr| > 1

By Proposition 7, Eq F (z) = 0 has two solutions z1 = −1 and z2 = 1 − A1/A2 As

|z1| = 1 it implies that z2 is a real number and z2 = zr = wr Therefore xr is given

by (38) From (28)-(31), (34), (37) and z2 = 1 − A1/A2 we immediately arrive at

(39), (40) and (41) The proof of Theorem 1 is completed

Remark 4: The case δ = 0 corresponds to the Rayleigh waves with free boundary condition whose velocity formula has been obtained by Malischewsky(2000), Vinh& Ogden (2004a) Since these formulas are algebraic expressions of γ,they are much more convenient in use than the integral formula {(38), (41)}

traction-As a checking example, a number of numerical values of xr are calculated by

using the the formulas (38)-(42) (denoted by x(1)r ) and by directly solving the secular

equation (9) in the domain 0 < x < 1 (denoted by x(2)r ) It is seen from Table 1

that they are the same.

The existence and uniqueness of Rayleigh waves are stated by the following theorem.Theorem 2: Suppose γ ∈ (0, 1) and δ ∈ R, then:

(i) A Rayleigh wave is always possible

(ii) If a Rayleigh wave exists, then it is unique

Proof:

(i) From Remark 1 and Proposition 7 it implies that the necessary and sufficientconditions for a Rayleigh wave to exist are: z2 ∈ R and z2 ∈ [−1, 1]./

The fact z2 is a real number is already stated in Remark 3 As F (1) = 4 it

follows z2 6= 1 According to Remark 2 (ii) we have z2 ∈ (−1, 1)./

To finish the proof of the statement (i) we need prove that z2 6= −1 Suppose

z2 = −1(= z1) From (24), Proposition 6 and (γ3) it follows lim

z→−1

F (z)(z + 1)2 = m,

|m| < ∞ This leads to lim

x2 = ∞ that is easily proved by using (9)

(ii) Suppose that there exist two different Rayleigh waves with corresponding ties x(1)r , x(2)r (x(1)r 6= x(2)r ) Then x(1)r and x(2)r are two different roots of Eq f (x) = 0

veloci-and 0 < x(1)r , x(2)r < 1 according to Remark 1 Since the transformation (10) is a

1 − 1 mapping from 0 < x < 1 to |w| > 1, it follows that Eq F (w) = 0 has twodifferent (real) roots lying in the domain |w| > 1, so does Eq F (z) = 0 From

this fact, the Proposition 6 and F (−1) = 0 it implies that the quadratic equation

P (z) = 0 has three different roots It is impossible and the proof of the statement(ii) is completed

Remark 5: Since Eq (9) is a linear equation for δ, it gives a unique value of δfor a given x ∈ (0 1) From this fact and Theorem 2, it follows immediately that thedimensionless Rayleigh wave velocity x(δ) is a monotonic function of δ ∈ (−∞ +∞)

as proved by Godoy et al (2012)

In this paper, an exact analytical formula for the velocity of Rayleigh waves agating in a compressible isotropic half-space subjected to impedance boundaryconditions has been derived by using the complex function method Based on theobtained formula, the existence and uniqueness of Rayleigh waves have been estab-lished immediately Since the obtained formula is exact and totally explicit, it is oftheoretical as well as practical interest

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