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

Acta Mech 223, 1537–1544 (2012) DOI 10.1007/s00707-012-0664-6 Pham Chi Vinh · Nguyen Thi Khanh Linh New results on Rayleigh waves in incompressible elastic media subjected to gravity Received: 11 October 2011 / Revised: 20 March 2012 / Published online: 19 May 2012 © Springer-Verlag 2012 Abstract In this paper, the following new results related to Rayleigh waves in incompressible elastic media under the influence of gravity are presented: (i) the exact formulas for the velocity of Rayleigh waves propagating along the free-surface of an incompressible isotropic elastic half-space under the gravity are derived, and (ii) two approximate formulas for the velocity of the Rayleigh waves are established and it is shown that their accuracy is very high To derive the exact formulas, we use the theory of cubic equation, and to establish the approximate formulas, we employ the best approximate second-order polynomials of the cubic power The obtained formulas are powerful tools for analyzing the effect of gravity on the propagation of Rayleigh waves and for solving the inverse problem Introduction Elastic surface waves in isotropic elastic solids, discovered by Rayleigh [1] more than 120 years ago, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science It would not be far-fetched to say that Rayleigh’s study of surface waves in an elastic half-space has had fundamental and far-reaching effects upon modern engineering, stretching from mobile phones to the study of earthquakes, as addressed by Adams et al [2] The problem of propagation of Rayleigh waves under the effect of gravity is an important problem in seismology and geophysics, and many investigations on this topic have been carried out, see for example [3–19] In most of these investigations, the material is assumed to be compressible, and among them, the studies by Vinh and Seriani [18] and Vinh [19] provide the explicit secular equations, and the remaining investigations provide the equations in an implicit form Bromwich [3], Biot [6] and Kuipers [12] assumed incompressibility in their studies for the sake of simplicity and derived the explicit secular equation of Rayleigh waves in an incompressible isotropic elastic half-space under the effect of gravity by employing different approaches The Rayleigh wave velocity is a physical quantity of great importance It is discussed in almost every text book and monograph on the subject of surface acoustic waves in solids Further, the Rayleigh wave velocity also appears in Green’s function of many elastodynamic problems for a half-space, and explicit formulas for the Rayleigh wave velocity are thus clearly of practical as well as theoretical interest It is worth noting that although the existence of Rayleigh waves has been discovered by Rayleigh [1] more than 120 years ago, the P C Vinh (B) Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam E-mail: pcvinh@vnu.edu.vn Tel.: +84-4-5532164 Fax: +84-4-8588817 N T K Linh Department of Engineering Mechanics, Water Resources University of Vietnam, 175 Tay Son Str., Hanoi, Vietnam 1538 P C Vinh, N T K Linh exact formulas for the velocity of Rayleigh waves in various elastic media were found only recently, see for example [20,21] and the references herein In this paper, we obtain the exact formulas for the velocity of Rayleigh waves in an incompressible isotropic elastic half-space by applying the theory of cubic equation Two approximate formulas for the Rayleigh wave velocity are established by employing the best approximate second-order polynomials of the cubic power It is shown that they are very good approximations The necessary and sufficient conditions for the existence and uniqueness of Rayleigh waves are also provided The obtained formulas might be powerful tools for evaluating the effect of gravity on Rayleigh waves and for solving inverse problems Secular equation In this section, we briefly recall the derivation of the secular equation of Rayleigh waves in an incompressible elastic half-space under gravity We follow Biot’s approach [6] but not employ the displacement potentials in the following analysis An incompressible isotropic elastic half-space subjected to gravity occupies the half-space x3 ≥ We are concerned with the plane strain u k = u k (x1 , x3 , t), k = 1, 3, u ≡ 0, (1) where u k are the displacement components, and t is time The equations of motion are then [6] (s11 + ρgu ),1 + s13,3 = uă , s13,1 + (s33 + gu ),3 = uă , (2) where ρ is the mass density of the medium, g is the acceleration due to gravity, si j are the stress components, commas indicate the differentiation with respect to spatial variables xk , and a superposed dot denotes the differentiation with respect to t The stress–strain relation is of the form [6] s11 − s = 2μu 1,1 , s33 − s = 2μu 3,3 , s13 = μ(u 1,3 + u 3,1 ), (3) where s = (s11 + s33 )/2, and μ is the Lame constant In addition to Eqs (2) and (3), the traction-free condition on the planar surface x3 = is s13 = s33 = on x3 = 0, (4) u = u = s11 = s13 = s33 = at x3 = +∞ (5) and the decay conditions require that Introducing the notion of fictitious stresses [6] s11 = s11 + ρgu , s33 = s33 + ρgu , s13 = s13 , (6) s11,1 + s13,3 = uă1 , s13,1 + s33,3 = uă3 , (7) s11 s = 2u 1,1 , s33 − s = 2μu 3,3 , s13 = μ(u 1,3 + u 3,1 ), (8) Eqs (2)–(5) become s13 = 0, s33 = ρgu on x3 = 0, u = u = s11 = s13 = s33 = at x3 = +∞, (9) (10) where s = (s11 + s33 )/2 = s + ρgu Substituting Eqs (8) into Eqs (7), and taking into account the incompressibility condition u 1,1 + u 3,3 = 0, (11) we obtain μ(u 1,11 + u 1,33 ) + s,1 = uă1 , (u 3,11 + u 3,33 ) + s,3 = uă3 (12) New results on Rayleigh waves 1539 Next, we consider propagation of a Rayleigh wave, traveling with the velocity c and the wave number k(> 0) in the x1 -direction and decaying in the x3 -direction Then, u , u , s are sought in the form u = A1 e−λkx3 eik(x1 −ct) , u = A3 e−λkx3 eik(x1 −ct) , s = A e−λkx3 eik(x1 −ct) , (13) where A1 , A3 and A are nonzero constants, and λ must have a positive real part to satisfy the decay condition (10) Substituting expressions (13) into Eqs (11) and (12), one obtains a homogeneous linear system of equations for A1 , A3 and A The vanishing of the determinant of the system provides an equation for λ (λ2 − 1)[λ2 − (1 − x)] = 0, where x = c2 /c22 , c22 = μ/ρ The Since Re(λ2 ) > 0, it follows that quadratic equation (14) for λ2 has two real roots (14) λ21 = and λ22 = − x < x < (15) √ Suppose that inequalities (15) hold, then λ1 = and λ2 = − x, and the general solution of the system of equations (11)–(12) that satisfies the decay condition (10) is then u = −(ikγ1 e−kx3 + ikλ2 γ2 e−λ2 kx3 )eik(x1 −ct) , u = (kγ1 e−kx3 + kγ2 e−λ2 kx3 )eik(x1 −ct) , 2 −kx3 ik(x1 −ct) s = γ1 ρk c e e (16) , where γ1 , γ2 are constants to be determined Substituting the relations (16) into the boundary conditions (9), we obtain 2γ1 + (2 − x)γ2 = 0, √ [(x − 2) − ρg/(kμ)]γ1 − [2 − x + ρg/(kμ)]γ2 = The vanishing of the determinant of the system of Eq (17) yields the secular equation √ (2 − x)2 − − x − x = 0, (17) (18) where = ρg/(kμ) ≥ Equation (18) is the secular equation of Rayleigh waves in an incompressible isotropic elastic half-space under gravity that was derived by Bromwich [3], Biot [5,6] and Kuipers [12] by applying different approaches Exact formulas for the velocity of Rayleigh waves Let x ∈ (0, 1) and define the function φ(x) as follows: √ (2 − x)2 − − x φ(x) = , x ∈ (0, 1) x (19) Then, Eq (18) can be written as φ(x) = , x ∈ (0, 1) It is readily to see that √ √ x − x φ (x) = (2 − x)[2 − − x(2 + x)] > ∀ x ∈ (0, 1) (20) (21) Therefore, φ (x) > ∀ x ∈ (0, 1), that is, φ(x) is strictly monotonously increasing in the interval (0, 1) Recalling Eq (20), using φ(+0) = −2 and φ(1) = and observing that the function φ(x) is strictly monotonously increasing in the interval (0, 1) (Fig 1), we formulate Proposition Proposition Let ≥ 0, then: (i) If ≤ < 1, then Eq (20) [also Eq (18)] has a unique real solution in the interval (0, 1), denoted by xr ( ) 1540 P C Vinh, N T K Linh 0.5 x0 φ(x) −0.5 −1 −1.5 −2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Fig Plot of the function φ(x)given in Eq (19) (ii) If ≥ 1, Eq (20) [also Eq (18)] has no real solution in the interval (0, 1) (iii) The function xr ( ) is strictly increasingly monotonous in the interval [0, 1) from x0 = xr (0) to (but not equal to 1) (Fig 1), where xr (0) is the velocity of Rayleigh waves in incompressible isotropic elastic half-spaces without the effect of gravity Note that the approximate value of x0 is 0.9126 (see, e.g., [22]) and its exact value is given by (see [20,23]) x0 = − η02 , (22) where η0 = 26 + 27 11 1/3 − 26 + 27 11 −1/3 − (23) It is not difficult to verify that for ∈ [0, 1), the function ϕ(x) = (2 − x)2 − x is strictly monotonously decreasing in the interval (0, 1), consequently ϕ(x) > ϕ(1) = − > ∀ x ∈ (0, 1) (24) Using the relations (24), one can show that, in the interval (0, 1), the secular equation (18) is equivalent to F(x) ≡ x + a2 x + a1 x + a0 = 0, (25) where a0 = −8(2 + ), a1 = 24 + + , a2 = −2(4 + ) (26) Therefore, from Proposition follows Proposition Proposition Let ≥ 0, then: (i) If ≤ < 1, Eq (25) has a unique real solution in the interval (0, 1), namely xr (ii) For ≥ 1, Eq (25) has no real root in the interval (0, 1) Introducing the new variable z defined by z = x + a2 , (27) z − 3q z + r = 0, (28) Equation (25) becomes where New results on Rayleigh waves 1541 q = (a22 − 3a1 )/9, r = 2a23 − 9a1 a2 + 27a0 /27 (29) Using the theory of cubic equation, three roots of Eq (28) are given by Cardan’s formula [24]: z = S + T, z = − (S + T ) + z = − (S + T ) − √ i 3(S − T ), √ i 3(S − T ), (30) where i = −1 and S= R+ √ D, T = R = − r, D = R2 + Q3, √ R− D, (31) Q = −q Remark The nature of the three roots of Eq (28) depends on the sign of its discriminant D, in particular: If D > 0, then Eq (28) has one real root and two complex conjugate roots; if D = 0, Eq (28) has three real roots, at least two of which are equal; if D < 0, then Eq (28) has three real distinct roots From Eqs (29) and (31), it follows that R = 9a1 a2 − 27a0 − 2a23 /54, (32) D = 4a0 a23 − a12 a22 − 18a0 a1 a2 + 27a02 + 4a13 /108 Using the relations (26), the first of Eqs (29) and (32) can be written as q2 = ( R = −( + − 8)/9, + 12 + 12 + 136)/27, D = 16( + 11)( (33) + 4)/27 It is clear from the third of Eqs (33) that D > 0∀ ∈ [0, 1) Thus, recalling Remark 1, Eq (28) has only one real root, denoted by zr , and with the aid of Eqs (30), it is given by zr = R+ √ D+ R− √ D (34) From Eqs (27) and (34), it follows that xr = 2(4 + ) − 3 16( + 11)( + 4)/27 + ( 8−8 − + 16( + 11)( + 4)/27 + ( + 12 + 12 + 136)/27 + 12 + 136)/27 + 12 , ∈ [0, 1) (35) The formula (35) is the exact expression for the velocity of Rayleigh waves in incompressible isotropic elastic half-spaces under the effect of gravity Figure shows its dependence on the gravity parameter = g/(kc22 ) One can see again from Fig that the Rayleigh wave velocity is a strictly monotonously increasing function of ∈ [0, 1) From the formula (35), we may derive an alternative exact expression for x0 17 x0 = − + 27 11 1/3 + 17 + 27 11 −1/3 (36) Using the transformation t= √ − x, < t ≤ t0 = − x0 ≈ 0.0.2956 (37) 1542 P C Vinh, N T K Linh 1.01 0.99 0.97 0.96 r x =c2/c2 0.98 0.95 0.94 0.93 0.92 0.91 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ε Fig Plot of the dimensionless Rayleigh wave velocity xr ( ) versus the gravity parameter ∈ (0, 1), = g/(kc22 ) Equation (18) takes the form t + (2 + )t − 4t + − = (38) It is readily to see that, in the interval (0, 1), Eq (38) is equivalent to t + t + (3 + )t − + = (39) We can thus formulate Proposition Proposition Suppose ≥ 0, then: (i) Equation (39) has a unique real solution in the interval (0, 1), denoted by tr , if and only if (ii) For ∈ [0, 1), tr is given by tr = 26 − + 27 ( + 11)( 27 + 4) 8+3 − 26 − 9 + 27 ( + 11)( 27 + 4) < 1 − ; (40) thus, the squared dimensionless Rayleigh wave velocity is ⎛ ⎜ ⎜ 26 − xr = − ⎜ + ⎜ 27 ⎝ ⎞2 ( + 11)( 27 + 4) 8+3 − 26 − 9 + 27 ( + 11)( 27 + 4) ⎟ 1⎟ − ⎟ 3⎟ ⎠ (41) We summarize the obtained results in Theorem Theorem Let ≥ 0, then: (i) A Rayleigh wave exists if and only if ≤ < (ii) If a Rayleigh wave exists, then it is unique, and its squared dimensionless velocity xr ( ) is given by Eqs (35) or (41) (iii) The squared dimensionless Rayleigh wave velocity xr ( ) is a strictly monotonously increasing function in the interval [0, 1), from x0 , given in Eqs (22) or (36), to (but not equal to 1) New results on Rayleigh waves 1543 Approximate formulas for the velocity of Rayleigh waves In this section, we set down approximate formulas for the velocity of Rayleigh waves by applying the best approximate second-order polynomials of the cubic power Since these approximate formulas are simpler than the exact ones and have a very high accuracy, they are useful in applications Following Vinh and Malischewsky [25], one can see that in the interval [x0 , 1], the best approximate second-order polynomial of x in the sense of least squares is 2.8689 x − 2.7424 x + 0.8734 (42) Introducing expression (42) into Eq (25), we obtain a quadratic equation (a2 + 2.8689)x − (2.7424 − a1 )x + a0 + 0.8734 = 0, whose solution corresponding to the Rayleigh wave is √ B − B − AC x1 = , 2A (43) (44) where A = −(5.1311 + ), B = −(21.2576 + + ), C = −(15.1266 + ) (45) Similarly, the best approximate second-order polynomial of t in the interval [0, t0 ] in the sense of least squares is 0.4434 t − 0.0524276 t + 0.00129147 (46) Introducing the expression (46) into Eq (39), we obtain a quadratic equation for t 1.4434t + (2.9475724 + )t − 0.99870853 + = 0, (47) whose solution corresponding to the Rayleigh wave is √ + 0.1215448 + 14.4543266 −(2.9475724 + ) + tr = 2.8868 (48) The squared dimensionless velocity is given by x2 = − tr2 Figure shows the dependence on the gravity parameter of the exact velocity xr ( ) and the approximate velocities x1 ( ) and x2 ( ) From Fig 3, one can see that the plots of xr ( ), x1 ( ) and x2 ( ) totally coincide with each other This means that the obtained approximate formulas have a very high accuracy 1.01 0.99 0.96 r x,x ,x 0.97 0.98 0.95 0.94 0.93 0.92 0.91 0.1 0.2 0.3 0.4 0.5 ε 0.6 0.7 0.8 0.9 Fig The plots of the exact velocity xr ( ) and the approximate velocities x1 ( ) and x2 ( ) They totally coincide with each other 1544 P C Vinh, N T K Linh Conclusions In this paper, the exact and highly accurate approximate formulas for the velocity of Rayleigh waves in an incompressible isotropic elastic half-space under gravity are derived These formulas are useful tools for evaluating the effect of gravity on propagation of Rayleigh waves and for solving the inverse problem as well Acknowledgments The work was supported by the Vietnam National 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F.: Elastic Waves in Layered Media McGraw-Hill Book Comp., New York (1957) 23 Ogden, R.W., Vinh, P.C.: On Rayleigh waves in incompressible orthotropic elastic solids J Acoust Soc Am 115, 530–533 (2004) 24 Cowles, W.H., Thompson, J.E.: Algebra D Van Nostrand Company, New York (1947) 25 Vinh, P.C., Malischewsky, P.: An approach for obtaining approximate formulas for the Rayleigh wave velocity Wave Motion 44, 549–562 (2007) ... that is, φ(x) is strictly monotonously increasing in the interval (0, 1) Recalling Eq (20), using φ(+0) = −2 and φ(1) = and observing that the function φ(x) is strictly monotonously increasing... dimensionless Rayleigh wave velocity xr ( ) is a strictly monotonously increasing function in the interval [0, 1), from x0 , given in Eqs (22) or (36), to (but not equal to 1) New results on Rayleigh. .. obtained formulas might be powerful tools for evaluating the effect of gravity on Rayleigh waves and for solving inverse problems Secular equation In this section, we briefly recall the derivation