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

Meccanica 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 propagation 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 secular 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 formula 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 secondorder approximate formula for the velocity is provided and an approximation, called global approximation, for it is derived by using the best approximate secondorder 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 Introduction Elastic surface waves in isotropic elastic solids, discovered 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 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 farreaching 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 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] Meccanica The propagation of Rayleigh waves in an incompressible isotropic elastic half-space underlying a nonviscous 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 dispersion equation of the wave by approximating directly the boundary conditions However, as illustrated below 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 equation, Bromwich derived a first-order approximate formula 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 general 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 indicate that it is not valid for all possible values of the Rayleigh wave velocity (ii) Based on the exact secular 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 exists 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 secular equation obtained by Bromwich is a special case (iv) For the case of thin layer and small effect of gravity, a second-order approximate formula for the velocity is created which recovers the first-order approximate formula obtained by Bromwich [3] (v) When only the layer being thin, a second-order approximate formula for the velocity is provided and an approximation, called global approximation, 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 researchers in various fields of science It is discussed Fig 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 Secular equation 2.1 Exact secular equation Consider an incompressible isotropic elastic halfspace x3 < that is overlaid with a layer of incompressible non-viscous water occupying the domain < x3 ≤ h (see Fig 1) The elastic half-space and the water layer is separated by the plane x3 = 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: uk = uk (x1 , x3 , t), p = p(x1 , x3 , t), k = 1, 3, u2 ≡ φ = φ(x1 , x3 , t) (1) where uk and p are respectively the displacement components and the hydrostatic pressure corresponding 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 u1 = uă , u1,1 + u3,3 = 0, p,3 + =0 u3 = uă (2) Meccanica where commas indicate differentiation with respect to spatial variables xk , a superposed dot denotes differentiation with respect to t, f = f,11 + f,33 , ρ and μ are the mass density and the Lame constant of the elastic solid Addition to Eqs (2) are required the boundary condition at x3 = h [3]: gφ,3 + ă = at x3 = h (3) the continuity conditions at x3 = [3]: φ,3 = u˙ at x3 = (4) μ(u1,3 + u3,1 ) = at x3 = (5) ˙ p + 2μu3,3 + g ρ − ρ u3 − ρ φ = at x3 = (6) and the decay condition at x3 = −∞: uk = (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 velocity 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 = Qekx3 exp(ikx1 + iωt) μk22 p,1 u1 = − + Aesx3 exp(ikx1 + iωt) μk2 p,3 u3 = − + Besx3 exp(ikx1 + iωt) μk2 (8) (9) (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 = k − k22 (> 0), Q, A, B, C, D are constants to be determined from the conditions (3), (4) and the relation ikA + sB = Using Eqs (8)– (10) into Eq (5) and taking into account ikA + sB = yield: 2Q + (x − 2)Bˆ = (12) where Bˆ = B/k, x = c2 /c22 called the squared dimensionless velocity of Rayleigh waves and < x < in order to satisfy the decay condition (7) From Eqs (6), (8), (10) and (11) we have: μk22 Q + 2μ sB − k Q + g ρ − ρ (B − kQ) − iρ ωC = (13) It follows from Eqs (4), (8), (10) and (11): D = iω(Bˆ − Q) (14) On use of (11) in (3) and taking into account (14) yield: (x − ε δ)C = iω(Bˆ − Q)(ε − x δ) (15) where ε = g/(kc22 ) (> 0) and δ = kh (> 0) Since < δ < 1, it follows from (15) that: x = ε δ, because otherwise either Bˆ = Q or ε − x δ = If Bˆ = Q then Bˆ = Q = D = C = A = by (12)–(14) and ikA + sB = It is impossible because this leads to a trivial solution If ε − x δ = 0, from x = ε δ we have immediately δ = From (15) and x = ε δ we have: C = iω(Bˆ − Q)f (x, ε, δ) (16) where: ε − x δ x − ε δ With the help of (16), Eq (13) becomes: f (x, ε, δ) = (x − 2) − ε(1 − r) − rf x Q √ + − x + ε(1 − r) + rf x Bˆ = (17) (18) where r = ρ /ρ (> 0) Equations (12) and (18) establish a homogeneous system of two linear equations for ˆ Vanishing the determinant of this system Q and B gives: √ (2 − x)2 − − x − εx + rεx − rf (x, ε, δ)x = 0, 0 ∀x ∈ (0, 1) (26) Therefore, φ (x) > ∀x ∈ (0, 1), i.e φ(x) is strictly increasingly monotonous in the interval (0, 1) Since (noting that < δ < 1): r δ[(x − ε)2 + 2εx(1 − δ)] >0 (x − ε δ)2 ∀x ∈ (0, 1), x = ε δ, ∀ε > (27) φ1 (x) = the function φ2 (x) is strictly increasingly monotonous in the intervals (0, ε δ) and (ε δ, 1) ∀δ, r, ε > It follows from (23)–(25) that: φ2 (+0) = −2 + ε(r − 1) (28) φ2 (−ε δ) = +∞ (29) φ2 (+ε δ) = −∞ (30) φ2 (1) = − ε + r δ(1 − ε ) (1 − ε δ) (31) Meccanica (i) Suppose < ε < 1, it follows that < ε δ < (due to < δ < 1) and φ2 (1) > (according to (31)) From φ2 (1) > and (30) it implies that Eq (19) has alway a unique real root in (ε δ, 1) From (28) and (29), if −2 + ε(r − 1) ≥ ↔ r ≥ + 2/ε Eq (19) has no real roots in (0, ε δ) and it has exactly one real root belong to (0, ε δ) if < r < + 2/ε The observation (i) is proved (ii) (+) Let ε ≥ and < ε δ < Then φ2 (1) ≤ by (31), therefore Eq (19) has no real root in the interval (ε δ, 1) due to (30) By (29), if φ2 (0) ≥ 0, Eq (19) thus has no real root in the interval (0, ε δ) and it has exactly one real root in (0, ε δ) if φ2 (0) < With the help of these facts and (28) the observation (ii) for < ε δ < is proved (+) Suppose ε ≥ and ε δ = 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) ≥ and it has exactly one real root in (0, 1) if φ2 (0) < These facts along with (28) leads to the observation (ii) for ε δ = (iii) Let ε ≥ and ε δ > Since φ2 (x) is continuous and strictly increasingly monotonous in the interval (0, 1), (⊂ (0, ε δ)), Eq (19) has a unique real root in the interval (0, 1) if φ2 (0) < and φ2 (1) > 0, and it has no real root in the interval (0, 1) if either φ2 (0) ≥ or φ2 (1) ≤ With these facts we arrive immediately at the observation (iii) (iii) There exists a unique Rayleigh wave, namely GRW, if either {ε ≥ 1, < ε δ ≤ 1, < r < + 2/ε} or {ε ≥ 1, ε δ > 1, m < r < + 2/ε} (iv) There exist exactly two Rayleigh waves, one CRW and one GRW, if {0 < ε < 1, < r < + 2/ε} Remark When ε → 0: x (1) → and x (2) → xr (δ), where xr (δ) is the unique real root of Eq (22) (see Remark 3) 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 gravity 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 Proposition and its proof we have the following theorem saying about the existence of Rayleigh waves 2.3 Approximate secular equations Theorem (i) A Rayleigh wave is impossible if either {ε ≥ 1, < ε δ ≤ 1, r ≥ + 2/ε} or {ε ≥ 1, ε δ > 1, r ∈ (0, m] ∪ [1 + 2/ε, +∞)} (ii) There exists a unique Rayleigh wave, namely CRW, if {0 < ε < 1, r ≥ + 2/ε} Remark By the same argument used for Proposition 1, one can prove that: (i) Equation (21) has a (unique) real solution in the interval (0, 1) if and only if ≤ ε < (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 (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 ≤ ε < + rδ Proof By the same argument used for Proposition We note that Bromwich [3] did not consider the existence and uniqueness of solution of Eq (20) Let < ε < 1, then according to Theorem 1, a (unique) CRW exists and its squared dimensionless velocity x (2) is determined by Eq (19) in the domain ε δ < x < Since < ε∗ δ < 1, ε∗ = ε/x, for x ∈ (ε δ, 1), the following expansion holds for x : ε δ < x < 1: (1 − ε∗ δ)−1 = + ε∗ δ + ε∗2 tanh2 δ + ε∗3 tanh3 δ + ε∗4 tanh4 δ + O δ (32) here δ is assumed to be sufficiently small From (17) and (32) we have: f (x, ε, δ) = (ε∗ − δ)(1 − ε∗ δ)−1 = ε∗ + ε∗2 − δ + ε∗ tanh2 δ + ε∗2 tanh3 δ + ε∗3 tanh4 δ + O δ (33) Meccanica Using the expansion δ = δ − δ /3 + O(δ ) into (33) leads to: f (x, ε, δ) = ε∗ + ε∗2 − δ + ε∗ δ + ε∗2 − 1/3 δ + ε∗3 − 2ε∗ /3 δ + O δ (34) Substituting (34) into Eq (19) yields the fourth-order approximate secular equation of the exact secular equation (19) in the domain ε δ < x < 1, namely: √ F (x, ε, δ) ≡ (2 − x)2 − − x − εx − r ε − x δ −r ε3 − εx δ x −r ε 4ε x − + δ 3 x2 −r ε5 x3 ε δ < x < − 5ε 3x + proximate formula for the dimensionless velocity of Rayleigh wave, namely: x = + 0.109ε − 0.099rδ x0 4+ or [23, 24]: x0 = − (35) In the first-order approximation, Eq (35) takes the form: √ F (x, ε, δ) ≡ (2 − x)2 − − x − εx − r ε − x δ = 0, ε δ < x < (36) If ε and δ are both sufficiently small, from (36) we immediately arrive at Eq (20) by neglecting −rε δ Remark (i) Equation (35) determines the approximation of x (2) , not of x (1) (ii) To obtain approximate equations for x (1) (i.e approximate secular equations for GRWs) we can start from: √ (x − ε δ) (2 − x)2 − − x − εx + rεx − rx (ε − x δ) = 0 < x < 1, x = ε δ (37) that is equivalent to Eq (19) (38) where x is the squared dimensionless velocity of Rayleigh waves propagating in an incompressible isotropic elastic half-space (i.e x = x(0, 0)) It is well-known that x is approximately 0.9126 (see [6]), and its exact value is given by [22]: x0 = 2εx δ =0 − √ −17 + 33 − 26 + 27 11 26 + 27 11 3 √ 17 + 33 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- (39) 1/3 −1/3 − (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) = F x(ε, δ), ε, δ ≡ (41) From (41) it follows: ϕε = 0, ϕεδ = 0, ϕδ = 0, ϕεε = ϕδδ = (42) here we use the notations fε = ∂f/∂ε, fδ = ∂f/∂δ, fεε = ∂ f/∂ε , fεδ = ∂ f/∂ε∂δ, fδδ = ∂ f/∂δ , f = f (ε, δ) Using (41) and (42) provides: xε = −Fε /Fx , xεε = − Fxx xε2 xδ = −Fδ /Fx + 2Fxε xε + Fεε /Fx xεδ = −(Fxx xε xδ + Fxδ xε + Fxε xδ + Fεδ )/Fx (43) xδδ = − Fxx xδ2 + 2Fxδ xδ + Fδδ /Fx here F = F [x(ε, δ), ε, δ] On the other hand, by expanding x(ε, δ) into Taylor series about the point (0, 0) up to the second order we have: x(ε, δ) = x(0, 0) + xε0 ε + xδ0 δ 0 + xεε ε + 2xεδ εδ + xδδ δ /2 Approximate formulas for the velocity √ x (44) where f = 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 − − x − εx + rδx (45) Meccanica From (43) and (45), after some manipulations we have: xε0 = 0.1988, xδ0 = −0.1814r = −0.2638, xεε xδδ = 0.2012r xεδ (46) = −0.1475r Substituting these results into (44) yields: x(ε, δ) = 0.9126 + 0.1988ε − 0.1814rδ − 0.1319ε + 0.2012rεδ − 0.0737r δ (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 approximation of the dimensionless velocity of Rayleigh waves, namely: x = + 0.1089ε − 0.0994rδ − 0.0782ε x0 + 0.1211rεδ − 0.0453r δ x0 = 2(4 + ε) − 3 (48) 3.2 Only δ being small Now suppose that δ is sufficiently small and < ε < 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 = we have: x(δ) = x0 + x (0)δ + O δ (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) = −Fδ0 /Fx0 (50) here f = f [x(0), 0], f = f [x(δ), δ] Note that x0 is determined by the following exact formula (see [25]): 16(ε + 11) ε + /27 + ε + 12ε + 12ε + 136 /27 − 8ε − ε + , 16(ε + 11)(ε + 4)/27 + (ε + 12ε + 12ε + 136)/27 or it is calculated by a very highly accurate approximation, namely (see [25]): √ B − B − 4AC (52) x0 = 2A where: A = −(5.1311 + 2ε) B = − 21.2576 + 8ε + ε − r ε2 − x δ (54) Using (54) in (50) gives: r(ε − x02 ) 2(x0 − 2) + 2(1 − x0 )−1/2 − ε x(δ) = x0 + r(ε − x02 )δ 2(x0 − 2) + 2(1 − x0 )−1/2 − ε (56) r(ε − x02 )δ + a2 δ a1 (57) where: a1 = 2(x0 − 2) + 2(1 − x0 )−1/2 − ε a2 = − − (55) (51) Following the same procedure one can see that the second-order approximation x(δ) is: x(δ) = x0 + For obtaining the first-order approximation of x(δ) we can ignore three last terms of F in (35), i.e the function F is taken as: √ F (x, ε, δ) = (2 − x)2 − − x − εx ε ∈ (0, 1) Therefore, at the first-order approximation x(δ) is given by: (53) C = −(15.1266 + 8ε) x (0) = that recovers the first-order approximation (38) Note that following the same procedure one can obtain the √ higher-orders approximations of x and x [2 + (1 − x0 )−3/2 ]r (ε − x02 )2 a13 4r x (ε a12 − x02 ) − (58) 2rε(x0 − and x0 given by (51) or (52) a1 ε2 x0 ) Meccanica 3.3 Global approximations Suppose < ε < and δ is sufficiently small Then, according to Theorem 1, 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 ∗ ε /x − ε = (59) ε δ < x < 1, δ ∗ = rδ > where φ3 (x, δ ∗ ) is given by: √ (2 − x)2 − − x φ3 (x) = + δ∗x x x ∈ (ε δ, 1) (60) Following the same procedure presented in Proposition 1, one can prove that: because as shown in Sect 2.1: ∂φ3 /∂x > ∀x ∈ (0, 1), δ ∗ > As xε = −Φε /Φx , from (63) we conclude that: xε > ∀ε > 0, δ ∗ > That means: x ε, δ ∗ > x 0, δ ∗ ∀ε > 0, δ ∗ > (64) where x(0, δ ∗ ) is the (unique) solution of the equation: √ (2 − x)2 − − x φ3 x, δ ∗ = + δ ∗ x = 0, x x ∈ (ε δ, 1) (65) On use of (65) it is not difficult to verify that dx(0, δ ∗ )/ dδ ∗ < 0, ∀δ ∗ > 0, therefore: x(0, δ ∗ ) > x(0, δ0∗ ) if < δ ∗ < δ0∗ From this fact and (64) we conclude that: x ε, δ ∗ > x 0, δ0∗ ∀ε > 0, ∀δ : < δ ∗ < δ0∗ (66) Now we want to have approximate expressions of the solution of Eq (36) by using the best approximate second-order polynomials of the powers x and x (see [26]) in the sense of least-square We call them the global approximations After squaring and rearranging Eq (36) is converted to: Inequality (66) says that the best interval on which we determine the best approximate second-order polynomials of the powers x and x is the interval [x(0, δ0∗ ), 1] Note that for a given value of δ0∗ it is easy to calculate x(0, δ0∗ ) by solving directly Eq (65) As an example, let r = 0.5, δ = 0.1, then δ0∗ = 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 x and x in the interval [0.9034, 1] in the sense of least-square are: a4 x + a3 x + a2 x + a1 x + a0 = x = 5.4364x − 6.8944x + 2.4578 (67) x = 2.8551x − 2.7158x + 0.8607 (68) Proposition Let < ε < and δ is sufficiently small Then Eq (36) has a unique real solution in the interval (ε δ, 1) (61) where: a0 = δ ∗ ε2 δ ∗ ε2 a1 = −16 + 8δ ∗ ε a2 = 8ε − 2δ ∗ ε −8 , + 2ε δ ∗ + 8δ ∗ − 8ε + ε2 (62) − 2δ ∗ ε + 24 a3 = −2(4 + ε) + δ ∗ , a4 = + δ ∗ δ ∗ ε2 ∂φ3 + >0 ∂x x 2δ ∗ ε 0, δ ∗ > Replacing x and x in Eq (61) by (67) and (68), respectively, we obtain a quadratic equation for x, namely: Ax + Bx + C = After replacing x and x by the best approximate second-order polynomials Eq (62) becomes a quadratic 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: Φx = (69) whose solution corresponding to Rayleigh waves is: √ −B + B − 4AC (70) x= 2A where < ε < 1, < δ ∗ < 0.05, and: A = 5.4364 + δ ∗ − 2δ ∗ ε + 24 − 2.8551(8 + 2ε) + δ ∗ B = −6.8944 + δ ∗ (63) + 8ε − 2δ ∗ ε + 8δ ∗ + ε 2 − 16 + 8δ ∗ ε + 2ε δ ∗ − 8ε + 2.7158(8 + 2ε) + δ ∗ C = 2.4578 + δ ∗ − 8δ ∗ ε − 0.8607(8 + 2ε) + δ ∗ + δ ∗ ε (71) Meccanica Fig Plots of x(ε, 0.04) calculated by the approximate formula (56), by the global approximation (70), (71) and by solving directly Eq (36) They most totally coincide with each other Figure shows the dependence on ε ∈ [0, 0.9] of x(ε, 0.04) which is calculated by the approximate formula (56), by the globally approximate formulae (70), (71) and by solving directly Eq (36) They most totally coincide with each other This says that the approximation (70) has a very high accuracy Remark (i) Since < x < 1, we can take the interval [0, 1] for determining the best approximate second-order polynomials of the powers x and x According to Vinh and Malischewsky [26], the best approximate second-order polynomials of the powers x and x in [0, 1] in the sense of least-square are: 12 32 x − x+ 35 35 x = 1.5x − 0.6x + 0.05 x4 = (72) in which < ε < and δ ∗ > 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 x and x , respectively, in the interval [x(0, δ0∗ ), 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 practical applications Conclusions (73) With the approximations (72), (73), x is given by (70) in which A, B, C are calculated by: 12 + δ ∗ + 8ε − 2δ ∗ ε + 8δ ∗ + ε − 2δ ∗ ε + 24 − (8 + 2ε) + δ ∗ 32 ∗ 1+δ B=− − 16 + 8δ ∗ ε + 2ε δ ∗ − 8ε 35 (74) + (8 + 2ε) + δ ∗ + δ ∗ − 8δ ∗ ε C= 35 − (8 + 2ε) + δ ∗ + δ ∗ ε 20 A= Fig Plots of x(ε, 0.04) calculated by the globally approximate formulae (70), (74) (dashed line), (70), (71) (solid line) and by solving directly Eq (36) (solid line) 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 Meccanica Acknowledgements The work was supported by the Vietnam National Foundation For Science and Technology Development (NAFOSTED) under 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Rayleigh waves in incompressible orthotropic elastic solids J Acoust Soc Am 115:530–533 24 Vinh PC (2010) On formulas for the velocity of Rayleigh waves in prestrained incompressible elastic solids Trans ASME J Appl Mech 77:021006 (9 pages) 25 Vinh PC, Linh NTK (2012 in press) New results on Rayleigh waves in incompressible elastic media subjected to gravity Acta Mech doi:10.1007/s00707-012-0664-6 26 Vinh PC, Malischewsky P (2007) An approach for obtaining approximate formulas for the Rayleigh wave velocity Wave Motion 44:549–562 ... 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. .. Equation (19) is the exact secular equation of Rayleigh waves propagating in an incompressible isotropic elastic half-space overlaid with a layer of incompressible non-viscous water of the finite...Meccanica The propagation of Rayleigh waves in an incompressible isotropic elastic half-space underlying a nonviscous incompressible fluid layer under the effect of gravity was studied also