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Wave Motion 44 (2007) 549–562 www.elsevier.com/locate/wavemoti An approach for obtaining approximate formulas for the Rayleigh wave velocity Pham Chi Vinh a a,* , Peter G Malischewsky b Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam b Institute for Geosciences, Friedrich-Schiller University Jena, Burgweg 11, 07749 Jena, Germany Received 25 July 2006; received in revised form February 2007; accepted February 2007 Available online 17 February 2007 Abstract In this paper, we introduce an approach for finding analytical approximate formulas for the Rayleigh wave velocity for isotropic elastic solids and anisotropic elastic media as well The approach is based on the least-square principle To demonstrate its application, we applied it in order to obtain an explanation for Bergmann’s approximation, the earliest known approximation of the Rayleigh wave velocity for isotropic elastic solids, and used it to establish a new approximation By employing this approach, the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1] were found By using the best approximate polynomial of the second order of the cubic power, we derived an approximate formula for the Rayleigh wave speed in isotropic elastic solids which is slightly better than the one given recently by Rahman and Michelitsch by employing Lanczos’s approximation Also by using this second order polynomial, analytical approximate expressions for orthotropic, incompressible and compressible elastic solids were found For incompressible case, it is shown that the approximation is comparable with Rahman and Michelitsch’s approximation, while for the compressible case, it is shown that our approximate formulas are more accurate than Mozhaev’s ones Remarkably, by using the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1], we derived an approximate formula of the Rayleigh wave velocity in incompressible monoclinic materials, where the explicit exact formulas of the Rayleigh wave velocity so far are not available Ó 2007 Elsevier B.V All rights reserved Keywords: Rayleigh waves; Rayleigh wave velocity; Rayleigh wave speed; Approach of least squares; The best approximation; Approximate formula; Approximate expression Introduction Elastic surface waves (i.e Rayleigh waves), first studied by Rayleigh [5] more than a century ago (in 1885), have been intensively studied and exploited, due to wide applications in seismology, acoustics, geophysics, materials science, nondestructive testing, telecommunication industry and so on * Corresponding author Tel.: +84 5532164; fax:+84 8588817 E-mail addresses: pcvinh@vnu.edu.vn (P.C Vinh), p.mali@uni-jena.de (P.G Malischewsky) 0165-2125/$ - see front matter Ó 2007 Elsevier B.V All rights reserved doi:10.1016/j.wavemoti.2007.02.001 550 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 For the Rayleigh wave, its velocity is a fundamental quantity which interests researchers in seismology, geophysics, and in other fields of physics and the material sciences It is discussed in almost every survey and monograph on the subject of surface acoustic waves in solids Due to the significance of the Rayleigh wave velocity in practical applications, researchers have attempted to find its analytical approximate expressions which are of simple forms and accurate enough for practical purposes That is why a lot of approximations of the Rayleigh wave speed appeared in the literature (see, e.g [3,4,6]) However, as indicated by Mozhaev [4], most of them were reported without any indication of the derivation procedure So it is interesting to have a method which can provide the explanations for these approximations In this paper, we present an approach originating from the principle of least squares (see, e.g [1]) for finding analytical approximate expressions for the Rayleigh wave speed This approach can give the derivation of previously proposed approximate formulas and establish new approximate formulas as well As a first application of this procedure let us present an explanation of Bergmann’s approximation, the earliest approximate expression of the Rayleigh wave speed in isotropic elastic solids In order to create new approximations we can start either from the explicit exact formulas for the Rayleigh wave speed, or from the secular equations of the Rayleigh waves For the first possibility, as an example, by using our approach we derived an approximate formula for the form of the third order polynomial, of the Rayleigh wave speed in isotropic elastic solids for the range [À1, 0.5] of Poisson’s ratio m It is shown that this result is a good approximation For the second possibility, by replacing the power z3 in the secular equations by its best approximate second order polynomial in the interval [0, 1] established by our approach, we have obtained: (i) An approximate formula of the Rayleigh wave speed in isotropic elastic solids and it is shown that this approximation is slightly more accurate than that given recently by Rahman and Michelitsch [3] (ii) An approximate expression of the Rayleigh wave speed in incompressible orthotropic elastic solids which is comparable with Rahman and Michelitsch’s [3] (iii) Approximate formulas of the Rayleigh wave velocity in compressible orthotropic elastic media and they are more accurate than those proposed by Mozhaev [4] (iv) Remarkably, by using the best approximate (in the sense of least squares) second order polynomials of z3 and z4 in the interval [0, 1], we derived an approximate formula of the Rayleigh wave speed for the materials with more complicated symmetry, namely the incompressible monoclinic materials with the plane of symmetry at x3 = 0, where the explicit exact formulas of the Rayleigh wave velocity so far are not available It is noted, that only recently explicit exact formulas of the Rayleigh wave speed have been published for isotropic compressible elastic solids (see [7–9,11]), for incompressible orthotropic elastic materials (see [10]) and for compressible orthotropic elastic ones (see [12,13]) Least-square approach As mentioned, there is a need for obtaining analytical approximate expressions of the Rayleigh-wave speed for the practical work in the laboratory or elsewhere These should be more simple than the exact one and sufficiently accurate This is, mathematically, related to the approximation problem of a given function which can be formulated as follows: Let X be a normed linear space and V be a subset of X For a given f X determine an element g V such that: kf À gk kf À hk for all h V ; ð1Þ here the symbol kuk denotes the norm of u X If the problem (1) has a solution then the element g is called a best approximation of f with respect to V If V is a finite dimensional linear subspace or a compact subset of X, then the problem (1) has a solution (see, e.g [14]) Moreover, if X is strictly convex (i.e ku ỵ wk < whenever kuk ¼ kwk ¼ and u w) and V is a finite dimensional linear subspace of X, then problem (1) has precisely one solution (see, e.g [14]) P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 551 The most applicable cases are the cases in which X = L2[a, b] or X = C[a, b], where L2[a, b] consists of all functions measurable in (a, b), whose squared value is integrable on [a, b] in the sense of Lebesgue, and C[a, b] contains continuous functions in [a, b] We recall that both L2[a, b] and C[a, b] are normed linear spaces, whose norms are defined, respectively, as follows: Z b 1=2 kuk ¼ u mị dm ; u L2 ẵa; b; 2ị a and kuk ẳ max jumịj; m2ẵa;b u Cẵa; b: In the present paper, we confine ourselves to the case of X = L2[a, b] Then the problem (1) becomes: Let V be a subset of L2[a, b] For a given function f L2[a, b], determine a function g V such that: Z b Z b ½f mị gmị2 dm ẳ ẵf mị hmị2 dm: ð3Þ h2V a a The Eq (3) expresses the principle of least squares The quantity Z b IðhÞ ẳ ẵf mị hmị dm; h V ; pffiffiffiffiffiffiffiffiffi IðhÞ where: ð4Þ a represents the deviation of the function h from the function f on the interval [a, b] or the distance between h and f in L2[a, b] The equality (3) shows that the best approximation g(m) (if exists) makes the deviation funcpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tional (4) minimum The quantity IðhÞ=ðb À aÞ is called the average error of the approximate solution of the problem (3) It is noted that L2[a, b] is a Hilbert space, so it is strictly convex (see [14]) Thus, the problem (3) has a unique solution in the case that V is a finite dimensional subspace of L2[a, b] The subset V of L2[a, b] is chosen such that g(m) has a simple form Since polynomials are considered as the simplest functions, V is normally taken as the set of polynomials of order not bigger than n À which is a linear subspace of L2[a, b] and has dimension n If V is a finite dimensional linear subspace with the basis h1(m), h2(m), , hn(m), for solving problem (3) we represent h(m) as a linear combination of h1(m), h2(m), , hn(m): n X hmị ẳ hi mị: 5ị iẳ1 Then the functional I(h) becomes a function of the n variables a1, a2, , an and problem (3) is leaded to a system of n linear equations for a1, a2, , an which has a unique solution In the case that V is a compact set of L2[a, b], for example, V contains functions having the form: n X hðm; yÞ ẳ yịhi mị; m; y ẵa; b; 6ị i¼1 where hi(m) are given elements of L2[a, b], ai(y) are prescribed differentiable functions of y in [a, b], the functional I(h) then becomes a differentiable function of y in the closed interval [a, b], so it attains its minimum in [a, b], and the problem (3) leads to solving the equation (non-linear in general): I yị ẳ 0; y ða; bÞ: ð7Þ The prime denotes here the first derivative Application of the least-square approach 3.1 Mathematical basis of Bergmann’s approximation Let c be the Rayleigh wave speed in compressible isotropic elastic solids and x ¼ ðc=bÞ , where b is the velocity of shear waves Then, it is well known that x is a solution of the equation (see, for instance, [5,11]): 552 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 x3 À 8x2 ỵ 83 2cịx 161 cị ẳ 0; ð8Þ which satisfies: ð9Þ < x < 1; where: c ẳ 2mị=21 mị ẳ b=aị2 ; ð10Þ and a is the velocity of longitudinal waves, m is Poisson’s ratio It is also noted that in the range (9) Eq (8) has precisely one real solution By Malischewsky [9] xðmÞ is expressed by the following formula: " # ffiffiffiffiffiffiffiffiffiffi 2ð1 À 6cÞ p ffiffiffiffiffiffiffiffiffiffi ; xmị ẳ h3 cị ỵ p 11ị 3 h3 cị where: h1 cị ẳ p 33 186c ỵ 321c2 192c3 ; h3 cị ẳ 17 45c ỵ h1 cị: 12ị In formula (11), the main values of the cubic roots are to be used Interestingly, only recently a convenient and simple form for xðmÞ, the derivation of which is not trivial, has been published by Malischewsky [8,9] and Pham Chi Vinh and Ogden [11], while analytical approximate p expressions of xmị ẳ xðmÞ started appearing in the literature long ago As we know, the earliest and wellknown approximate expression of x(m) was proposed by Bergmann [2] without explanation, and it has the form: xb mị ẳ 0:87 ỵ 1:12m ; 1ỵm m ẵ0; 0:5: 13ị Now, by using the approach of least squares, we prove that xb(m) is the best approximation with respect to V whose elements have the form: a ỵ bm ; 1ỵm hmị ẳ m ẵ0; 0:5; ð14Þ where a and b are constants It is noted that V is a linear subspace of L2[0, 0.5] which has dimension with a basis, for example, h1(m), h2(m) as follows: h1 mị ẳ ; 1ỵm h2 mị ẳ m ; m ẵ0; 0:5: 1ỵm As mentioned above, in this case problem (3) has unique solution In order to find this solution we substitute (14) into (4) The functional I(h) then becomes a function of two variables a, b, denoted by I(a, b), which is of the form: Ia; bị ẳ m0 a2 ỵ 2m1 ab ỵ m2 b2 2m3 a 2m4 b ỵ m5 ; 15ị where: mi ẳ Z 0:5 mi dm ỵ mị ; i ẳ 0; 1; 2; m3ỵi ẳ Z 0:5 xmịmi dm ỵ mị ; i ¼ 0; 1; m5 ¼ Z 0:5 x2 ðmÞ dm: From the conditions: oI/oa = 0, oI/ob = 0, we obtain two linear equations: m0 a þ m1 b ¼ m3 ; m1 a þ m2 b ẳ m4 : Using (10)(12) and noting that xmị ¼ m0 ¼ 0:333333; m1 ¼ 0:0721318; ð16Þ ð17Þ p xmị, from (16) we have: m2 ẳ 0:0224031; m3 ¼ 0:371013; m4 ¼ 0:0878859: ð18Þ P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 553 It is easy to verify that the system (17) with coefficients defined by (18) has precisely one solution, namely: a ¼ 0:87096; b ¼ 1:11869: ð19Þ So, the best approximation of x(m) with respect to V (in the sense of least squares) in the interval [0, 0.5] is: gb mị ẳ 0:87096 ỵ 1:11869m ; 1ỵm m ẵ0; 0:5: 20ị It is clear from (13) and (20) that, xb(m) and gb(m) are almost totally identical with each other, so we can say that Bergmann’s approximation is the best approximation of x(m) (in the sense of least squares), in the interval [0, 0.5], with respect to the class of all functions expressed by (14) From (4), (13) and (20) it is shown that the squared deviation of xb(m) and gb(m) from x(m) are 7.5 · 10À7 and 5.85 · 10À7, respectively Remark The approximation of Bergmann is very good for positive values of Poisson’s ratio, but completely fails for negative values Materials with negative values of Poisson’s ratio, so-called auxetic materials, really exist (see, e.g a new review by Yang et al [15]) and become increasingly interesting in material sciences This motivated Malischewsky [6] to find an approximation pffiffiffiffiffiffiffiffithat is good throughout the whole range of physically possible values of Poisson’s ratio By expanding xðmÞ (xðmÞ is defined by (11)) at a certain value of Poisson’s ratio, which was found by trial and error, he has found the following approximation: xm ðmÞ ẳ 0:874 ỵ 0:196m 0:043m2 0:055m3 ; m ẵ1; 0:5: 21ị An explanation for his result, which originates from the least-square approach, was given by Pham Chi Vinh and Malischewsky [16] It was proven that the approximation (21) is almost totally identical with the best approximation of x(m) (in the sense of least squares), in the interval [À1, 0.5], with respect to the class of Taylor expansions of x(m) up to the third power at the values belong to the interval [À1, 0.5] In this case, the set V contains functions: hm; yị ẳ xyị ỵ x1ị ðyÞ xð2Þ ðyÞ xð3Þ ðyÞ ðm À yÞ þ ðm À yÞ þ ðm À yÞ ; 1! 2! 3! ð22Þ in which y [À1, 0.5] is considered as a parameter Here by x(k)(y) we denote the derivative of order k of x(y) with respect to y It is easy to observe that, in this case V is a compact subset of L2[À1, 0.5] 3.2 A new approximation for the Rayleigh wave speed To show the effectiveness of the least-square approach, in this subsection we give a new approximation of x(m) on the interval [À1, 0.5] in the form of a polynomial of the third order For this purpose, naturally we choose V as the set of all polynomials of order not bigger than 3: hmị ẳ am3 ỵ bm2 ỵ cm ỵ d; 23ị where a, b, c, d are constants In this case, V is a four-dimensional linear subspace of L [À1, 0.5], so problem (3) has precisely one solution In order to find this solution, analogously as above, we substitute (23) into (4) The functional (4) then is converted to a function of four variables: a, b, c, d, denoted by I(a, b, c, d) From the condition: oI=oa ¼ 0; oI=ob ¼ 0; oI=oc ¼ 0; oI=od ¼ 0; we obtain the following system of linear equations: 2=7ị27 ỵ 1ịa ỵ 1=3ị26 1ịb ỵ 2=5ị25 ỵ 1ịc ỵ 1=2ị24 1ịd ẳ 0:33139; > > > < 1=3ị26 1ịa þ ð2=5Þð2À5 þ 1Þb þ ð1=2Þð2À4 À 1Þc þ ð2=3Þð2À3 ỵ 1ịd ẳ 0:56514; > 2=5ị25 ỵ 1ịa ỵ 1=2ị24 1ịb ỵ 2=3ị23 ỵ 1ịc ỵ 22 1ịd ẳ 0:51276; > > : 1=2ị24 1ịa ỵ 2=3ị23 ỵ 1ịb ỵ 22 1ịc ỵ 3d ẳ 2:47152: It is not difficult to verify that the solution of system (25) is: ð24Þ ð25Þ 554 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 a ¼ À0:0439059; b ¼ À0:0350168; c ¼ 0:192422; d ¼ 0:87384: ð26Þ Thus, the best approximation of x(m) in this case is: g3 mị ẳ 0:87384 ỵ 0:192422m 0:0350168m2 0:0439059m3 ; m ẵ1; 0:5: 27ị Substituting (21) and (27) into (4) leads to: I(xm) = 3.3 · 10 and I(g3) = 2.53367 · 10 That means the squared distance between xm(m) and x(m) in the space L2[À1, 0.5] is 3.3 · 10À6, while the one between g3(m) and x(m) is 2.53367 · 10À7, i.e g3(m) approximates x(m) better than does xm(m) (in the sense of least squares) It is evidence because the class of Taylor expansions of x(m) up to the third power at the values which belong to the interval [À1, 0.5] is a subset of the set of all polynomials of order not bigger than Fig shows plots of x(m) and its approximation g3(m) defined by (27) in the interval [À1, 0.5] It is very difficult to distinguish one from the other It is clear from Fig 2, that the approximation g3(m) is more accurate than Malischewsky’s xm(m) and Rahman and Michelitsch’s published recently in [3] (formula (7)), and xm(m) is extraordinary good in the range about (0, 0.4) which is very important for geophysical applications Remark Analogously, we can obtain the best approximation gn(m) of x(m) in the interval [À1, 0.5] with respect to Pn+1, n = 4, 5, , by using the least-square approach, where by Pn we signify the set of all polynomials of order not bigger than n À As Pn & Pn+1 " n P 1, it is clear that "n P 1, gn+1(m) approximates x(m) better than does gn(m), in the sense of least squares 3.3 The best approximate second order polynomials of the powers z3 and z4 in the interval [0, 1] In this subsection, first we want to find a second order polynomial which is the best approximation of the power z3 with respect to P3, in the interval [0, 1] by using the method of least squares In this case, we consider problem (1) in which X = L2[0, 1], V = P3 and a = 0, b = 1, f = z3 An element h(z) of P3 is expressed as follows: hzị ẳ az2 ỵ bz ỵ c; ð28Þ where a, b, c are constants In order to find constants a, b, c corresponding to the best approximation, we substitute (28) into (4) The functional I(h) then becomes a function I(a, b, c) given by: 0.95 0.9 x=c/β 0.85 0.8 0.75 0.7 0.65 —1 —0.5 0.5 Poisson’s Ratio ν Fig Plots of x(m) (solid line) and its approximation g3(m) (dotted line) in the interval [À1, 0.5] P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 555 0.45 0.4 Percentage Error (\%) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 —1 —0.5 0.5 Poisson’s Ratio ν Fig Percentage errors of g3(m) (solid line), xm(m) (dashed line) and Rahman–Michelitsch’s approximation (dash-dot line) in the interval [À1, 0.5] Percentage error = |1 À g(m)/x(m)| · 100%, g(m) is an approximation of x(m) Iða; b; cÞ ẳ a2 b2 ab 2ac a 2b c ỵ bc ỵ : ỵ ỵ c2 ỵ þ 3 ð29Þ From the condition: oI=oa ¼ 0; oI=ob ¼ 0; @I=@c ẳ 0; it follows: > < 2=5ịa ỵ 1=2ịb ỵ 2=3ịc ẳ 1=3; 1=2ịa ỵ 2=3ịb ỵ c ẳ 2=5; > : 2=3ịa ỵ b ỵ 2c ẳ 1=2: ð30Þ ð31Þ It is easy to verify that the system (31) has an unique solution: a ¼ 1; 5; b ¼ À0:6; c ¼ 0:05: ð32Þ Hence, the desired second order polynomial is: pzị ẳ 1:5z2 0:6z ỵ 0:05: ð33Þ Analogously, we can find the best approximation second order polynomial of the power zn with respect to P3, in the interval [0, 1], in the sense of least squares For example, the best approximation second order polynomial of the power z4 is: pzị ẳ 12 32 z zỵ : 35 35 34ị Remark Consider the problem (1) for the case: X = C[0, 1], V = P3, a = 0, b = and f = z3 We call it the problem (1*) The problem (1*) has a unique solution although C[0, 1] is not strictly convex (see, for instance [14,17]) Now we show that the unique solution of (1*) is: pà ðzÞ ẳ 1:5z2 0:5625z ỵ 0:03125: First, we observe that: among all polynomials q(z) of the nth degree whose leading coefficient is unity: ð35Þ 556 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549562 qzị ẳ zn ỵ an1 zn1 ỵ ỵ a1 z ỵ a0 ; ð36Þ nÀ1 the Chebysev polynomial Tn(z)/2 (see [1]) deviates the least from zero in C[À1, 1] Indeed, suppose there exists a polynomial q0(z) of the form (36) such that: T n ðzÞ max jq0 ðzÞj < max n1 ẳ n1 : z2ẵ1;1 z2ẵ1;1 2 37ị Then G(z) = Tn(z)/2nÀ1 À q0(z) is a polynomial of degree not exceeding n 1, and: Gzi ị ẳ ặ nÀ1 À q0 ðzi Þ; ð38Þ where zi ¼ cosðip=nÞ; i ¼ 0; n From (38) it follows that G(z) has n zeroes, so G(z) 0, i.e q0(z) Tn(z)/2nÀ1 But this contradicts (37), and the observation is proven By the transformation z = 2t À 1, this observation leads to the conclusion: among all polynomials of the nth order whose leading coefficient is unity, the shifted Chebysev polynomial T Ãn ðzÞ=22nÀ1 (see [1]) deviates the least from zero in C[0, 1], where T n zị ẳ T n ð2z À 1Þ That means the following proposition is valid: the polynomial pn1 zị ẳ zn T Ãn ðzÞ=22nÀ1 deviates the least from zn in C[0, 1] Applying the proposition for n = we have p2(z) = p*(z) and (35) is demonstrated It is noted that Rahman and Michelitsch [3] called p*(z) Lanczos’s approximation 3.4 An approximation for the Rayleigh wave speed in compressible isotropic elastic solids Now, following Rahman and Michelitsch [3], we use the approximation (33) of z3 in order to obtain a approximate formula of the Rayleigh wave speed in compressible isotropic elastic media By replacing x3 by pðxÞ, the cubic Eq (8) is reduced to the following quadratic: 6:5x2 À ð23:4 À 16cịx 16c 15:95ị ẳ 0; 39ị whose solution satisfying (9) is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 23:4 À 16c À ð23:4 16cị ỵ 2616c 15:95ị x ẳ : 13 ð40Þ pffiffiffiffiffiffiffiffi Using (10), from (40) we obtain the following approximate formula of xmị ẳ xmị in term of Poissons ratio: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 15:4 À 7:4m À 56:06m2 À 22:52m ỵ 30:46 x ẳ x ẳ ; m ½À1; 0:5; ð41Þ 13ð1 À mÞ and the velocity of the Rayleigh waves is c = bx* Fig shows that approximate expression (41) is slightly more accurate than that obtained recently by Rahman and Michelitsch [3] using Lanczos’s approximation (35), for the whole interval [À1, 0.5] By using the interval [0.474572, 0.912622], instead of the one [0, 1], analogously as above, we obtain the following approximation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 29:171 13:171m 203:188m2 70:194m ỵ 123 xvm2 ẳ ; m ẵ1; 0:5; 23:6771 mị which is more accurate than x* (see Fig 3) Remark (i) By using the interval [0.763932, 0.912622], instead of the one [0, 1], quite analogously, we obtain the following approximation for the range m [0, 0.5]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 27:7904 À 11:7904m 190:467m2 56:1246m ỵ 121:6576 xvm1 ẳ ; m ẵ0; 0:5: 21:940671 mị P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 557 x 10 —3 3.5 Absolute Error 2.5 1.5 0.5 —1 —0.5 0.5 Poisson Ratio ν Fig Absolute errors of the approximation x* (solid line), xvm2 (dash-dot line) and Rahman–Michelitsch’s approximation given by (7) in [3] (dashed line) Absolute error = |x(m) À g(m)|, g(m) is an approximation of x(m) (ii) Independently, Li [21] has found the approximation (41), also using (33) (iii) By using the best approximate second order polynomial of z3 in the interval [0.47, 1] ([0.75, 1]) in the sense of least squares, Li [21] has established an approximation for the range m [À1, 0.5] (m [0, 0.5]), defined by (20) ((22)) in [21], whose percentage error does not exceed 0.16% (0.004%) Because the maximum of the percentage error of xvm2, xvm1 is 0.09%, 0.0019%, respectively, it is shown that the approximation xvm1, xvm2 is better than corresponding Li’s one 3.5 An approximation for the Rayleigh wave speed in incompressible orthotropic elastic solids For incompressible orthotropic elastic solids, the secular equation of the Rayleigh waves is of the form (see, for instance [10]): g3 ỵ g2 ỵ 1ịg ẳ 0; 42ị < g < 1; where: g¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi À x2 ; ð0 < x < 1ị; x ẳ c= q=c66 ; ẳ c11 ỵ c22 À 2c12 > 0: c66 ð43Þ Here c11, c22, c12 and c66 are the material constants, q is the mass density, c is the Rayleigh wave speed By using the approximation (33) for g3, Eq (42) is simplified to the equation: 2:5g2 1:6 ịg 0:95 ẳ 0; whose solution belonging to (0, 1) is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:6 ỵ 1:6 ị ỵ 9:5 : gà ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi From (45) and the fact xÃi ¼ À g2à we obtain: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10:38 þ 6:4 À 22 À 2ð1:6 À Þ ð1:6 À ị ỵ 9:5 ; xi ẳ 44ị 45ị 46ị 558 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 where x*i() is the approximation of x() which is defined by the explicit formula (38) in [10] From Fig 4, it is shown that the approximation (46) is comparable with the one given by Rahman and Michelitsch recently ((9) in [3]), using Lanczos’s approximation (33) 3.6 An approximation for the Rayleigh wave speed in compressible orthotropic elastic solids For compressible orthotropic elastic solids, the secular equation of the Rayleigh waves is of the form (see, for instance, [12,13]): z3 þ e2 z2 þ e1 z þ e0 ¼ 0; 47ị < z < minf1; rg; where: z ẳ qc2 =c55 ; e0 ¼ ar2 d2 ; cÀa e1 ẳ ardrd ỵ 2ị ; ca e2 ẳ a ỵ 2ard ; ca 48ị c is the Rayleigh wave speed, q is the mass density The dimensionless material parameters are defined by: a ¼ c33 =c11 ; c ¼ c55 =c11 ; d ¼ À c213 =c11 c33 ; r ¼ 1=c; a > 0; c > 0; < d < 1; ð49Þ where cij are material constants Replacing z3 by p(z) defined by (33) reduces the cubic Eq (47) to the following quadratic equation: e2 ỵ 1:5ịz2 ỵ e1 0:6ịz ỵ e0 ỵ 0:05 ẳ 0; 50ị whose solution satisfying < z < min{1, r} is: q 0:6 e1 ỵ 0:6 e1 ị 4e2 ỵ 1:5ịe0 ỵ 0:05ị : z1 ẳ 2e2 ỵ p The Rayleigh wave velocity is given by: c ¼ x1à c55 =q, where: v q u u0:6 e ỵ 0:6 e ị2 4e ỵ 1:5ịe ỵ 0:05ị t 1 : x1 a; r; dị ẳ 2e2 ỵ ð51Þ ð52Þ 0.18 0.16 0.14 Absolute Error 0.12 0.1 0.08 0.06 0.04 0.02 0 10 Epsilon Fig Absolute errors of the approximation x*i defined by (46) (solid line) and Rahman–Michelitsch’s approximation given by (9) in [3] (dashed line) P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 559 Fig shows that the approximation (52) is better than Mozhaev’s ones given by (16) and (20) in [4] in the indicated range of the parameters Now we start from another secular equation of the Rayleigh waves in compressible orthotropic elastic materials, namely (see [12]): t3 ỵ a2 t2 t ỵ a0 ẳ 0; 53ị where: r À cz ; t¼ 1Àz z ¼ qc2 =c55 ; p a0 ẳ a1 dị; a2 ẳ p að1 À rdÞ; ð54Þ and ð55Þ < t < if < r < 1; t > if r > 1; pffiffiffi pffiffiffi (when r ¼ 1; qc2 =c55 ẳ adị=1 ỵ aị, see [12]) Case 1: < t < 1(0 < r < 1): In this case, taking into account (33), Eq (53) becomes: a2 ỵ 1:5ịt2 1:6t ỵ a0 ỵ 0:05 ẳ 0: ð56Þ The solution of (56) belonging to (0, 1) is: p p p 1; ỵ 2:56 4ẵ a1 rdị ỵ 1:5ẵ0:05 a1 dị p ; t ẳ a1 rdị ỵ pffiffiffiffiffiffiffiffiffiffiffi and the Rayleigh wave speed is given by: c ¼ x2à c55 =q, here: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rð1 À t2Ã Þ : x2 a; r; dị ẳ rt2 ị ð57Þ ð58Þ It is clear from Fig 6, that the approximate formula (58) is much more accurate than Mozhaev’s ones defined by (16) and (20) in [4], in the indicated range of the parameters Case 2: t > 1(r > 1): In terms of the new variable u = 1/t Eq (53), for the case t > 1, is of the form: Dimensionless Rayleigh wave speed 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 δ Fig Plots of the approximation x1*(1, 2, d) (solid line), Mozhaev’s given by (16) (dotted line with points), (20) (dash-dot line) in [4] at a = 1,r = 2,d [0.4, 0.95] and their exact values defined by (3.28) in [12] (dashed line) 560 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 Dimensionless Rayleigh wave speed 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 δ Fig Plots of the approximation x2* (1, 0.5, d) (solid line), Mozhaev’s given by (16) (dotted line with points), (20) (dash-dot line) in [4] at a = 1,r = 0.5,d [0.4, 0.95] and their exact values defined by (3.28) in [12] (dashed line) a0 u3 u2 ỵ a2 u ỵ ẳ 0; ð59Þ < u < 1; which is simplified to the following equation using the approximation (33): ð1 À 1:5a0 ịu2 a2 0:6a0 ịu ỵ 0:05a0 ị ẳ 0: 60ị The solution of (60) satisfying < u < is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 À 0:6a0 ỵ a2 0:6a0 ị ỵ 41 ỵ 0:05a0 Þð1 À 1:5a0 Þ uà ¼ : À 3a0 pffiffiffiffiffiffiffiffiffiffiffi The Rayleigh wave speed is given by: c ¼ x3à c55 =q, in which: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðu2à À 1Þ : x3 ẳ u2 rị 61ị 62ị 3.7 An approximation for the Rayleigh wave speed in incompressible monoclinic elastic materials Consider incompressible monoclinic elastic materials with the plane of symmetry at x3 = (see [20]) For these materials, the dispersion equation of the Rayleigh waves in the explicit form was found recently by Destrade [18], namely: z4 ỵ b3 z3 ỵ b2 z2 ỵ b1 z ỵ b0 ẳ 0; ð63Þ where z = qc /a, < z < 1,c is the Rayleigh wave velocity and: b0 ẳ 1=2ị4b ỵ c2 ị ; b1 ẳ 2b ỵ 2ị4b ỵ c2 ị ; 2 64ị b2 ẳ 1=2ị16b ỵ 20 3c ị4b ỵ c ị; b3 ẳ 5=2ị4b ỵ c ị; s0 s0 s0 a ¼ 0 11 ; b ¼ 660 À 1; c ¼ À 16 ; 4s11 s011 s11 s66 À ðs16 Þ s0ij are reduced elastic compliances (see [20]) On use of approximations (33) and (34) of z3 and z4, respectively, we simplify Eq (63) to: ð65Þ ð66Þ Dimensionless Rayleigh wave speed P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 561 0.97 0.96 γ = 0.1 0.95 0.94 0.93 0.92 0.91 β 0.9 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Fig Plots of the approximation x4* at c = 0.1, b [0, 0.5] (dashed line) and its exact values (solid line) b2 ỵ 1:5b3 ỵ 1:714ịz2 ỵ b1 0:6b3 0:914ịz ỵ b0 ỵ 0:05b3 ỵ 0:086 ẳ 0: ð67Þ The solution of (67) satisfying the condition < z < is: zà ¼ À ðb1 À 0:6b3 0:914ị 2b2 ỵ 1:5b3 ỵ 1:714ị q b1 0:6b3 0:914ị 4b2 ỵ 1:5b3 ỵ 1:714ịb0 þ 0:05b3 þ 0:086Þ : ð68Þ 2ðb2 þ 1:5b3 þ 1:714Þ pffiffiffiffiffiffiffiffi pffiffiffiffi The Rayleigh wave speed is given by: c ¼ x4à a=q, here: x4à ¼ zà From (68) we have: x4* = 0.9551 at b = 0.3, c = 0.1, while by numerically directly solving (63), Nair [19] and Destrade [18] obtained x = 0.94671 for this case That means we have a good agreement Fig shows the plots of the approximation x4* at c = 0.1, b [0, 0.5] and its exact values which were obtained by numerical solution of (63) The absolute error does not exceed 0.01, so the accuracy is acceptable À Conclusions In this paper, the approach of least squares is recommended for obtaining analytical approximate expressions of the Rayleigh wave speed By employing this method we can give the mathematical basis to the previously proposed approximations and establish new approximate formulas as well As examples, we used it in order to explain Bergmann’s approximation, the oldest known approximation of the Rayleigh wave speed in isotropic elastic materials, and create a new approximate formula for these materials in the interval [À1, 0.5], and it is shown that it is a good approximation It is noted that, starting from the explicit exact formulas for the Rayleigh wave speed, we can construct approximate expressions in different forms corresponding to chosen different sets V By this method we have found the best approximate second order polynomials of the powers z3 and z4 in the interval [0, 1] By replacing z3 and z4 in the secular equations by these second order polynomials, we have obtained new approximate formulas of the Rayleigh wave speed for isotropic elastic solids, incompressible and compressible orthotropic elastic materials, especially for the incompressible monoclinic materials with the plane of symmetry at x3 = 0, where the explicit exact formulas for the Rayleigh wave speed so far are not available It is shown that all these approximate formulas are more accurate than those proposed previously, except the incompressible orthotropic case where our result is comparable with the one obtained recently by Rahman and Michelitsch [3] It is noted that we can use the best approximate second order polynomials of the powers z3 and z4 in the interval [0, 1], in the space C[0, 1] (which Rahman and Michelitsch referred to Lanczos’ approximations) to get analogous approximations 562 P.C Vinh, P.G Malischewsky / Wave Motion 44 (2007) 549–562 Acknowledgements The work was done partly during the first author’s visit of two months to the Institute for Geosciences, Friedrich-Schiller University Jena, which was supported by a DAAD Grant No A/05/58097 PGM kindly acknowledges the support of BMBF in the joint project ‘‘WTZ Deutsch-Israel’’ Grant No 03F0448A and of DFG Grant No MA 1520/6-2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] C Lanczos, Applied Analysis, Prentice-Hall Inc., New Jersy, 1956 L 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