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Improved Approximations of the Rayleigh Wave Velocity PHAM CHI VINH* Faculty of Mathematics, Mechanics and Informatics Hanoi University of Science, Thanh Xuan, Hanoi, Vietnam PETER G MALISCHEWSKY Institute for Geosciences, Friedrich-Schiller University Jena, Jena 07749, Germany ABSTRACT: In this article we have derived some approximations for the Rayleigh wave velocity in isotropic elastic solids which are much more accurate than the ones of the same form, previously proposed In particular: (1) A second (third)-order polynomial approximation has been found whose maximum percentage error is 29 (19) times smaller than that of the approximate polynomial of the second (third) order proposed recently by Nesvijski [Nesvijski, E G., J Thermoplas Compos Mat 14 (2001), 356–364] (2) Especially, a fourth-order polynomial approximation has been obtained, the maximum percentage error of which is 8461 (1134) times smaller than that of Nesvijski’s second (third)-order polynomial approximation (3) For Brekhovskikh–Godin’s approximation [Brekhovskikh, L M., Godin, O A 1990, Acoustics of Layered Media: Plane and Quasi-Plane Waves Springer-Verlag, Berlin], we have created an improved approximation whose maximum percentage error decreases 313 times (4) For Sinclair’s approximation [Malischewsky, P G., Nanotechnology 16 (2005), 995–996], we have established improved approximations which are times, 6.9 times and 88 times better than it in the sense of maximum percentage error In order to find these approximations the method of least squares is employed and the obtained approximations are the best ones in the space L2[0, 0.5] with respect to its corresponding subsets KEY WORDS: Rayleigh wave velocity, the best approximation, method of least squares *Author to whom correspondence should be addressed E-mail: pcvinh@vnu.edu.vn Journal of THERMOPLASTIC COMPOSITE MATERIALS, Vol 21—July 2008 0892-7057/08/04 0337–16 $10.00/0 DOI: 10.1177/0892705708089479 ß SAGE Publications 2008 Los Angeles, London, New Delhi and Singapore Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 337 338 P C VINH AND P G MALISCHEWSKY INTRODUCTION 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 material science, for example For the Rayleigh wave, its velocity is a fundamental quantity which is of interest to researchers in all these areas of application, and due to its significance in practical applications, researchers have attempted to find its analytical approximate expressions which are of simple forms and accurate enough for practical purposes Let c be the Rayleigh wave velocity in isotropic elastic solids and x(v) ¼ c/ , where is the velocity of shear waves and v is Poisson’s ratio Perhaps, the earliest known approximate formula of x(v) was proposed by Bergmann [2], namely: E LASTIC SURFACE WAVES xb ị ẳ 0:87 ỵ 1:12 , 1ỵ ẵ0, 0:5: 1ị In the form of the second-order polynomial, the approximate formula: xn2 ị ẳ 0:874 ỵ 0:198 0:0542 , ẵ0, 0:5, 2ị given by Nesvijski [3], while in form of the third-order polynomial he proposed the following approximation [3]: xn3 ị ẳ 0:874 ỵ 0:196 À 0:0432 À 0:0523 , ½0, 0:5: 3ị In terms of the parameter ẳ (1 )/4(1 ỵ ), Brekhovskikh and Godin [4] established the approximate expression: 27 1 , : 4ị xbg ị ẳ ỵ , 2 16 12 In form of the inverse of a polynomial of the second order, Sinclair developed the following approximate formula (see [5,6]): xsc ị ẳ , 1:14418 0:25771 ỵ 0:126612 ẵ0, 0:5: 5ị It is noted that, for isotropic materials, apart from (1) to (5), there exists a number of other approximations of the Rayleigh wave velocity (see, for example [7–10]) Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 Improved Approximations of the Rayleigh Wave Velocity 339 As addressed by Nesvijski [3], nondestructive testing of composites is a complex problem because components of materials may have very similar physical-mechanical properties In order to distinguish one component from another we need highly accurate approximations of the Rayleigh wave velocity Some recent experimental results cannot be explained unambiguously by existing approximate formulas This motivates the authors to improve previously proposed approximations The present article is devoted to the improvement of the approximations (2)–(5) In particular (1) we derive a second (third)-order polynomial approximation that is 29 (19) times better than approximation (2) (approximation (3)) proposed recently by Nesvijski [3], in the sense of maximum percentage (relative) error (2) Especially, a fourth-order polynomial approximation is established whose maximum percentage error is 8461 (1134) times smaller than that of Nesvijski’s second (third)-order polynomial approximation given by formula (2) (approximation (3)) (3) We create a new approximation which is 313 times more accurate than Brekhovskikh–Godin’s approximation (4) (4) For Sinclair’s approximation (5), we establish improved approximations which are 4, 6.9, and 88 times better than it In order to find these approximations the method of least squares is employed and the obtained approximations are the best ones in the space L2[0, 0.5] with respect to its corresponding subsets The method can be used to create more accurate approximations It is noted that approximations (1)–(5) were reported without indicating the derivation procedure (see [3,7]) Recently, it was proved by Vinh and Malischewsky [9] that Bergmann’s approximation is the best approximation of x(), in the sense of least squares, in the interval [0, 0.5], with respect to the class of all functions expressed by hị ẳ a ỵ b , 1ỵ ẵ0, 0:5, 6ị where a, b are constants It is noted that V is a linear subspace of L2[0, 0.5] which has dimension It will be shown in this article that the inverse of Sinclair’s approximation is the best approximation of 1/x() in the interval [0, 0.5], in the sense of least squares, with respect to the set of all Taylor expansions of 1/x() up to the second power at the values y ½0, 0:5 (Proposition 4) FORMULAS FOR THE RAYLEIGH WAVE VELOCITY Interestingly that, only recently explicit and exact formulas of a convenient and simple form for xðÞ, the derivation of which is not trivial, has been published by Malischewsky [11,12] and Vinh and Ogden [13], Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 340 P C VINH AND P G MALISCHEWSKY pffiffiffiffiffiffiffiffiffi while analytical approximate expressions of xị ẳ xị started appearing in the literature long ago In Malischewsky’s notation [12], the Rayleigh wave velocity is expressed by: " # ffiffiffiffiffiffiffiffiffiffiffi 2ð1 À 6 Þ pffiffiffiffiffiffiffiffiffi p , xị ẳ , h3 ị ỵ p xị ẳ c= ẳ xị 7ị 3 h3 ð Þ where 2 À 2 ¼ ¼ , 2ð1 À Þ ð8Þ and with the auxiliary functions: p h1 ị ẳ 33 186 ỵ 321 192 , h3 ị ẳ 17 45 ỵ h1 ị: 9ị Here a is the velocity of longitudinal waves For the inverse of x() (dimensionless slowness), it is convenient to use the following formula given by Vinh and Ogden [13]: " # ffiffiffiffiffiffiffiffiffiffi þ ð4 À 3Þ2 p pffiffi 1 p sị ẳ ẳ s, s ẳ ỵ V ị ỵ , 10ị xị 41 ị V ị where: V ị ẳ 27 90 ỵ 99 32 ị 27 p ỵ p ị 11 62 ỵ 107 À 64 3 ð11Þ In formulas (7) and (10) the main values of the cubic roots are to be used It should be noted that Rahman and Barber [14], Nkemzi [15], Romeo [16] have also found explicit formulas for the Rayleigh wave speed in isotropic solids, but they are not simple as (7) and (10) It is also noted that explicit exact formulas of the Rayleigh wave speed in orthotropic elastic materials have been found recently by Vinh and Ogden [17–19] LEAST-SQUARE APPROACH In order to obtain the improved approximations of the Rayleigh wave velocity we will use the least-square method which was presented in detail in [9] Here we recall it shortly Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 341 Improved Approximations of the Rayleigh Wave Velocity Let V be a subset of the space L2[a, b] (that consists of all functions measurable in (a, b), whose squared values are integrable on [a, b] in the sense of Lebesgue), and f is a given function of L2[a, b] A function g V is called the best approximation of f with respect to V, in the sense of least squares, if it satisfies the equation Z b ẵfị gị2 d ẳ Ihị, 12ị h2V a where Z b ẵfị hị2 d: Ihị ẳ 13ị a If V is a finite-dimensional linear subspace (a compact set) of L2[a, b], then the problem (12) has a unique solution (a solution) (see [20]) By Pn we denote the set of polynomials of order not bigger than n À 1, that is a linear subspace of L2[a, b] and has dimension n When V Pn , its basic functions can be chosen as the orthogonal Chebyshev polynomials Tk tịị, k ẳ 0, n À defined as follows (see [21,22]): hk ðÞ ẳ Tk tịị ẳ cos ẵk arccos tị, tị ẳ k ¼ 0, n À 1, 2 À a À b , ba 14ị 15ị where ẵa, b and t ½À1, 1 In this case, the best approximate polynomial of f() with respect to Pn, in the interval [a, b] is (see [22]): pn1 ị ẳ n1 X ck Tk tịị, 16ị kẳ0 where: c0 ẳ Z fẵ ịd , ck ẳ Z fẵ ịcos k d , k ẳ 1, n 1, 17ị in which ị ẳ ba aỵb cos þ : 2 Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 ð18Þ 342 P C VINH AND P G MALISCHEWSKY Noting that cos ðn Ỉ 1ị ẳ cos n cos ầ sin sin, from Equation (14) the recursion formula is deduced for the Chebyshev polynomials: Tkỵ1 tị ẳ 2tTk tị Tk1 tị, 19ị starting with: T0 tị ẳ 1, T1 tị ¼ t: ð20Þ Applying successively formula (19) and taking into account starting condition (20), the first five Chebyshev polynomials are (see also [21]): T0 tị ẳ 1, T1 tị ẳ t, T2 tị ẳ 2t2 1, 21ị T3 tị ¼ 4t3 À 3t, T4 ðtÞ ¼ 8t4 À 8t2 þ 1: Remark 1: It is obvious that if gi() is the best approximation of f() with respect to Vi & L2 ẵa, b; i ẳ 1, 2ị in the interval [a, b], and V1 & V2 , then the approximation g2() is more accurate than g1() in the sense of least squares, i.e., Iðg2 Þ Iðg1 Þ In order to evaluate an approximation’s accuracy we use the maximum percentage (relative) error I defined as follows: gðÞ Â 100% I ẳ max ẵa, b xị 22ị where g() is an approximation of x() in the interval [a, b] IMPROVED NESVIJSKI’S APPROXIMATIONS As mentioned above, among poplynomials of second order, Nesvijski [3] proposed xn2(), given by formula (2), as an approximation of x() in the interval [0, 0.5] Now, we check that whether it is the best approximation of x() or not, in this range, in the sense of least squares, with respect to P3 By using Equations (17) and (18) in which f() ¼ x(), x() defined by Equations (7)–(9) and a ¼ 0, b ¼ 0.5, we obtain: c0 ẳ 0:91701093855671, c1 ẳ 0:04074468783261, 23ị c2 ¼ À0:00236434432626: Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 343 Improved Approximations of the Rayleigh Wave Velocity From Equation (15) with a ¼ 0, b ¼ 0.5 and Equation (21), it follows: T0 ðtðÞÞ ¼ 1, T1 tịị ẳ 4 1, T2 tịị ẳ 322 16 ỵ 1: 24ị Substituting Equations (23) and (24) into Equation (16) leads to: p2 ị ẳ 0:8739 þ 0:2008 À 0:075662 : ð25Þ Thus, we have the following proposition: Proposition 1: The best approximate polynomial of the second order of x() in the interval [0, 0.5], in the sense of least squares, with respect to P3 is the polynomial p2() given by formula (25) It is clear, from formulas (2) and (25), that Nesvijski’s approximation xn2() is not the best approximation of x() in the interval [0, 0.5], in the sense of least squares, with respect to P3 This fact can be seen from Figure From Equations (2), (7), (22), and (25) it follows: n2 Þ ¼ 0:44%, Iðp Þ ¼ 0:015%: Iðx ð26Þ 0.45 0.4 Percentage error (%) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ν Figure Percentage errors of approximations: xn2() (dash-dot line), p2() (solid line), p4() (dashed line: almost coincides with the -axis) Percentage error ¼ |1 À g()/x()|  100%, g() is an approximation of x() Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 344 P C VINH AND P G MALISCHEWSKY i.e., the maximum percentage error of p2() is 29 times smaller than that of xn2() Analogously as above, and taking into account c3 ẳ 1:045818847690321 104 , T3 tịị ẳ 2563 1922 ỵ 36 1, 27ị it is deduced from Equations (16), (23), (24), and (27): p3 ị ẳ 0:874006 ỵ 0:19704 0:055582 0:026773 , ð28Þ and the following conclusion is valid: Proposition 2: The best approximate polynomial of the third order of x() in the interval [0, 0.5], in the sense of least squares, with respect to P4 is the polynomial p3() defined by formula (28) From Equations (3) and (28) it is obvious that xn3() is not the best approximation of x() in the interval [0, 0.5], with respect to P4 This fact can be observed by Figure In view of Equations (3), (7), (22), and (28) we have: n3 Þ ¼ 0:059%, Iðp Þ ¼ 3:1  10À3 %: Iðx ð29Þ 0.06 Percentage error (%) 0.05 0.04 0.03 0.02 0.01 0 0.05 0.1 0.15 0.2 0.25 ν 0.3 0.35 0.4 0.45 0.5 Figure Percentage errors of approximations: xn3 (dash-dot line), p3 (solid line), p4 (dashed line: almost coincides with the -axis) Percentage error ¼ |1 À g()/x()|  100%, g() is an approximation of x() Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 345 Improved Approximations of the Rayleigh Wave Velocity This says that the maximum percentage error of p3() is 19 times smaller than that of xn3() Analogously, by using the fact c4 ẳ 2:602481661645599 105 , T4 tịị ẳ 20484 20483 ỵ 6402 64 ỵ 1, 30ị it follows from Equations (16), (23), (24), (27) and (30): p4 ị ẳ 0:0532994 0:0800723 0:0389232 ỵ 0:1953777 þ 0:8740325: ð31Þ Using Equations (7), (22), and (31) we have: ị ẳ 5:2 105 %: Ip ð32Þ From Equations (26), (29), and (32) it follows that the approximation p4() is 8461 (1134) times better than xn2() (xn3()) in the sense of maximum percentage error IMPROVED BREKHOVSKIKH–GODIN’S APPROXIMATIONS Considering the Rayleigh wave velocity as a function of ẳ (1 )/ 4(1 ỵ ), Brekhovskikh and Godin [4] proposed the approximate formula (4) which is a polynomial of the third order in terms of We shall point out that it is not the best approximate third-order polynomial of x() in the sense of least squares, in the interval [1/12, 1/4], with respect to P4: the set of all polynomials of order not bigger than three in terms of Even, it is less accurate than the best approximate second-order polynomial obtained by the presented approach In view of Equations (14)–(18), in this case, the best approximate (n À 1) th order polynomial of x() with respect to Pn, in the interval [1/12, 1/4] is: pÃðnÀ1Þ ị ẳ n1 X ck Tk tịị, 33ị kẳ0 where: c0 ẳ Z xẵ ịd , ck ẳ Z xẵ ịcos k d , k ẳ 1, n À 1, Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 ð34Þ 346 P C VINH AND P G MALISCHEWSKY in which ị ẳ 1 cos ỵ : 12 35ị Replacing by in Equation (15) and putting a ¼ 1/12, b ẳ 1/4 yield: tị ẳ 12 2, ẵ1=12, 1=4: 36ị Employing Equations (21), (36) leads to: T0 tịị ẳ 1, T1 tịị ẳ 12 2, T2 tịị ẳ 2882 96 ỵ 7, T3 tịị ẳ 69123 34562 ỵ 540 26: 37ị It follows from Equations (34) and (35): c0 ¼ 0:91287085775639, c1 ¼ À0:04079848265769, c2 ¼ 0:00183775883442, c3 ¼ 1:566525867870697  10À4 : ð38Þ By using Equations (33), (37) and (38) we have: p3 ị ẳ 1:00326 0:58141 0:012122 ỵ 1:082783 : ð39Þ Thus, the following conclusion is true Proposition 3: Among the third-order polynomials of , p*3() is the best approximation of x() in the sense of least squares, in the interval [1/12, 1/4] It is clear from formulas (4) and (39) that Brekhovskikh–Godin’s approximation xbg() is not the best approximate third-order polynomial of x() in the sense of least squares, in the interval [1/12, 1/4], with respect to P4 This is also demonstrated by Figure In view of Equations (4), (7), (22), and (39) we have: bg ị ẳ 1:35%, Ip ị ẳ 0:0043%: Iðx ð40Þ That means the approximation p*3() is 313 times better than xbg() in the sense of maximum percentage error By applying Equation (33) for n ¼ and using Equations (37) and (38) we obtain the best approximate second-order polynomial of x(), namely: p2 ị ẳ 1:00733 0:666 þ 0:529272 : Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 ð41Þ 347 Improved Approximations of the Rayleigh Wave Velocity 0.014 0.012 Percentage error (%) 0.01 0.008 0.006 0.004 0.002 0.1 0.15 0.2 0.25 δ Figure Percentage errors of approximations: xbg (dash-dot line), p*2 (solid line), p*3 (dashed line: almost coincides with the -axis) Percentage error ¼ |1 À g()/x()|  100%, g() is an approximation of x() Astonishingly, the approximation p*2() is also more accurate than Brekhovskikh–Godin’s approximation xbg() (see Figure 3) ị ẳ 0:021%, hence from In view of Equations (4), (7), (22), and (41) Iðp Equation (40), the approximation p*2() is 64 times more accurate than xbg() Remark 2: Following the same procedure we obtain the best approximate polynomial of the fourth order of x(), namely: p4 ị ẳ 6:1049074 ỵ 5:1527213 0:9872062 0:482492 ỵ 0:999689: 42ị which is 4804 times accurate than Brekhovskikh–Godin’s approximation in the sense of maximum percentage error IMPROVED SINCLAIR’S APPROXIMATIONS First we prove the following proposition: Proposition 4: The inverse of Sinclair’s approximation defined by formula (5) is the best approximation of s() in the interval [0, 0.5], in the sense of Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 348 P C VINH AND P G MALISCHEWSKY least squares, with respect to the set P3* of all Taylor expansions of s() up to the second power at the values y ½0, 0:5 That means, in order to find xsc() we first seek the best approximate second-order polynomial of s() in [0, 0.5], denoted by q2*(), in L2[0, 0.5], with respect to the set V ¼ P3* (a compact subset of L2[0, 0.5]), and then take its inverse In other words, q2*(() is the solution of the problem (12), in which a ¼ 0, b ¼ 0.5, f() ¼ s() and the elements of V are given by: h, yị ẳ syị ỵ s1ị yị yị ỵ s2ị yị yị2 , , y ½0, 0:5, ð43Þ where y is considered as a parameter and by sðkÞ ðyÞ we denote the derivative of order k of s(y) with respect to y When h(v, y) defined by Equation (43), the functional I(h) becomes a function of y, denoted by I(y) Taking into account Equations (8), (10), (11), (13), and (43), it is not difficult to verify that I(y) is a differentiable function of y in the interval [0, 0.5], so it has a minimum in [0, 0.5] (this is also observed by the fact that V is a compact subset of L2[0, 0.5]) By using Equations (8), (10), (11), (13), and (43) we have: Iyị ẳ X fi yị, 44ị iẳ1 where f1 yị ẳ ẵs2ị yị2 ẵ0:5 yị5 ỵ y5 =20; f2 yị ẳ ẵs1ị yị2 ẵ0:5 yị3 ỵ y3 =3, f4 yị ẳ s2ị yịs1ị yịẵ0:5 yị4 y4 =4, f3 yị ẳ ẵsyị2 =2; f5 yị ẳ s2ị yịsyịẵ0:5 yị3 ỵ y3 =3; f7 yị ẳ s1ị yịsyịẵ0:5 yị2 y2 ; f6 yị ẳ sð2Þ ðyÞð2m1 y À m2 À m0 y2 Þ, f8 yị ẳ 2s1ị yịm0 y m1 ị, f9 yị ẳ 2m0 syị ỵ m , 45ị where Z Z 0:5 i mi ẳ sịd, i ẳ 0, 1, 2, m ẳ 0:5 ẵsị2 d: 46ị In order to find the minimum of the function I(y) in the compact interval [0, 0.5] we have to find critical points of I(y) (the roots of equation I=yị ẳ 0) Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 Improved Approximations of the Rayleigh Wave Velocity 349 in the open interval (0, 0.5), then compare the values of I(y) at these points with I(0) and I(0.5) By using Equations (8), (10), (11), and (44)–(46) we numerically solve the equation I=yị ẳ 0, and three its roots in the open (0, 0.5) are: y1 ¼ 0:14584942510293, y2 ¼ 0:21907305699270, 47ị y3 ẳ 0:29539595383846: It follows from Equations (8), (10), (11), and (44)(47): I0ị ẳ 9:2 107 , Iy1 ị ẳ 6:8 1010 , Iy2 ị ẳ 4:7 109 , Iy3 ị ẳ 1:9 109 , I0:5ị ẳ 1:2 106 : 48ị It is clear, from Equation (48), that in the interval [0, 0.5] function I(y) attains minimum at y1 Substituting y ¼ y1 ¼ 0.14584942510293 into Equation (43) leads to: q2à ðÞ ¼ 1:14421 0:25788 ỵ 0:126512 : 49ị It is seen, from formulas (5) and (49), that the denominator of xsc() and q2*() is almost totally identical to each other, so the Proposition is demonstrated Now we find approximations of Sinclairs form with higher accuracy Because 05xị518 ẵ0, 0:5, it follows that sị ẳ 1=xị418 ẵ0, 0:5 This leads to: x À 1 5 s À g 8g L2 ẵ0, 0:5 : gị41 8 ẵ0, 0:5: g ð50Þ The inequality (50) is valid for both L2[0, 0.5] -norm and C[0, 0.5] -norm It follows form inequality (50) that if g() is a good approximation of s() then 1/g() is a good one of x() A more accurate approximation of s() likely leads to a corresponding more accurate one of x() by this way Following this idea, in order to improve the accuracy of Sinclair’s approximation (5) we find approximations of s() which are better than q2*() given by formula (49) Because P3à & P3 & P4 & P5 , according to the remark 1, the best approximations of s() to P3, P4, P5, denoted respectively by q2(), q3(), q4() are more accurate than q2*(), in the sense of least squares (and q4() is better than q3(), q3() is more accurate than q2()) Now we find these polynomials using Chebyshev orthogonal basis Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 350 P C VINH AND P G MALISCHEWSKY By using Equations (17) and (18) in which f() ¼ s(), a ¼ 0, b ¼ 0.5 we have: c0 ¼1:09158536649509, c1 ¼ À0:04865048166307, c2 ¼ 0:00389293077594, c3 ¼ À2:339167226160535  10À5 , c4 ẳ 2:827216331577937 105 : 51ị From Equations (15) and (21) it follows: T4 tịị ẳ 20484 20483 þ 6402 À 64 þ 1: ð52Þ Employing Equation (16) for n ¼ 3, 4, and taking into account Equations (24), and second part of equation (27), (51), (52) we have: q2 ị ẳ 0:124572 0:25689 ỵ 1:14413, 53ị q3 ị ẳ 0:005993 ỵ 0:129062 0:25773 ỵ 1:14415, 54ị q4 ị ẳ 0:05794 ỵ 0:051913 ỵ 0:110972 0:255922 ỵ 1:144124: 55ị It is shown from Figure that the approximations 1/q4() is much more accurate than Sinclair’s approximation Substituting Equations (53), (54), (55) into Equation (22) and taking into account Equation (7) lead to: Ixsc ị ẳ 0:02%, I ẳ 0:0048%, I q2 q3 ¼ 0:0029%, I ¼ 2:28  10À4 % q4 ð56Þ It follows from (56) that approximations 1/q4(), 1/q3(), 1/q2() are 88, 6.9, and times, respectively, better than approximation xsc() in the sense of maximum percentage error CONCLUSIONS In this article, by using the method of least squares, we have obtained some improved approximations for the Rayleigh wave velocity in isotropic elastic solids which are much more accurate than the ones of the same form, previously proposed These improved approximations may be useful for distinguishing components of composites which have very similar Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 351 Improved Approximations of the Rayleigh Wave Velocity 0.02 0.018 Percentage error (%) 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0.05 0.1 0.15 0.2 0.25 ν 0.3 0.35 0.4 0.45 0.5 Figure Percentage errors of approximations: xsc() (dashed line), 1/q4() (solid line) Percentage error ¼ |1 À g()/x()|  100%, g() is an approximation of x() physical-mechanical properties, and for other applications of the Rayleigh waves It is noted that the technique used here can be employed to improve other approximations ACKNOWLEDGMENTS The work was done partly during the first author’s visit to the Institute for Geosciences, Friedrich-Schiller University Jena, which was supported by a DAAD grant No A/05/58097, and it was finished during his visit to the University of California-Berkeley financed by a grant No 06-30776 of the Fulbright Scholar Program He is very grateful to the DAAD and the Fulbright Scholar Program for the financial support REFERENCES Rayleigh, L (1985) On Waves Propagating Along the Plane Surface of an Elastic Solid, Proc R Soc Lond., A17: 4–11 Bergmann, L (1948) Ultrasonics and their Scientific and Technical Applications, John Wiley Sons, New York Nesvijski, E G (2001) On Rayleigh Equation and Accuracy of Its Real Roots Calculations, J Thermoplas Compos Mat., 14: 356–364 Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 352 P C VINH AND P G MALISCHEWSKY Brekhovskikh, L.M and Godin, O.A (1990) Acoustics of Layered Media: Plane and QuasiPlane Waves, Springer-Verlag, Berlin Briggs, G.A.D (1992) Acoustic Microscopy, Clarendon Press, Oxford Malischewsky, P.G (2005) Comparison of Approximated Solutions for the Phase Velocity of Rayleigh Waves (Comment on ‘Characterization of Surface Damage Via Surface Acoustic Waves’), Nanotechnology, 16: 995–996 Mozhaev, V.G (1991) Approximate Analytical Expressions for the Velocity of Rayleigh Waves in Isotropic Media and on the Basal Plane in High-symmetry Crystals, Sov Phys Acoust., 37(2): 186–189 Li, X.-F (2006) On Approximate Analytic Expressions for the Velocity of Rayleigh Waves, Wave Motion, 44: 120–127 Vinh, P.C and Malischewsky, P.G (2007) An approach for Obtaining Approximate Formulas for the Rayleigh Wave Velocity, Wave Motion, 44: 549–562 10 Vinh, P.C and Malischewsky, P.G (2007) An Improved Approximation of Bergmann’s Form for the Rayleigh Wave Velocity, Ultrasonics, 47: 49–54 11 Malischewsky, P.G (2000) Comment to ‘A New Formula for the Velocity of Rayleigh Waves’ by D Nkemzi [Wave Motion 26 (1997) 199–205], Wave Motion, 31: 93–96 12 Malischewsky Auning, P G (2004) A Note on Rayleigh-Wave Velocities as a Function of the Material Parameters, Geofisica Internacional, 43: 507–509 13 Vinh, P.C and Ogden, R.W (2004) On Formulas for the Rayleigh Wave Speed, Wave Motion, 39: 191–197 14 Rahman, M and Barber, J.R (1995) Exact Expression for the Roots of the Secular Equation for Rayleigh Waves, ASME J Appl Mech., 62: 250–252 15 Nkemzi, D (1997) A New Formula for the Velocity of Rayleigh Waves, Wave Motion, 26: 199–205 16 Romeo, M (2001) Rayleigh Waves in a Viscoelastic Solid Half-space, J Acoust Soc Am., 110(1): 59–67 17 Ogden, R.W and Vinh, P.C (2004) On Rayleigh Waves in Incompressible Orthotropic Elastic Solids, J Acoust Soc Am., 115(2): 530–533 18 Vinh, P.C and Ogden, R.W (2004) Formulas for the Rayleigh Wave Speed in Orthotropic Elastic Solids, Arch Mech., 56: 247–265 19 Vinh, P.C and Ogden, R.W (2005) On the Rayleigh Wave Speed in Orthotropic Elastic Solids, Meccanica, 40: 147–161 20 Meinardus, G (1967) Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, Berlin, Heidelberg, New York 21 Lanczos, C (1956) Applied Analysis, Prentice-Hall Inc., New Jersy 22 Achieser, N.I (1956) Theory of Approximation, Frederick Ungar, New York Downloaded from jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 ... March 24, 2015 Improved Approximations of the Rayleigh Wave Velocity 349 in the open interval (0, 0.5), then compare the values of I(y) at these points with I(0) and I(0.5) By using Equations (8),... jtc.sagepub.com at MICHIGAN STATE UNIV LIBRARIES on March 24, 2015 345 Improved Approximations of the Rayleigh Wave Velocity This says that the maximum percentage error of p3() is 19 times smaller than that... that, for isotropic materials, apart from (1) to (5), there exists a number of other approximations of the Rayleigh wave velocity (see, for example [7–10]) Downloaded from jtc.sagepub.com at