Ultrasonics 45 (2006) 77–81 www.elsevier.com/locate/ultras Explanation for Malischewsky’s approximate expression 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, Viet Nam b Institute for Geosciences, Friedrich-Schiller University Jena, Burgweg 11, 07749 Jena, Germany Available online 28 July 2006 Abstract An approach for obtaining approximations of the Rayleigh wave velocity created by the principle of least squares is introduced In view of this approach, Malischewsky’s approximation of the Rayleigh wave velocity for Poisson ratios m [À1, 0.5] proposed quite recently [3] is explained It is shown that Malischewsky’s approximation obtained by trial and error is (almost) identical with the one established by this approach Ó 2006 Elsevier B.V All rights reserved PACS: 43.20.Jr; 43.35.Pt Keywords: Rayleigh wave velocity; The principle of least squares, The best approximation Introduction Rayleigh waves propagating over the surface of elastic half-spaces are a well-known and prominent feature of the wave theory Its velocity c is a fundamental quantity which interests researchers in ultrasonics, seismology, and in other fields of physics and material sciences In the homogeneous half-space, c is frequency-independent, it depends only on the velocity a of longitudinal waves and b of shear waves According to Lord Rayleigh [1] the velocity of these surface waves follows from the solution of a cubic equation, which has been text-book knowledge for many years In the meantime, a lot of approximations of this velocity, which is without a doubt fundamental and essential, appeared in the literature (see e.g [2] and [3]) It is therefore surprising, that only recently a convenient and simple form of the exact solution, which is possible but not trivial, has been published by Malischewsky [4,5] * 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) 0041-624X/$ - see front matter Ó 2006 Elsevier B.V All rights reserved doi:10.1016/j.ultras.2006.07.001 and Pham Chi Vinh and Ogden [6] The existence of an explicit formula for the Rayleigh wave velocity in a halfspace is useful as a test case in inverting geophysical data [7] and for different applications in non-destructive testing [3] Because this formula is not yet well-known within the ultrasonic community, it would be useful to present this in the framework of this article The exact solution which contains cubic roots, will not render superfluous approximate solutions for the practical work in the laboratory or elsewhere The oldest known approximation of Bergmann [2] is very good for positive Poisson ratios m, but completely fails for negative m Materials with negative Poisson ratios, so-called auxetic materials, really exist (see e.g a new review by Yang et al [8]) and may become increasingly interesting in material sciences This was the motivation of Malischewsky [3] to search for an approximation being very good within the whole range of physically possible Poisson ratios (À1 m 0.5) by expanding his exact formula into a Taylor series It turned out by trial and error that this expansion has to be carried out at a strange Poisson ratio of about m0 % 0.12 in order to get the best result Most of the approximations in use are good enough for practical purposes However, scientific exactness requires 78 P.C Vinh, P.G Malischewsky / Ultrasonics 45 (2006) 77–81 a straightforward and mathematically well founded procedure for obtaining the best result in all cases under consideration We present here a simple approach created by the least-square principle, which fulfils these requirements, and demonstrate its application to Malischewsky’s approximation [3] It should be noted that Rahman and Michelitsch [9] have recently published an alternative approximation on the basis of Lanczos’ approximation [11] The final formula is more complicated than Malischewsky’s one We postpone a comparison of both formulas with the exact solution to the end of paragraph A paper with discussion of all existing approximations in the light of the approach proposed here is in preparation Exact formula for the Rayleigh wave velocity in a halfspace value is integrable on [a, b] in the sense of Lebesgue, we can consider the case when X = L2[a, b] We recall that L2[a, b] is a normed linear space whose norm is defined as follows: Z b 12 u ðmÞdm ; u L2 ẵa; b 4ị kuk ẳ a Then the problem (3) 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 ẵf mị gmị dm ẳ ẵf mị hmị2 dm 5ị h2V a a The Eq (5) ffiexpresses the principle of least squares The p quantity Ihị where Z b Ihị ẳ ẵf ðmÞ À hðmÞ dm; h V ð6Þ a By using Malischewsky’s notation [5] we obtain the following expression for the Rayleigh wave velocity: " # pffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi 2ð1 À 6cị p xmị ẳ c=b ẳ xmị; xmị ẳ h3 cị ỵ p 3 h3 ðcÞ ð1Þ where c = (1 À 2m)/2(1 À m) = (b/a)2 and with the auxiliary functions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 cị ẳ 33 186c ỵ 321c2 192c3 ; h3 cị ẳ 17 45c ỵ h1 cị ð2Þ In formula (1), the main values of the cubic roots are to be used Least-square approach As mentioned, there is a need to obtain analytical approximate expressions of the Rayleigh wave speed x(m) for the practical work in the laboratory or elsewhere, which are reasonably simpler than the exact one and yet are accurate enough 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 ð3Þ here the symbol kuk denotes the norm of u X If the problem (3) 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 (3) has a solution (see e.g [10]) Moreover, if X is strictly convex (i.e ku + wk < whenever kuk = kwk = and u w) and V is a finite dimensional linear subspace of X, then problem (3) has precisely one solution (see e.g [10]) Since x(m) can be considered as an element of the space L2[a, b], (À1 a b 0.5) which consists of all functions measurable in (a, b), whose squared 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 (5) shows that the best approximation g(m) (if exists) makes the deviation functional (6) minimum pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The quantity I ẳ Igị=b aị is called the average error of the approximate solution g(m) of the problem (5) on [a, b] It is noted that L2[a, b] is a Hilbert space, so it is strictly convex (see [10]) Thus, the problem (5) 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 type 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 (5) we represent h(m) as a linear combination of h1(m), h2(m), , hn(m): n X hi mị 7ị hmị ẳ iẳ1 Then the functional I(h) becomes a function of the n variables a1, a2, , an and problem (5) 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 yịhi mị; m; y ẵa; b 8ị hm; yị ẳ iẳ1 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 (5) leads to solving the equation (non-linear in general): I yị ẳ 0; y a; bị The prime denotes here the first derivative ð9Þ P.C Vinh, P.G Malischewsky / Ultrasonics 45 (2006) 77–81 79 2 Mathematical basis of Malischewskys approximation f12 yị ẳ xyịx1ị yịẵ0:5 yị ỵ yị As known, by trial and error Malischewsky [3] has obtained a good approximation of the Rayleigh wave velocity in the range of Poisson values [1, 0.5] which is f13 yị ẳ 2x1ị yịm0 y m1 ị f14 yị ẳ 2m0 xyị; f 15 yị ẳ m Z 0:5 Z mi ẳ mi xmịdm; i ẳ 0; 1; 2; 3; m ẳ xm mị ẳ 0:874 ỵ 0:196m 0:043m2 0:055m3 10ị Now we apply the least-square approach presented above to give an explanation for this approximation Since the approximation (10) is obtained by carrying out a Taylor expansion of function x(m) defined by (1) up to the third power at the value m0 % 0.12, and by trial and error it is shown that among polynomials of third order obtained by expanding x(m) into a Taylor series up to the term of third power at values y [À1, 0.5], xm(m) deviates the least from function x(m) in the interval [À1, 0.5], it can be said that among these polynomials, xm(m) is the best approximation of x(m) in the interval [À1, 0.5] (in the sense of least squares) That means that xm(m) is the solution of the problem (5) in which f(m) = x(m), a = À1, b = 0.5 and V is a set of elements h(m, y) having the form: x1ị yị x2ị yị m yị ỵ m yị hm; yị ẳ xyị ỵ 1! 2! x3ị yị m yị ỵ 11ị 3! 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], whose elements are of the form (11) that similar to (8) When h(m, y) taking the form (11), the functional I(h) becomes a function of y, denoted by I(y) Taking into account (1), (2), (6) and (11), it is not difficult to verify that I(y) is a differentiable function of y in the interval [À1, 0.5], so it has a minimum in [À1, 0.5] (this is also observed by the fact that V is a compact subset of L2[À1, 0.5]) By using (1), (6) and (11) we have Iyị ẳ 15 X 13ị 0:5 ẵxmị2 dm In order to find the minimum of the function I(y) in the compact interval [À1, 0.5] we have to find critical points of I(y) (the roots of equation I (y) = 0) in the open interval (À1, 0.5), then compare the values of I(y) at these values with I(À1) and I(0.5) From (12) and (13) we have I yị ẳ 14 X ui yị 14ị iẳ1 where u1 yị ẳ x3ị yịx4ị yịẵ0:5 yị7 ỵ ỵ yị7 =126 ẵx3ị yị2 ẵ0:5 yị6 ỵ yị6 =36 u2 yị ẳ x2ị yịx3ị yịẵ0:5 yị5 ỵ ỵ yị5 =10 ẵx2ị yị2 ẵ0:5 yị4 ỵ yị4 =4 u3 yị ẳ 2x1ị yịx2ị yịẵ0:5 yị3 ỵ ỵ yị3 =3 ẵx1ị yị2 ẵ0:5 yị2 ỵ yị2 u4 yị ẳ 3xyịx1ị yị u5 yị ẳ ẵx2ị yịx4ị yị ỵ x3ị yịị2 ẵ0:5 yị6 ỵ yị6 =36 x2ị yịx3ị yịẵ0:5 yị5 ỵ ỵ yị5 =6 u6 yị ẳ ẵx1ị yịx4ị yị ỵ x2ị yịx3ị yịẵ0:5 yị5 ỵ ỵ yị5 =15 x1ị yịx3ị yịẵ0:5 yị4 ỵ yị4 =3 u7 yị ẳ ẵxyịx4ị yị ỵ x1ị yịx3ị yịẵ0:5 yị4 ỵ yị4 =12 3 xyịx3ị yịẵ0:5 yị ỵ ỵ yị =3 u8 yị ẳ x4ị yịm0 y 3m1 y ỵ 3m2 y m3 ị=3 ỵ x3ị yị3m0 y 6m1 y ỵ 3m2 ị=3 fi yị 12ị iẳ1 u9 yị ẳ ẵx1ị yịx3ị yị ỵ x2ị yịị2 ẵ0:5 yị4 ỵ yị4 =4 x1ị yịx2ị yịẵ0:5 yị3 ỵ ỵ yị3 where f1 yị ẳ ẵx3ị yị2 ẵ0:5 yị7 ỵ ỵ yị7 =252 5 3 f2 yị ẳ ẵx2ị yị ẵ0:5 yị ỵ ỵ yị =20 u10 yị ẳ ẵxyịx3ị yị ỵ x1ị yịx2ị yịẵ0:5 yị3 ỵ ỵ yị3 =3 xyịx2ị yịẵ0:5 yị2 ỵ yị2 u11 yị ẳ x3ị yị2m1 y m2 m0 y ị ỵ 2x2ị yịm1 m0 yị f3 yị ẳ ẵx1ị yị ẵ0:5 yị ỵ ỵ yị =3 u12 yị ẳ ẵxyịx2ị yị ỵ x1ị yịị2 ẵ0:5 yị2 ỵ yị2 f4 yị ẳ 3ẵxyị =2 6 5 f5 yị ẳ x2ị yịx3ị yịẵ0:5 yị ỵ yị =36 3xyịx1ị yị f6 yị ẳ x1ị yịx3ị yịẵ0:5 yị ỵ ỵ yị =15 u13 yị ẳ 2x2ị yịm0 y m1 ị ỵ 2m0 x1ị yị f7 yị ẳ xyịx3ị yịẵ0:5 yị4 ỵ yị4 =12 u14 yị ẳ 2m0 x1ị yị f8 yị ẳ x3ị yịm0 y 3m1 y ỵ 3m2 y m3 ị=3 15ị f9 yị ẳ x1ị yịx2ị yịẵ0:5 yị4 ỵ yị4 =4 f10 yị ẳ xyịx2ị yịẵ0:5 yị ỵ ỵ yị =3 By using (1), (2), (14), (15) for numerically solving equation I (y) = in the interval (À1, 0.5) we nd its three roots: f11 yị ẳ x2ị yị2m1 y À m2 À m0 y Þ y ¼ À0:28100; 3 y ¼ À0:04788; y ¼ 0:10644: 80 P.C Vinh, P.G Malischewsky / Ultrasonics 45 (2006) 77–81 By using (12) and (13): À6 Iðy Þ ¼ 8:9 Â 10 ; À6 Iðy Þ ¼ 1:81 Â 10 ; À5 Iðy Þ ¼ 3:85 10 ; I1ị ẳ 0:00072; I0:5ị ẳ 0:0052: so y3 is the value which makes I(y) minimum in the interval [À1, 0.5] Substituting y = y3 = 0.10644 into (11) we have gmị ẳ hm; y ị ẳ 0:874027 ỵ 0:195608m 0:0425231m2 0:0569549m3 16ị It is shown from (10) and (16) that xm(m) and g(m) are almost totally identical with each other So Malischewsky’s approximate expression (10) can be considered as the best approximation of the Rayleigh wave velocity in the interval [À1, 0.5], in the sense of least squares, with respect to the compact set V shown above We have calculated the average error and the absolute error’s maximum for Malischewsky’s approximation Table Average error I* of the approximations in different intervals ½a; b : I ẳ p I=b aị, I defined by (6) Intervals of m Malischewsky’s I* Rahman–Michelitsch’s I* [À1, À0.5] [À0.5, 0] [0, 0.5] [À1, 0.5] 0.00239 0.00098 0.00025 0.0015 0.0012 0.0032 0.0020 0.0023 according to (10) and Rahman–Michelitsch’s one, respectively, for different subintervals of Poisson’s ratio and for the whole interval as well (see Tables and 2) It is obvious that Rahman–Michelitsch’s approximation is better for the interval [À1, À0.5] but worse for the other intervals Malischewsky’s is especially good for non-auxetic materials as already noted by Rahman and Michelitsch [9] For completeness, also the percentage error d = j1 À g(m)/x(m)j · 100% was calculated for both approximations and is presented as a function of Poisson’s ratio in Fig 1, here g(m) is the approximation of x(m) The method of Rahman and Michelitsch can be profitably applied in those cases where the exact solution is not available Conclusions By the least-square approach it is shown that Malischewsky’s approximation can be considered as the best approximation of the Rayleigh wave velocity in the interval [À1, 0.5], in the sense of least squares, with respect to the class of Taylor expansions of x(m) up to the third power at the values y [À1, 0.5] It should be noted that the least-square approach presented here is applicable for not only the case where the explicit exact formulas of the Rayleigh wave speed are known, but also the case in which the explicit exact formulas of the Rayleigh wave speed are not available This will be discussed separately elsewhere Acknowledgements Table Absolute error’s maximum bI of the approximations in dierent intervals: bI ẳ maxẵa;b jxmị À gðmÞj, g(m) is the approximation of x(m) Intervals of m Malischewsky’s bI Rahman–Michelitsch’s bI [À1, À0.5] [À0.5, 0] [0, 0.5] [À1, 0.5] 0.0031 0.0021 0.00094 0.0031 0.0023 0.0035 0.00325 0.0035 The work was done during the first author’s visit of two months to the Institute for Geosciences, FriedrichSchiller University Jena, which was supported by a DAAD Grant No A/05/58097 He is very grateful to the DAAD for the financial support Additionally, he thanks Prof Peter G Malischewsky and all colleagues of the Institute for Geosciences who helped to make his stay a success References 0.45 Percentage Error δ (%) 0.4 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 error d of Malischewsky’s (solid line) and Rahman– Michelitsch’s (dashed line) approximation, d = j1 À g(m)/x(m)j · 100%, g(m) is the approximation of x(m) [1] L Rayleigh, On waves propagated along the plane surface of an elastic solid, Proc R Soc Lond A17 (1885) 4–11 [2] L Bergmann, Ultrasonics and their Scientific and Technical Applications, John Wiley & Sons, New York, 1948 [3] P.G Malischewsky, Comparison of approximated solutions for the phase velocity of Rayleigh waves (Comment on ’Characterization of surface damage via surface acoustic waves’), Nanotechnology 16 (2005) 995–996 [4] P.G Malischewsky, Comment to ‘‘A new formula for the velocity of Rayleigh waves’’ by D Nkemzi [Wave Motion 26 (1997) 199–205], Wave Motion 31 (2000) 93–96 [5] P.G Malischewsky Auning, A note on Rayleigh-wave velocities as a function of the material parameters, Geofı´s Int 43 (2004) 507–509 [6] Pham Chi Vinh, R.W Ogden, On formulas for the Rayleigh wave speed, Wave Motion 39 (2004) 191–197 [7] I.G Roy, Iteratively adaptive regularization in inverse modeling with Bayesian outlook – application on geophysical data, Inverse Probl Sci Eng 13 (2005) 655–670 P.C Vinh, P.G Malischewsky / Ultrasonics 45 (2006) 77–81 [8] W Yang, Z.-M Li, W Shi, B-H Xie, M-B Yang, On auxetic materials, J Mater Sci 39 (2004) 3269–3279 [9] M Rahman, T Michelitsch, A note on the formula for the Rayleigh wave speed, Wave Motion 43 (2006) 272–276 81 [10] Gunter Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, Berlin/Heidelberg/New York, 1967 [11] C Lanczos, Applied Analysis, Prentice-Hall Inc., New Jersy, 1956 ... is applicable for not only the case where the explicit exact formulas of the Rayleigh wave speed are known, but also the case in which the explicit exact formulas of the Rayleigh wave speed are... that xm(m) and g(m) are almost totally identical with each other So Malischewsky’s approximate expression (10) can be considered as the best approximation of the Rayleigh wave velocity in the. .. mentioned, there is a need to obtain analytical approximate expressions of the Rayleigh wave speed x(m) for the practical work in the laboratory or elsewhere, which are reasonably simpler than the exact