Explanation for Malischewsky’s approximate expressionfor the Rayleigh wave velocity a Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str
Trang 1Explanation for Malischewsky’s approximate expression
for the Rayleigh wave velocity
a 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 m2 [1, 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
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
1 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
bof shear waves According to Lord Rayleigh[1]the
veloc-ity 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]
and Pham Chi Vinh and Ogden [6] The existence of an explicit formula for the Rayleigh wave velocity in a half-space 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 approxi-mate 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 (1 6 m 6 0.5) by expanding his exact for-mula into a Taylor series It turned out by trial and error that this expansion has to be carried out at a strange Pois-son 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
0041-624X/$ - see front matter 2006 Elsevier B.V All rights reserved.
doi:10.1016/j.ultras.2006.07.001
*
Corresponding author Tel.: +84 4 5532164; fax: +84 4 8588817.
E-mail addresses: pcvinh@vnu.edu.vn (P.C Vinh), p.mali@uni-jena.de
(P.G Malischewsky).
www.elsevier.com/locate/ultras
Trang 2a straightforward and mathematically well founded
proce-dure for obtaining the best result in all cases under
consid-eration We present here a simple approach created by the
least-square principle, which fulfils these requirements, and
demonstrate its application to Malischewsky’s
approxima-tion[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
for-mula is more complicated than Malischewsky’s one We
postpone a comparison of both formulas with the exact
solution to the end of paragraph 4
A paper with discussion of all existing approximations
in the light of the approach proposed here is in
preparation
2 Exact formula for the Rayleigh wave velocity in a
half-space
By using Malischewsky’s notation[5]we obtain the
fol-lowing expression for the Rayleigh wave velocity:
xðmÞ ¼ c=b ¼ ffiffiffiffiffiffiffiffi
ðmÞ
p
; ðmÞ ¼2
3 4 ffiffiffiffiffiffiffiffiffiffi
h3ðcÞ 3 p
þ2ð1 6cÞffiffiffiffiffiffiffiffiffiffi h3ðcÞ 3 p
ð1Þ where c = (1 2m)/2(1 m) = (b/a)2and with the auxiliary
functions:
33 186c þ 321c2 192c3
p
;
In formula(1), the main values of the cubic roots are to
be used
3 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
accu-rate enough This is, mathematically, related to the
approx-imation 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 f2 X determine an element g 2 V such that
here the symbol kuk denotes the norm of u 2 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 < 2 whenever
kuk = kwk = 1 and u 5 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], (1 6 a 6 b 6 0.5) which
consists of all functions measurable in (a, b), whose squared
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:
kuk ¼
Z b a
u2ðmÞdm
Then the problem (3)becomes:
Let V be a subset of L2[a, b] For a given function
f2 L2[a, b], determine a function g2 V such that
Z b a
½f ðmÞ gðmÞ2dm¼ min
h2V
Z b a
½f ðmÞ hðmÞ2dm ð5Þ The Eq.(5)expresses the principle of least squares The quantity ffiffiffiffiffiffiffiffiffi
IðhÞ
p where IðhÞ ¼
Z b 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(5)shows that the best approximation g(m) (if exists) makes the deviation functional(6)minimum The quantity I¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
IðgÞ=ðb aÞ
p
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 un-ique solution in the case that V is a finite dimensional sub-space 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 1 which is a lin-ear 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 lin-ear combination of h1(m), h2(m), , hn(m):
hðmÞ ¼Xn i¼1
Then the functional I(h) becomes a function of the n variables a1, a2, , an and problem(5) is leaded to a sys-tem 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:
hðm; yÞ ¼Xn
i¼1
where hi(m) are given elements of L2[a, b], ai(y) are pre-scribed 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-lin-ear in general):
The prime denotes here the first derivative
Trang 34 Mathematical basis of Malischewsky’s approximation
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
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
Tay-lor 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 y2 [1, 0.5], xm(m) deviates the least
from function x(m) in the interval [1, 0.5], it can be said
that among these polynomials, xm(m) is the best
approxima-tion of x(m) in the interval [1, 0.5] (in the sense of least
squares) That means that xm(m) is the solution of the
prob-lem(5) in which f(m) = x(m), a =1, b = 0.5 and V is a set
of elements h(m, y) having the form:
hðm; yÞ ¼ xðyÞ þx
1! ðm yÞ þx
2! ðm yÞ2
þx
in which y2 [1, 0.5] is considered as a parameter Here by
x(k)(y) we denote the derivative of order k of x(y) with
re-spect to y It is easy to observe that, in this case, V is a
com-pact subset of L2[1, 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
dif-ficult to verify that I(y) is a differentiable function of y in
the interval [1, 0.5], so it has a minimum in [1, 0.5] (this
is also observed by the fact that V is a compact subset of
L2[1, 0.5]) By using (1), (6) and (11)we have
IðyÞ ¼X15
i¼1
where
f3ðyÞ ¼ ½xð1ÞðyÞ2½ð0:5 yÞ3þ ð1 þ yÞ3=3
f5ðyÞ ¼ xð2ÞðyÞxð3ÞðyÞ½ð0:5 yÞ6 ð1 þ yÞ6=36
f9ðyÞ ¼ xð1ÞðyÞxð2ÞðyÞ½ð0:5 yÞ4 ð1 þ yÞ4=4
f11ðyÞ ¼ xð2ÞðyÞð2m1y m2 m0y2Þ
f12ðyÞ ¼ xðyÞxð1ÞðyÞ½ð0:5 yÞ2 ð1 þ yÞ2
mi¼
Z 0:5
1
mixðmÞdm; i ¼ 0; 1; 2; 3; m¼
Z 0:5
1
½xðmÞ2dm
In order to find the minimum of the function I(y) in the compact interval [1, 0.5] we have to find critical points of I(y) (the roots of equation I0(y) = 0) in the open interval (1, 0.5), then compare the values of I(y) at these values with I(1) and I(0.5) From(12) and (13)we have
I0ðyÞ ¼X14
i¼1
where
u1ðyÞ ¼ xð3ÞðyÞxð4ÞðyÞ½ð0:5 yÞ7þ ð1 þ yÞ7=126
½xð3ÞðyÞ2½ð0:5 yÞ6 ð1 þ yÞ6=36
u2ðyÞ ¼ xð2ÞðyÞxð3ÞðyÞ½ð0:5 yÞ5þ ð1 þ yÞ5=10
½xð2ÞðyÞ2½ð0:5 yÞ4 ð1 þ yÞ4=4
u3ðyÞ ¼ 2xð1ÞðyÞxð2ÞðyÞ½ð0:5 yÞ3þ ð1 þ yÞ3=3
½xð1ÞðyÞ2½ð0:5 yÞ2 ð1 þ yÞ2
u4ðyÞ ¼ 3xðyÞxð1ÞðyÞ
u5ðyÞ ¼ ½xð2ÞðyÞxð4ÞðyÞ þ ðxð3ÞðyÞÞ2½ð0:5 yÞ6 ð1 þ yÞ6=36
xð2ÞðyÞxð3ÞðyÞ½ð0:5 yÞ5þ ð1 þ yÞ5=6
u6ðyÞ ¼ ½xð1ÞðyÞxð4ÞðyÞ þ xð2ÞðyÞxð3ÞðyÞ½ð0:5 yÞ5þ ð1 þ yÞ5=15
xð1ÞðyÞxð3ÞðyÞ½ð0:5 yÞ4 ð1 þ yÞ4=3
u7ðyÞ ¼ ½xðyÞxð4ÞðyÞ þ xð1ÞðyÞxð3ÞðyÞ½ð0:5 yÞ4 ð1 þ yÞ4=12
xðyÞxð3ÞðyÞ½ð0:5 yÞ3þ ð1 þ yÞ3=3
u8ðyÞ ¼ xð4ÞðyÞðm0y3 3m1y2þ 3m2y m3Þ=3
þ xð3ÞðyÞð3m0y2 6m1yþ 3m2Þ=3
u9ðyÞ ¼ ½xð1ÞðyÞxð3ÞðyÞ þ ðxð2ÞðyÞÞ2½ð0:5 yÞ4 ð1 þ yÞ4=4
xð1ÞðyÞxð2ÞðyÞ½ð0:5 yÞ3þ ð1 þ yÞ3
u10ðyÞ ¼ ½xðyÞxð3ÞðyÞ þ xð1ÞðyÞxð2ÞðyÞ½ð0:5 yÞ3þ ð1 þ yÞ3=3
xðyÞxð2ÞðyÞ½ð0:5 yÞ2 ð1 þ yÞ2
u11ðyÞ ¼ xð3ÞðyÞð2m1y m2 m0y2Þ þ 2xð2ÞðyÞðm1 m0yÞ
u12ðyÞ ¼ ½xðyÞxð2ÞðyÞ þ ðxð1ÞðyÞÞ2½ð0:5 yÞ2 ð1 þ yÞ2
3xðyÞxð1ÞðyÞ
u13ðyÞ ¼ 2xð2ÞðyÞðm0y m1Þ þ 2m0xð1ÞðyÞ
u14ðyÞ ¼ 2m0xð1ÞðyÞ
ð15Þ
By using (1), (2), (14), (15) for numerically solving equation I0(y) = 0 in the interval (1, 0.5) we find its three roots:
y ¼ 0:28100; y ¼ 0:04788; y ¼ 0:10644:
Trang 4By using(12) and (13):
Iðy1Þ ¼ 8:9 106; Iðy2Þ ¼ 3:85 105;
Iðy3Þ ¼ 1:81 106; Ið1Þ ¼ 0:00072; Ið0:5Þ ¼ 0:0052:
so y3is the value which makes I(y) minimum in the interval
[1, 0.5] Substituting y = y3= 0.10644 into(11)we have
gðmÞ ¼ hðm; y3Þ
¼ 0:874027 þ 0:195608m 0:0425231m2 0:0569549m3
ð16Þ
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
[1, 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
according to (10)and Rahman–Michelitsch’s one, respec-tively, for different subintervals of Poisson’s ratio and for the whole interval as well (seeTables 1 and 2) It is obvious that Rahman–Michelitsch’s approximation is better for the interval [1, 0.5] but worse for the other intervals Malis-chewsky’s is especially good for non-auxetic materials as already noted by Rahman and Michelitsch[9]
For completeness, also the percentage error d = j1 g(m)/x(m)j · 100% was calculated for both approxima-tions 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
5 Conclusions
By the least-square approach it is shown that Malis-chewsky’s approximation can be considered as the best approximation of the Rayleigh wave velocity in the interval [1, 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 y2 [1, 0.5]
It should be noted that the least-square approach pre-sented 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 formu-las of the Rayleigh wave speed are not available This will
be discussed separately elsewhere
Acknowledgements The work was done 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 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
[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 Appli-cations, 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.
Table 1
Average error I ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *of the approximations in different intervals ½a; b : I ¼
I=ðb aÞ
p
, I defined by (6)
Intervals of m Malischewsky’s I* Rahman–Michelitsch’s I*
Table 2
Absolute error’s maximum b I of the approximations in different intervals:
bI ¼ max ½a;b jxðmÞ gðmÞj, g(m) is the approximation of x(m)
Intervals of m Malischewsky’s b I Rahman–Michelitsch’s b I
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Poisson’s Ratio ν
Fig 1 Percentage error d of Malischewsky’s (solid line) and Rahman–
Michelitsch’s (dashed line) approximation, d = j1 g(m)/x(m)j · 100%, g(m)
is the approximation of x(m).
Trang 5[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.
[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.