DSpace at VNU: Improved approximations of the Rayleigh wave velocity

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of the Rayleigh Wave Velocity

PHAMCHIVINH* Faculty of Mathematics, Mechanics and Informatics

Hanoi University of Science, Thanh Xuan, Hanoi, Vietnam

PETERG 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 4 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

ELASTIC SURFACE WAVESin 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:

xbðÞ ¼0:87 þ 1:12

In the form of the second-order polynomial, the approximate formula:

xn ðÞ ¼0:874 þ 0:198  0:0542,  2 ½0, 0:5
, ð2Þ given by Nesvijski [3], while in form of the third-order polynomial he proposed the following approximation [3]:

xn ðÞ ¼0:874 þ 0:196  0:04320:0523,  2 ½0, 0:5
: ð3Þ

In terms of the parameter  ¼ (1 – )/4(1 þ ), Brekhovskikh and Godin [4] established the approximate expression:

xbgðÞ ¼1 1

2 

5

8

16

12,

1 4

In form of the inverse of a polynomial of the second order, Sinclair developed the following approximate formula (see [5,6]):

1:14418  0:25771 þ 0:126612,  2 ½0, 0:5
: ð5Þ

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])

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 unambigu-ously 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 approxima-tion (5), we establish improved approximaapproxima-tions 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

hðÞ ¼a þ b

where a, b are constants It is noted that V is a linear subspace of L2[0, 0.5] which has dimension 2 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 2 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],

while analytical approximate expressions of xðÞ ¼ ffiffiffiffiffiffiffiffiffi

xðÞ

p started appearing

in the literature long ago

In Malischewsky’s notation [12], the Rayleigh wave velocity is expressed by: xðÞ ¼ c=
¼ ffiffiffiffiffiffiffiffiffi

xðÞ

p ,
xðÞ ¼2

3 4 

ffiffiffiffiffiffiffiffiffiffiffi

h3ðÞ

3

p

þ2ð1  6Þ ffiffiffiffiffiffiffiffiffiffiffi

h3ðÞ

3

p

where

 ¼ 1  2

2ð1  Þ¼



 2

and with the auxiliary functions:

h1ðÞ ¼3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

33  186 þ 32121923

p

, h3ðÞ ¼17  45 þ h1ðÞ: ð9Þ 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]:

sðÞ ¼ 1

xðÞ¼

ffiffi s

p

4ð1  Þ 2 

4

3 þ

ffiffiffiffiffiffiffiffiffiffi VðÞ

3

p

þ3 þ ð4  3Þ2

9 ffiffiffiffiffiffiffiffiffiffi VðÞ

3

p

, ð10Þ

where:

VðÞ ¼ 2

27ð27  90 þ 99

2323Þ

3 ffiffiffi 3

p ð1  Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

11  62 þ 1072643

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

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 2 V is called the best approximation of f with respect to V, in the sense of least squares, if it satisfies the equation

Zb a

½fðÞ  gðÞ
2d ¼ min

where

IðhÞ ¼

Zb 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  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

TkðtðÞÞ, k ¼ 0, n  1 defined as follows (see [21,22]):

hkðÞ ¼ TkðtðÞÞ ¼cos ½k arccos tðÞ
, k ¼0, n  1, ð14Þ

tðÞ ¼2  a  b

where  2 ½a, b
and t 2 ½1, 1
In this case, the best approximate polynomial of f() with respect to Pn, in the interval [a, b] is (see [22]):

pn1ðÞ ¼Xn1

k¼0

where:

c0¼1



Z

0

f½ðÞ
d, ck¼2



Z  0

f½ðÞ
cos k d, k ¼1, n  1, ð17Þ

in which

ðÞ ¼b  a

2 cos  þ

a þ b

Noting that cos ðn  1Þ ¼ cos n  cos   sin  sin, from Equation (14) the recursion formula is deduced for the Chebyshev polynomials:

Tkþ1ðtÞ ¼2tTkðtÞ  Tk1ðtÞ, ð19Þ starting with:

Applying successively formula (19) and taking into account starting condition (20), the first five Chebyshev polynomials are (see also [21]):

T0ðtÞ ¼1, T1ðtÞ ¼ t, T2ðtÞ ¼2t21,

T3ðtÞ ¼4t33t, T4ðtÞ ¼8t48t2þ1:

ð21Þ

Remark 1: It is obvious that if gi() is the best approximation of f() with respect to ViL2½a, b
; ði ¼ 1, 2Þ 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., Iðg2Þ Iðg1Þ

In order to evaluate an approximation’s accuracy we use the maximum percentage (relative) error I defined as follows:

I ¼max

½a, b
1 gðÞ

xðÞ



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 xn(), 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:

c0¼0:91701093855671, c1¼0:04074468783261,

c2¼ 0:00236434432626:

ð23Þ

From Equation (15) with a ¼ 0, b ¼ 0.5 and Equation (21), it follows:

T0ðtðÞÞ ¼1, T1ðtðÞÞ ¼4  1, T2ðtðÞÞ ¼32216 þ 1: ð24Þ Substituting Equations (23) and (24) into Equation (16) leads to:

p2ðÞ ¼0:8739 þ 0:2008  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 P3is the polynomial p2() given by formula (25)

It is clear, from formulas (2) and (25), that Nesvijski’s approximation xn()

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 1

From Equations (2), (7), (22), and (25) it follows:

Iðxn Þ ¼0:44%,
Iðp2Þ ¼0:015%: ð26Þ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

ν

Figure 1 Percentage errors of approximations: x n2 () (dash-dot line), p 2 () (solid line), p 4 () (dashed line: almost coincides with the -axis) Percentage error ¼ |1  g()/x()|  100%, g() is an approximation of x().

i.e., the maximum percentage error of p2() is 29 times smaller than that

of xn()

Analogously as above, and taking into account

c3¼ 1:045818847690321  104, T3ðtðÞÞ ¼25631922þ36  1, ð27Þ

it is deduced from Equations (16), (23), (24), and (27):

p3ðÞ ¼0:874006 þ 0:19704  0:0555820:026773, 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 P4is the polynomial p3() defined by formula (28)

From Equations (3) and (28) it is obvious that xn () is not the best approximation of x() in the interval [0, 0.5], with respect to P4 This fact can

be observed by Figure 2

In view of Equations (3), (7), (22), and (28) we have:

Iðxn Þ ¼0:059%,
Iðp3Þ ¼3:1  103%: ð29Þ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

0.01

0.02

0.03

0.04

0.05

0.06

ν Figure 2 Percentage errors of approximations: x n3 (dash-dot line), p 3 (solid line), p 4

(dashed line: almost coincides with the -axis) Percentage error ¼ |1  g()/x()|  100%, g() is an approximation of x().

This says that the maximum percentage error of p3() is 19 times smaller than that of xn()

Analogously, by using the fact

c4¼2:602481661645599  105,

T4ðtðÞÞ ¼2048420483þ640264 þ 1,

ð30Þ

it follows from Equations (16), (23), (24), (27) and (30):

p4ðÞ ¼0:05329940:08007230:0389232þ0:1953777 þ 0:8740325:

ð31Þ Using Equations (7), (22), and (31) we have:

From Equations (26), (29), and (32) it follows that the approximation p4()

is 8461 (1134) times better than xn () (xn ()) in the sense of maximum percentage error

IMPROVED BREKHOVSKIKH–GODIN’S APPROXIMATIONS Considering the Rayleigh wave velocity as a function of  ¼ (1  )/ 4(1 þ ), 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  1) th order polynomial of x() with respect to Pn, in the interval [1/12, 1/4] is:

p ðn1ÞðÞ ¼Xn1

k¼0

where:

c0¼1



Z

0

x½ðÞ
d, ck¼2



Z  0

x½ðÞ
cos k d, k ¼1, n  1, ð34Þ

in which

ðÞ ¼ 1

12cos  þ

1

Replacing  by  in Equation (15) and putting a ¼ 1/12, b ¼ 1/4 yield:

Employing Equations (21), (36) leads to:

T0ðtðÞÞ ¼1, T1ðtðÞÞ ¼12  2, T2ðtðÞÞ ¼288296 þ 7,

It follows from Equations (34) and (35):

c0¼0:91287085775639, c1¼ 0:04079848265769,

c2¼0:00183775883442, c3¼1:566525867870697  104: ð38Þ

By using Equations (33), (37) and (38) we have:

p 3ðÞ ¼1:00326  0:58141  0:012122þ1:082783: 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 3

In view of Equations (4), (7), (22), and (39) we have:

IðxbgÞ ¼1:35%,
Iðp 3Þ ¼0:0043%: 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 3 and using Equations (37) and (38) we obtain the best approximate second-order polynomial of x(), namely:

p 2ðÞ ¼1:00733  0:666 þ 0:529272: ð41Þ