DSpace at VNU: Improved approximations of the Rayleigh wave velocity

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 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:

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

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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

 2 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 ,

 2 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 0:0432 0:0523 ,

 2 ẵ0, 0:5:

3ị

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


1
5 2 27 3
1 1
, :
4ị
xbg ị ẳ 1   ỵ  ,  2
2
8
16
12 4
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
,
1:14418 0:25771 ỵ 0:126612

 2 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]).

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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

 2 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 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, 0:5Š
(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],

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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 Malischewskys notation [12], the Rayleigh wave velocity is expressed by:
"
#
ffiffiffiffiffiffiffiffiffiffiffi 2ð1 6
ị
p
p
2
3
 , xị
 ẳ
,
4 h3
ị ỵ p
xị ẳ c=
ẳ xị
7ị
3
3
h3
ị
where
 2

1 2


ẳ
ẳ
,
21 ị


8ị

and with the auxiliary functions:
p
h1
ị ẳ 3 33 186
ỵ 321
2 192
3 ,

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]:
"

#
3 ỵ 4
3ị2
p
p
1
1
4
3
p
sị ẳ
ẳ s, s ẳ
2
ỵ V
ị ỵ
, 10ị
xị
41
ị
3
9 3 V
ị
where:
V
ị ẳ

2
27 90
ỵ 99
2 32

3 ị
27
p
2
ỵ p 1
ị 11 62
þ 107
2 À 64
3
3 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.

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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 2 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 min 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 1 defined as follows (see [21,22]):
hk ị ẳ Tk tịị ẳ cos ẵk arccos tị,

tị ẳ

k ẳ 0, n 1,

2 a b
,
ba

14ị

15ị

where  2 ẵa, b and t 2 ½À1, 1Š. 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

1


Z



fẵịd, ck ẳ
0

2


Z



fẵịcos k d,

k ẳ 1, n 1,

17ị

0

in which
ị ẳ

ba
aỵb
cos  ỵ
:
2
2

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ð18Þ


342

P. C. VINH AND P. G. MALISCHEWSKY

Noting that cos ðn ặ 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 ẳ t:

20ị

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 ¼ t, T2 ðtÞ ¼ 2t2 À 1,
ð21Þ
T3 ðtÞ ¼ 4t3 3t, T4 tị ẳ 8t4 8t2 ỵ 1:
Remark 1: It is obvious that if gi() is the best approximation of f() with
respect to Vi & L2 ½a, 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


 100%
I ¼ max

1
ẵa, b
xị

ð22Þ

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 ¼ 0:91701093855671, c1 ¼ 0:04074468783261,
23ị
c2 ẳ 0:00236434432626:

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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 tịị ẳ 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 1.
From Equations (2), (7), (22), and (25) it follows:
 n2 Þ ¼ 0:44%, Iðp
 2 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

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ν

Figure 1. 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()| Â 100%,
g() is an approximation of x().

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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 ¼ À1:045818847690321 Â 10À4 , T3 tịị ẳ 2563 1922 ỵ 36 1, ð27Þ
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 2.
In view of Equations (3), (7), (22), and (28) we have:
 n3 Þ ¼ 0:059%, Iðp
 3 Þ ¼ 3:1 Â 10À3 %:
Iðx

ð29Þ

0.06

Percentage error (%)

0.05

0.04

0.03

0.02

0.01

0
0

0.05

0.1

0.15

0.2

0.25
ν

0.3

0.35

0.4

0.45

0.5

Figure 2. 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()| Â 100%,
g() is an approximation of x().

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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 ¼ 2:602481661645599 Â 10À5 ,
T4 tịị ẳ 20484 20483 ỵ 6402 64 þ 1,

ð30Þ

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:
 4 Þ ¼ 5:2 Â 10À5 %:
Iðp

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

1


Z



xẵịd, ck ẳ
0

2


Z



xẵịcos k d,

k ẳ 1, n À 1,

0

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ð34Þ

346

P. C. VINH AND P. G. MALISCHEWSKY

in which
ị ẳ

1
1
cos  þ :
12
6

ð35Þ

Replacing  by  in Equation (15) and putting a ẳ 1/12, b ẳ 1/4 yield:
tị ẳ 12 2,  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 Â 10À4 :

ð38Þ

By using Equations (33), (37) and (38) we have:
p3 ị ẳ 1:00326 0:58141 0:012122 þ 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 3.
In view of Equations (4), (7), (22), and (39) we have:
 bg ị ẳ 1:35%, Ip
 3 ị ẳ 0:0043%:
Ix

40ị

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:
p2 ị ẳ 1:00733 0:666 ỵ 0:529272 :

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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

0.1

0.15

0.2

0.25

δ

Figure 3. 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()| Â 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).
 2 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

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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 2 ½0, 0:5Š.
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:
1
h, yị ẳ syị ỵ s1ị yị yị ỵ s2ị yị yị2 , , y 2 ẵ0, 0:5,
2

43ị

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

9
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 s2ị yị2m1 y m2 m0 y2 ị,
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,
0

i ẳ 0, 1, 2,

m ẳ

0:5

ẵsị2 d:

46ị

0

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)

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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 ẳ 0:21907305699270,
47ị
y3 ẳ 0:29539595383846:
It follows from Equations (8), (10), (11), and (44)(47):
I0ị ẳ 9:2 107 , Iy1 ị ¼ 6:8 Â 10À10 , Iðy2 Þ ¼ 4:7 Â 109 ,
Iy3 ị ẳ 1:9 109 , I0:5ị ẳ 1:2 Â 10À6 :

ð48Þ

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:
q2Ã ị ẳ 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 4 is
demonstrated.
Now we find approximations of Sinclair’s form with higher accuracy.
Because 05xị518 2 ẵ0, 0:5, it follows that sị ẳ 1=xị418
 2 ẵ0, 0:5. This leads to:






x 1

5
s g
8g 2 L2 ẵ0, 0:5 : gị41 8 2 ẵ0, 0:5:


g

50ị

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 P3Ã & 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.

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