DSpace at VNU: An approach for obtaining approximate formulas for the Rayleigh wave velocity

Wave Motion 44 (2007) 549–562
www.elsevier.com/locate/wavemoti

An approach for obtaining approximate formulas
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, Vietnam
b
Institute for Geosciences, Friedrich-Schiller University Jena, Burgweg 11, 07749 Jena, Germany
Received 25 July 2006; received in revised form 1 February 2007; accepted 6 February 2007
Available online 17 February 2007

Abstract
In this paper, we introduce an approach for finding analytical approximate formulas for the Rayleigh wave velocity for
isotropic elastic solids and anisotropic elastic media as well. The approach is based on the least-square principle. To demonstrate its application, we applied it in order to obtain an explanation for Bergmann’s approximation, the earliest known
approximation of the Rayleigh wave velocity for isotropic elastic solids, and used it to establish a new approximation. By
employing this approach, the best approximate polynomials of the second order of the cubic power and the quartic power
in the interval [0, 1] were found. By using the best approximate polynomial of the second order of the cubic power, we
derived an approximate formula for the Rayleigh wave speed in isotropic elastic solids which is slightly better than the
one given recently by Rahman and Michelitsch by employing Lanczos’s approximation. Also by using this second order
polynomial, analytical approximate expressions for orthotropic, incompressible and compressible elastic solids were found.
For incompressible case, it is shown that the approximation is comparable with Rahman and Michelitsch’s approximation,
while for the compressible case, it is shown that our approximate formulas are more accurate than Mozhaev’s ones.

Remarkably, by using the best approximate polynomials of the second order of the cubic power and the quartic power
in the interval [0, 1], we derived an approximate formula of the Rayleigh wave velocity in incompressible monoclinic materials, where the explicit exact formulas of the Rayleigh wave velocity so far are not available.
Ó 2007 Elsevier B.V. All rights reserved.
Keywords: Rayleigh waves; Rayleigh wave velocity; Rayleigh wave speed; Approach of least squares; The best approximation;
Approximate formula; Approximate expression

1. Introduction
Elastic surface waves (i.e. Rayleigh waves), first studied by Rayleigh [5] more than a century ago (in 1885),
have been intensively studied and exploited, due to wide applications in seismology, acoustics, geophysics,
materials science, nondestructive testing, telecommunication industry and so on.

*

Corresponding author. Tel.: +84 4 5532164; fax:+84 4 8588817.
E-mail addresses: (P.C. Vinh), (P.G. Malischewsky).

0165-2125/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.wavemoti.2007.02.001


550

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

For the Rayleigh wave, its velocity is a fundamental quantity which interests researchers in seismology, geophysics, and in other fields of physics and the material sciences. It is discussed in almost every survey and
monograph on the subject of surface acoustic waves in solids.
Due to the significance of the Rayleigh wave velocity in practical applications, researchers have attempted
to find its analytical approximate expressions which are of simple forms and accurate enough for practical
purposes. That is why a lot of approximations of the Rayleigh wave speed appeared in the literature (see,
e.g. [3,4,6]). However, as indicated by Mozhaev [4], most of them were reported without any indication of

the derivation procedure. So it is interesting to have a method which can provide the explanations for these
approximations.
In this paper, we present an approach originating from the principle of least squares (see, e.g. [1]) for finding
analytical approximate expressions for the Rayleigh wave speed. This approach can give the derivation of previously proposed approximate formulas and establish new approximate formulas as well. As a first application
of this procedure let us present an explanation of Bergmann’s approximation, the earliest approximate expression of the Rayleigh wave speed in isotropic elastic solids.
In order to create new approximations we can start either from the explicit exact formulas for the Rayleigh
wave speed, or from the secular equations of the Rayleigh waves. For the first possibility, as an example, by
using our approach we derived an approximate formula for the form of the third order polynomial, of the Rayleigh wave speed in isotropic elastic solids for the range [À1, 0.5] of Poisson’s ratio m. It is shown that this result
is a good approximation. For the second possibility, by replacing the power z3 in the secular equations by its
best approximate second order polynomial in the interval [0, 1] established by our approach, we have obtained:
(i) An approximate formula of the Rayleigh wave speed in isotropic elastic solids and it is shown that this
approximation is slightly more accurate than that given recently by Rahman and Michelitsch [3].
(ii) An approximate expression of the Rayleigh wave speed in incompressible orthotropic elastic solids
which is comparable with Rahman and Michelitsch’s [3].
(iii) Approximate formulas of the Rayleigh wave velocity in compressible orthotropic elastic media and they
are more accurate than those proposed by Mozhaev [4].
(iv) Remarkably, by using the best approximate (in the sense of least squares) second order polynomials of
z3 and z4 in the interval [0, 1], we derived an approximate formula of the Rayleigh wave speed for the
materials with more complicated symmetry, namely the incompressible monoclinic materials with the
plane of symmetry at x3 = 0, where the explicit exact formulas of the Rayleigh wave velocity so far
are not available.
It is noted, that only recently explicit exact formulas of the Rayleigh wave speed have been published for
isotropic compressible elastic solids (see [7–9,11]), for incompressible orthotropic elastic materials (see [10])
and for compressible orthotropic elastic ones (see [12,13]).
2. Least-square approach
As mentioned, there is a need for obtaining analytical approximate expressions of the Rayleigh-wave speed
for the practical work in the laboratory or elsewhere. These should be more simple than the exact one and
sufficiently accurate. 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 2 X determine an element g 2 V such

that:
kf À gk 6 kf À hk

for all h 2 V ;

ð1Þ

here the symbol kuk denotes the norm of u 2 X. If the problem (1) 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 (1) has a solution (see, e.g. [14]). Moreover, if X is strictly convex (i.e. ku ỵ wk < 2 whenever
kuk ¼ kwk ¼ 1 and u 5 w) and V is a finite dimensional linear subspace of X, then problem (1) has precisely
one solution (see, e.g. [14]).


P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

551

The most applicable cases are the cases in which X = L2[a, b] or X = C[a, b], where L2[a, b] consists of all
functions measurable in (a, b), whose squared value is integrable on [a, b] in the sense of Lebesgue, and
C[a, b] contains continuous functions in [a, b]. We recall that both L2[a, b] and C[a, b] are normed linear spaces,
whose norms are dened, respectively, as follows:
Z b
1=2
2
kuk ẳ
u mị dm
; u 2 L2 ẵa; b;
2ị
a

and
kuk ẳ max jumịj;
m2ẵa;b

u 2 C½a; bŠ:

In the present paper, we confine ourselves to the case of X = L2[a, b]. Then the problem (1) becomes:
Let V be a subset of L2[a, b]. For a given function f 2 L2[a, b], determine a function g 2 V such that:
Z b
Z b
ẵf mị gmị2 dm ẳ min
ẵf mị hmị2 dm:
3ị
h2V

a

a

The Eq. (3) expresses the principle of least squares. The quantity
Z b
2
Ihị ẳ
ẵf mị hðmފ dm; h 2 V ;

pffiffiffiffiffiffiffiffiffi
IðhÞ where:
ð4Þ

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 (3) shows
that the best approximation g(m) (if exists) makes the deviation funcpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tional (4) minimum. The quantity IðhÞ=ðb À aÞ is called the average error of the approximate solution of the
problem (3). It is noted that L2[a, b] is a Hilbert space, so it is strictly convex (see [14]). Thus, the problem (3)
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 form. Since polynomials are considered as the simplest functions, V is normally taken as the set of polynomials of order not bigger than n À 1 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 (3) we represent h(m) as a linear combination of h1(m), h2(m), . . . , hn(m):
n
X
hmị ẳ
ai hi mị:
5ị
iẳ1

Then the functional I(h) becomes a function of the n variables a1, a2, . . . , an and problem (3) 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
hðm; yị ẳ
ai yịhi mị; m; y 2 ẵa; b;
6ị
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 (3) leads to solving the equation (non-linear in general):

I 0 yị ẳ 0;

y 2 a; bị:

7ị

The prime denotes here the first derivative.
3. Application of the least-square approach
3.1. Mathematical basis of Bergmann’s approximation
2
Let c be the Rayleigh wave speed in compressible isotropic elastic solids and x ẳ c=bị , where b is the
velocity of shear waves. Then, it is well known that x is a solution of the equation (see, for instance, [5,11]):


552

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

x3 8x2 ỵ 83 2cịx 161 cị ¼ 0;

ð8Þ

which satisfies:
ð9Þ

0 < x < 1;
where:
c ¼ ð1 À 2mị=21 mị ẳ b=aị2 ;

10ị

and a is the velocity of longitudinal waves, m is Poisson’s ratio. It is also noted that in the range (9) Eq. (8) has
precisely one real solution.
By Malischewsky [9] xðmÞ is expressed by the following formula:
"
#
21 6cị
p
2
3
;
xmị ẳ
4 h3 cị ỵ p
11ị
3
3
h3 cị
where:
h1 cị ẳ 3

p
33 186c ỵ 321c2 192c3 ;

h3 cị ẳ 17 45c ỵ h1 cị:

12ị

In formula (11), the main values of the cubic roots are to be used.
Interestingly, only recently a convenient and simple form for xðmÞ, the derivation of which is not trivial, has
been published by Malischewsky

[8,9] and Pham Chi Vinh and Ogden [11], while analytical approximate
p
expressions of xmị ẳ xmị started appearing in the literature long ago. As we know, the earliest and wellknown approximate expression of x(m) was proposed by Bergmann [2] without explanation, and it has the
form:
xb mị ẳ

0:87 ỵ 1:12m
;
1ỵm

m 2 ẵ0; 0:5:

13ị

Now, by using the approach of least squares, we prove that xb(m) is the best approximation with respect to V
whose elements have the form:
a ỵ bm
;
1ỵm

hmị ẳ

m 2 ½0; 0:5Š;

ð14Þ

where a and b are constants. It is noted that V is a linear subspace of L2[0, 0.5] which has dimension 2 with a
basis, for example, h1(m), h2(m) as follows:
h1 mị ẳ

1
;
1ỵm

h2 mị ẳ

m
; m 2 ẵ0; 0:5:
1ỵm

As mentioned above, in this case problem (3) has unique solution. In order to find this solution we substitute
(14) into (4). The functional I(h) then becomes a function of two variables a, b, denoted by I(a, b), which is of
the form:
Iða; bÞ ẳ m0 a2 ỵ 2m1 ab ỵ m2 b2 2m3 a 2m4 b ỵ m5 ;

15ị

where:
mi ẳ

Z
0

0:5

mi dm
1 ỵ mị

2

; i ẳ 0; 1; 2;

m3ỵi ẳ

Z

0:5
0

xmịmi dm
1 ỵ mị

2

; i ẳ 0; 1;

m5 ẳ

Z

0:5

x2 mị dm:

From the conditions: oI/oa = 0, oI/ob = 0, we obtain two linear equations:

m0 a ỵ m1 b ẳ m3 ;
m1 a ỵ m2 b ẳ m4 :
Using (10)(12) and noting that xmị ¼
m0 ¼ 0:333333;

m1 ¼ 0:0721318;

ð16Þ

0

ð17Þ

pffiffiffiffiffiffiffiffi
xðmÞ, from (16) we have:
m2 ¼ 0:0224031;

m3 ẳ 0:371013;

m4 ẳ 0:0878859:

18ị


P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

553

It is easy to verify that the system (17) with coefficients defined by (18) has precisely one solution, namely:
a ¼ 0:87096;

b ¼ 1:11869:

ð19Þ

So, the best approximation of x(m) with respect to V (in the sense of least squares) in the interval [0, 0.5] is:
gb mị ẳ

0:87096 ỵ 1:11869m
;
1ỵm

m 2 ẵ0; 0:5:

20ị

It is clear from (13) and (20) that, xb(m) and gb(m) are almost totally identical with each other, so we can say
that Bergmann’s approximation is the best approximation of x(m) (in the sense of least squares), in the interval
[0, 0.5], with respect to the class of all functions expressed by (14). From (4), (13) and (20) it is shown that the
squared deviation of xb(m) and gb(m) from x(m) are 7.5 · 10À7 and 5.85 · 10À7, respectively.
Remark 1. The approximation of Bergmann is very good for positive values of Poisson’s ratio, but completely
fails for negative values. Materials with negative values of Poisson’s ratio, so-called auxetic materials, really
exist (see, e.g. a new review by Yang et al. [15]) and become increasingly interesting in material sciences. This
motivated Malischewsky [6] to find an approximation
pffiffiffiffiffiffiffiffithat is good throughout the whole range of physically
possible values of Poisson’s ratio. By expanding xðmÞ (xðmÞ is defined by (11)) at a certain value of Poisson’s
ratio, which was found by trial and error, he has found the following approximation:
xm ðmÞ ẳ 0:874 ỵ 0:196m 0:043m2 0:055m3 ;

m 2 ½À1; 0:5Š:

ð21Þ

An explanation for his result, which originates from the least-square approach, was given by Pham Chi Vinh

and Malischewsky [16]. It was proven that the approximation (21) is almost totally identical with the best
approximation of x(m) (in the sense of least squares), in the interval [À1, 0.5], with respect to the class of Taylor
expansions of x(m) up to the third power at the values belong to the interval [À1, 0.5]. In this case, the set V
contains functions:
hm; yị ẳ xyị þ

xð1Þ ðyÞ
xð2Þ ðyÞ
xð3Þ ðyÞ
2
3
ðm À yÞ þ
ðm À yÞ þ
ðm À yÞ ;
1!
2!
3!

ð22Þ

in which y 2 [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].
3.2. A new approximation for the Rayleigh wave speed
To show the effectiveness of the least-square approach, in this subsection we give a new approximation of
x(m) on the interval [À1, 0.5] in the form of a polynomial of the third order. For this purpose, naturally we
choose V as the set of all polynomials of order not bigger than 3:
hmị ẳ am3 ỵ bm2 ỵ cm ỵ d;

23ị

2

where a, b, c, d are constants. In this case, V is a four-dimensional linear subspace of L [À1, 0.5], so problem
(3) has precisely one solution. In order to find this solution, analogously as above, we substitute (23) into (4).
The functional (4) then is converted to a function of four variables: a, b, c, d, denoted by I(a, b, c, d). From the
condition:
oI=oa ¼ 0;

oI=ob ¼ 0;

oI=oc ¼ 0;

oI=od ¼ 0;

we obtain the following system of linear equations:
8
2=7ị27 ỵ 1ịa ỵ 1=3ị26 1ịb ỵ 2=5ị25 ỵ 1ịc ỵ 1=2ị24 1ịd ẳ 0:33139;
>
>
>
< 1=3ị26 1ịa ỵ 2=5ị25 ỵ 1ịb ỵ 1=2ị24 1ịc ỵ 2=3ị23 ỵ 1ịd ẳ 0:56514;
>
2=5ị25 ỵ 1ịa ỵ 1=2ị24 1ịb ỵ 2=3ị23 ỵ 1ịc ỵ 22 1ịd ẳ 0:51276;
>
>
:
1=2ị24 1ịa ỵ 2=3ị23 ỵ 1ịb ỵ 22 1ịc ỵ 3d ẳ 2:47152:
It is not difficult to verify that the solution of system (25) is:

ð24Þ

ð25Þ


554

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

a ¼ À0:0439059;

b ¼ À0:0350168;

c ¼ 0:192422;

d ¼ 0:87384:

ð26Þ

Thus, the best approximation of x(m) in this case is:
g3 mị ẳ 0:87384 ỵ 0:192422m 0:0350168m2 0:0439059m3 ;

m 2 ẵ1; 0:5:

6

27ị
7

Substituting (21) and (27) into (4) leads to: I(xm) = 3.3 · 10 and I(g3) = 2.53367 · 10 . That means the
squared distance between xm(m) and x(m) in the space L2[À1, 0.5] is 3.3 · 10À6, while the one between g3(m)

and x(m) is 2.53367 · 10À7, i.e. g3(m) approximates x(m) better than does xm(m) (in the sense of least squares).
It is evidence because the class of Taylor expansions of x(m) up to the third power at the values which belong
to the interval [À1, 0.5] is a subset of the set of all polynomials of order not bigger than 3.
Fig. 1 shows plots of x(m) and its approximation g3(m) defined by (27) in the interval [À1, 0.5]. It is very difficult to distinguish one from the other.
It is clear from Fig. 2, that the approximation g3(m) is more accurate than Malischewsky’s xm(m) and Rahman and Michelitsch’s published recently in [3] (formula (7)), and xm(m) is extraordinary good in the range
about (0, 0.4) which is very important for geophysical applications.
Remark 2. Analogously, we can obtain the best approximation gn(m) of x(m) in the interval [À1, 0.5] with
respect to Pn+1, n = 4, 5, . . . , by using the least-square approach, where by Pn we signify the set of all
polynomials of order not bigger than n À 1. As Pn & Pn+1 " n P 1, it is clear that "n P 1, gn+1(m)
approximates x(m) better than does gn(m), in the sense of least squares.

3.3. The best approximate second order polynomials of the powers z3 and z4 in the interval [0, 1]
In this subsection, first we want to find a second order polynomial which is the best approximation of the
power z3 with respect to P3, in the interval [0, 1] by using the method of least squares. In this case, we consider
problem (1) in which X = L2[0, 1], V = P3 and a = 0, b = 1, f = z3. An element h(z) of P3 is expressed as
follows:
hzị ẳ az2 ỵ bz ỵ c;

28ị

where a, b, c are constants. In order to find constants a, b, c corresponding to the best approximation, we substitute (28) into (4). The functional I(h) then becomes a function I(a, b, c) given by:
1

0.95

0.9

x=c/β

0.85

0.8

0.75

0.7

0.65
—1

—0.5

0

0.5

Poisson’s Ratio ν

Fig. 1. Plots of x(m) (solid line) and its approximation g3(m) (dotted line) in the interval [À1, 0.5].


P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

555

0.45
0.4

Percentage Error (\%)

0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
—1

—0.5

0

0.5

Poisson’s Ratio ν

Fig. 2. Percentage errors of g3(m) (solid line), xm(m) (dashed line) and Rahman–Michelitsch’s approximation (dash-dot line) in the interval
[À1, 0.5]. Percentage error = |1 À g(m)/x(m)| · 100%, g(m) is an approximation of x(m).

Ia; b; cị ẳ

a2 b2
ab 2ac
a 2b c 1
ỵ bc ỵ :
ỵ ỵ c2 ỵ ỵ
2
3

3 5 2 7
5
3

29ị

From the condition:
oI=oa ẳ 0;

oI=ob ẳ 0;

@I=@c ẳ 0;

it follows:
8
>
< 2=5ịa ỵ 1=2ịb ỵ 2=3ịc ẳ 1=3;
1=2ịa ỵ 2=3ịb ỵ c ẳ 2=5;
>
:
2=3ịa ỵ b ỵ 2c ẳ 1=2:

30ị

31ị

It is easy to verify that the system (31) has an unique solution:
a ¼ 1; 5;

b ẳ 0:6;

c ẳ 0:05:

32ị

Hence, the desired second order polynomial is:
pzị ẳ 1:5z2 0:6z ỵ 0:05:

33ị

Analogously, we can nd the best approximation second order polynomial of the power zn with respect to P3,
in the interval [0, 1], in the sense of least squares. For example, the best approximation second order polynomial of the power z4 is:

pzị ẳ

12 2 32
3
z zỵ :
7
35
35

34ị

Remark 3. Consider the problem (1) for the case: X = C[0, 1], V = P3, a = 0, b = 1 and f = z3. We call it the
problem (1*). The problem (1*) has a unique solution although C[0, 1] is not strictly convex (see, for instance
[14,17]). Now we show that the unique solution of (1*) is:
pà zị ẳ 1:5z2 0:5625z ỵ 0:03125:
First, we observe that: among all polynomials q(z) of the nth degree whose leading coefficient is unity:

ð35Þ


556

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

qðzÞ ẳ zn ỵ an1 zn1 ỵ ỵ a1 z ỵ a0 ;

36ị

n1

the Chebysev polynomial Tn(z)/2
(see [1]) deviates the least from zero in C[À1, 1].
Indeed, suppose there exists a polynomial q0(z) of the form (36) such that:





T n ðzÞ

1
max jq0 ðzÞj < max

nÀ1

ẳ n1 :
z2ẵ1;1

z2ẵ1;1 2
2

37ị

Then G(z) = Tn(z)/2n1 q0(z) is a polynomial of degree not exceeding n À 1, and:
Gzi ị ẳ ặ

1
2

n1

q0 zi ị;

38ị

where zi ẳ cosip=nị; i ¼ 0; n. From (38) it follows that G(z) has n zeroes, so G(z)  0, i.e. q0(z)  Tn(z)/2nÀ1.
But this contradicts (37), and the observation is proven. By the transformation z = 2t À 1, this observation
leads to the conclusion: among all polynomials of the nth order whose leading coefficient is unity, the shifted
Chebysev polynomial T Ãn ðzÞ=22nÀ1 (see [1]) deviates the least from zero in C[0, 1], where T n zị ẳ T n 2z 1ị.
That means the following proposition is valid: the polynomial pnÀ1 ðzÞ ¼ zn À T Ãn ðzÞ=22nÀ1 deviates the least
from zn in C[0, 1]. Applying the proposition for n = 3 we have p2(z) = p*(z) and (35) is demonstrated.
It is noted that Rahman and Michelitsch [3] called p*(z) Lanczos’s approximation.
3.4. An approximation for the Rayleigh wave speed in compressible isotropic elastic solids
Now, following Rahman and Michelitsch [3], we use the approximation (33) of z3 in order to obtain a
approximate formula of the Rayleigh wave speed in compressible isotropic elastic media.
By replacing x3 by pðxÞ, the cubic Eq. (8) is reduced to the following quadratic:
6:5x2 À ð23:4 À 16cÞx À ð16c 15:95ị ẳ 0;

39ị

whose solution satisfying (9) is:
q
2
23:4 16c 23:4 16cị ỵ 2616c 15:95ị
x ẳ
:
13

40ị
p
Using (10), from (40) we obtain the following approximate formula of xðmÞ ¼ xðmÞ in term of Poisson’s ratio:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
15:4 À 7:4m À 56:06m2 22:52m ỵ 30:46
x ẳ x ẳ
; m 2 ½À1; 0:5Š;
ð41Þ
13ð1 À mÞ

and the velocity of the Rayleigh waves is c = bx*.
Fig. 3 shows that approximate expression (41) is slightly more accurate than that obtained recently by Rahman and Michelitsch [3] using Lanczos’s approximation (35), for the whole interval [À1, 0.5].
By using the interval [0.474572, 0.912622], instead of the one [0, 1], analogously as above, we obtain the following approximation:
s
p
29:171 13:171m 203:188m2 70:194m ỵ 123
xvm2 ẳ
; m 2 ẵ1; 0:5;

23:6771 mị
which is more accurate than x* (see Fig. 3).
Remark 4.
(i) By using the interval [0.763932, 0.912622], instead of the one [0, 1], quite analogously, we obtain the following approximation for the range m 2 [0, 0.5]:
s
p
27:7904 11:7904m 190:467m2 56:1246m ỵ 121:6576
xvm1 ẳ
; m 2 ẵ0; 0:5:
21:940671 mị


P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

4

557

x 10 —3

3.5

Absolute Error

3
2.5
2
1.5
1
0.5

0
—1

—0.5

0

0.5

Poisson Ratio ν

Fig. 3. Absolute errors of the approximation x* (solid line), xvm2 (dash-dot line) and Rahman–Michelitsch’s approximation given by (7) in
[3] (dashed line). Absolute error = |x(m) À g(m)|, g(m) is an approximation of x(m).

(ii) Independently, Li [21] has found the approximation (41), also using (33).
(iii) By using the best approximate second order polynomial of z3 in the interval [0.47, 1] ([0.75, 1]) in the
sense of least squares, Li [21] has established an approximation for the range m 2 [À1, 0.5]
(m 2 [0, 0.5]), defined by (20) ((22)) in [21], whose percentage error does not exceed 0.16% (0.004%).
Because the maximum of the percentage error of xvm2, xvm1 is 0.09%, 0.0019%, respectively, it is shown
that the approximation xvm1, xvm2 is better than corresponding Li’s one.
3.5. An approximation for the Rayleigh wave speed in incompressible orthotropic elastic solids
For incompressible orthotropic elastic solids, the secular equation of the Rayleigh waves is of the form (see,
for instance [10]):
g3 ỵ g2 ỵ  1ịg 1 ẳ 0;

42ị

0 < g < 1;

where:

gẳ

p
p
1 x2 ; 0 < x < 1ị; x ẳ c= q=c66 ;

ẳ

c11 ỵ c22 2c12
> 0:
c66

43ị

Here c11, c22, c12 and c66 are the material constants, q is the mass density, c is the Rayleigh wave speed.
By using the approximation (33) for g3, Eq. (42) is simplified to the equation:
2:5g2 1:6 ịg 0:95 ẳ 0;
whose solution belonging to (0, 1) is:
q
2
1:6  ỵ 1:6 ị ỵ 9:5
:
g ẳ
5
p
From (45) and the fact xi ẳ 1 g2 we obtain:
r
q
2
10:38 ỵ 6:4 22 21:6 ị 1:6 ị ỵ 9:5

;
xi ẳ
5

44ị

45ị

46ị


558

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

where x*i() is the approximation of x() which is defined by the explicit formula (38) in [10].
From Fig. 4, it is shown that the approximation (46) is comparable with the one given by Rahman and
Michelitsch recently ((9) in [3]), using Lanczos’s approximation (33).
3.6. An approximation for the Rayleigh wave speed in compressible orthotropic elastic solids
For compressible orthotropic elastic solids, the secular equation of the Rayleigh waves is of the form (see,
for instance, [12,13]):
z3 ỵ e2 z2 ỵ e1 z ỵ e0 ¼ 0;

ð47Þ

0 < z < minf1; rg;

where:
z ¼ qc2 =c55 ;

e0 ẳ

ar2 d2
;
ca

e1 ẳ

ardrd ỵ 2ị
;
ca

e2 ẳ

a 1 þ 2ard
;
cÀa

ð48Þ

c is the Rayleigh wave speed, q is the mass density. The dimensionless material parameters are defined by:
a ¼ c33 =c11 ;

c ¼ c55 =c11 ;

d ¼ 1 À c213 =c11 c33 ;

r ¼ 1=c;

a > 0;

c > 0;

0 < d < 1;

ð49Þ

where cij are material constants.
Replacing z3 by p(z) defined by (33) reduces the cubic Eq. (47) to the following quadratic equation:
e2 ỵ 1:5ịz2 ỵ e1 0:6ịz ỵ e0 ỵ 0:05 ẳ 0;

50ị

whose solution satisfying 0 < z < min{1, r} is:
q
2
0:6 e1 ỵ 0:6 e1 ị 4e2 ỵ 1:5ịe0 ỵ 0:05ị
:
z1 ẳ
2e2 ỵ 3
pffiffiffiffiffiffiffiffiffiffiffi
The Rayleigh wave velocity is given by: c ¼ x1Ã c55 =q, where:
v
q
u
u0:6 e ỵ 0:6 e ị2 4e ỵ 1:5ịe ỵ 0:05ị
t
1
1
2

0
:
x1 a; r; dị ẳ
2e2 þ 3

ð51Þ

ð52Þ

0.18
0.16
0.14

Absolute Error

0.12
0.1
0.08
0.06
0.04
0.02
0

0

1

2

3

4

5

6

7

8

9

10

Epsilon

Fig. 4. Absolute errors of the approximation x*i defined by (46) (solid line) and Rahman–Michelitsch’s approximation given by (9) in [3]
(dashed line).


P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

559

Fig. 5 shows that the approximation (52) is better than Mozhaev’s ones given by (16) and (20) in [4] in the
indicated range of the parameters.
Now we start from another secular equation of the Rayleigh waves in compressible orthotropic elastic
materials, namely (see [12]):
t3 ỵ a2 t2 t ỵ a0 ẳ 0;

53ị

where:
r
1 À cz
;
t¼
1Àz

z ¼ qc2 =c55 ;

pffiffiffi
a0 ¼ À að1 À dị;

a2 ẳ

p
a1 rdị;

54ị

and
55ị

0 < t < 1 if 0 < r < 1; t > 1 if r > 1;
p
p
(when r ẳ 1; qc2 =c55 ẳ adị=1 ỵ aÞ, see [12]).
Case 1: 0 < t < 1(0 < r < 1):

In this case, taking into account (33), Eq. (53) becomes:
a2 ỵ 1:5ịt2 1:6t ỵ a0 ỵ 0:05 ¼ 0:

ð56Þ

The solution of (56) belonging to (0, 1) is:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
pffiffiffi
pffiffiffi
1; 6 ỵ 2:56 4ẵ a1 rdị ỵ 1:5ẵ0:05 a1 dị
p
;
t ẳ
2 a1 rdị ỵ 3
p
and the Rayleigh wave speed is given by: c ¼ x2Ã c55 =q, here:
s
r1 t2 ị
:
x2 a; r; dị ẳ
1 À rt2Ã Þ

ð57Þ

ð58Þ

It is clear from Fig. 6, that the approximate formula (58) is much more accurate than Mozhaev’s ones defined
by (16) and (20) in [4], in the indicated range of the parameters.
Case 2: t > 1(r > 1):

In terms of the new variable u = 1/t Eq. (53), for the case t > 1, is of the form:

Dimensionless Rayleigh wave speed

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6
0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

δ

Fig. 5. Plots of the approximation x1*(1, 2, d) (solid line), Mozhaev’s given by (16) (dotted line with points), (20) (dash-dot line) in [4] at
a = 1,r = 2,d 2 [0.4, 0.95] and their exact values defined by (3.28) in [12] (dashed line).


560

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

Dimensionless Rayleigh wave speed

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3
0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

δ

Fig. 6. Plots of the approximation x2* (1, 0.5, d) (solid line), Mozhaev’s given by (16) (dotted line with points), (20) (dash-dot line) in [4] at
a = 1,r = 0.5,d 2 [0.4, 0.95] and their exact values defined by (3.28) in [12] (dashed line).

a0 u3 u2 ỵ a2 u ỵ 1 ẳ 0;

59ị

0 < u < 1;

which is simplified to the following equation using the approximation (33):
ð1 À 1:5a0 Þu2 À ða2 À 0:6a0 Þu À ð1 ỵ 0:05a0 ị ẳ 0:

60ị

The solution of (60) satisfying 0 < u < 1 is:
q
2
a2 0:6a0 ỵ a2 0:6a0 ị ỵ 41 ỵ 0:05a0 ị1 1:5a0 ị
u ¼
:
2 À 3a0
pffiffiffiffiffiffiffiffiffiffiffi
The Rayleigh wave speed is given by: c ¼ x3Ã c55 =q, in which:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rðu2Ã À 1Þ
:
x3Ã ¼
ðu2Ã À rÞ

ð61Þ

ð62Þ

3.7. An approximation for the Rayleigh wave speed in incompressible monoclinic elastic materials
Consider incompressible monoclinic elastic materials with the plane of symmetry at x3 = 0 (see [20]). For
these materials, the dispersion equation of the Rayleigh waves in the explicit form was found recently by Destrade [18], namely:
z4 ỵ b3 z3 ỵ b2 z2 ỵ b1 z ỵ b0 ẳ 0;

ð63Þ

2

where z = qc /a, 0 < z < 1,c is the Rayleigh wave velocity and:
3

2

b0 ẳ 1=2ị4b ỵ 4 c2 ị ;

b1 ẳ 2b ỵ 2ị4b ỵ 4 c2 ị ;
2

2

64ị
2

b2 ẳ 1=2ị16b ỵ 20 3c ị4b ỵ 4 c ị; b3 ẳ 5=2ị4b ỵ 4 c ị;

s0
s0
s0
a ẳ 0 0 11 0 2 ; b ¼ 660 À 1; c ¼ À 16
;
4s11
s011
s11 s66 À ðs16 Þ
s0ij are reduced elastic compliances (see [20]).
On use of approximations (33) and (34) of z3 and z4, respectively, we simplify Eq. (63) to:

ð65Þ
ð66Þ


Dimensionless Rayleigh wave speed

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

561

0.97
0.96

γ = 0.1

0.95
0.94
0.93
0.92

0.91

β

0.9
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Fig. 7. Plots of the approximation x4* at c = 0.1, b 2 [0, 0.5] (dashed line) and its exact values (solid line).

b2 ỵ 1:5b3 ỵ 1:714ịz2 ỵ b1 0:6b3 0:914ịz ỵ b0 ỵ 0:05b3 ỵ 0:086 ẳ 0:

67ị

The solution of (67) satisfying the condition 0 < z < 1 is:
zà ¼ À

ðb1 À 0:6b3 0:914ị
2b2 ỵ 1:5b3 ỵ 1:714ị
q
2
b1 0:6b3 0:914ị 4b2 ỵ 1:5b3 ỵ 1:714ịb0 ỵ 0:05b3 ỵ 0:086ị

:
68ị
2b2 ỵ 1:5b3 ỵ 1:714ị
p
p
The Rayleigh wave speed is given by: c ¼ x4à a=q, here: x4à ¼ zà .
From (68) we have: x4* = 0.9551 at b = 0.3, c = 0.1, while by numerically directly solving (63), Nair [19]
and Destrade [18] obtained x = 0.94671 for this case. That means we have a good agreement.
Fig. 7 shows the plots of the approximation x4* at c = 0.1, b 2 [0, 0.5] and its exact values which were
obtained by numerical solution of (63). The absolute error does not exceed 0.01, so the accuracy is
acceptable.
À

4. Conclusions
In this paper, the approach of least squares is recommended for obtaining analytical approximate
expressions of the Rayleigh wave speed. By employing this method we can give the mathematical basis
to the previously proposed approximations and establish new approximate formulas as well. As examples,
we used it in order to explain Bergmann’s approximation, the oldest known approximation of the Rayleigh

wave speed in isotropic elastic materials, and create a new approximate formula for these materials in the
interval [À1, 0.5], and it is shown that it is a good approximation. It is noted that, starting from the explicit exact formulas for the Rayleigh wave speed, we can construct approximate expressions in different
forms corresponding to chosen different sets V. By this method we have found the best approximate second order polynomials of the powers z3 and z4 in the interval [0, 1]. By replacing z3 and z4 in the secular
equations by these second order polynomials, we have obtained new approximate formulas of the Rayleigh
wave speed for isotropic elastic solids, incompressible and compressible orthotropic elastic materials, especially for the incompressible monoclinic materials with the plane of symmetry at x3 = 0, where the explicit
exact formulas for the Rayleigh wave speed so far are not available. It is shown that all these approximate
formulas are more accurate than those proposed previously, except the incompressible orthotropic case
where our result is comparable with the one obtained recently by Rahman and Michelitsch [3]. It is noted
that we can use the best approximate second order polynomials of the powers z3 and z4 in the interval
[0, 1], in the space C[0, 1] (which Rahman and Michelitsch referred to Lanczos’ approximations) to get
analogous approximations.


562

P.C. Vinh, P.G. Malischewsky / Wave Motion 44 (2007) 549–562

Acknowledgements
The work was done partly 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. PGM kindly
acknowledges the support of BMBF in the joint project ‘‘WTZ Deutsch-Israel’’ Grant No. 03F0448A and
of DFG Grant No. MA 1520/6-2.
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[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]

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