World Applied Sciences Journal 15 (12): 1761-1769, 2011
ISSN 1818-4952
© IDOSI Publications, 2011
Speed of Longitudinal and Transverse Plane Elastic Waves
in Rotating and Non-Rotating Anisotropic Mediums
A. Khan, S. Islam, M. Khan and I. Siddiqui
Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan
Abstract: In this article, we study the rotational effects on the speed of longitudinal and transverse wave
propagates along the coordinate planes and axes. The mediums considered are isotropic, transversely
isotropic, orthotropic, monoclinic and general anisotropic. It is observed that results from rotating to nonrotating mediums cannot be obtained neither results of one medium can be derived from the others. It is
further observed that rotational effect decreases the number of the longitudinal and transverse waves and
the number of the longitudinal and transverse waves decreases with the increase of anisotropy.
Key words: Isotropic, Transversely isotropic, Orthotropic, Monoclinic, Longitudina and transverse waves
INTRODUCTION
Speed of longitudinal and transverse waves in
elastic bodies were discussed by a number of authors
from the beginning of 19th century but Schoenberg and
Censor [1] have consider the effect of rotation on plane
waves propagating in an isotropic medium. They
showed that in a rotating isotropic medium three waves
can propagate. They also showed that longitudinal or
transverse wave can exist only if the direction of
propagation and axis of rotation are either parallel or
perpendicular, which is evident from Table 1.1 of
section 1.
Propagation of waves in the transversely isotropic
medium has been discussed by a number of
authors/researches but Chadwick [2] has discussed in
detail in non-rotating medium. He showed that three
waves propagate in transversely isotropic medium. In
section 2, we extend the case for rotating medium. In
sections 3, 4 and 5 rotational effects on the speed of the
longitudinal and transverse waves are discussed in
the orthotropic, monoclinic and anisotropic mediums
respectively.
ISOTROPIC MEDIUM
Wave propagation in the non-rotating isotropic
medium: The constitutive equations of motion in the
absence of body forces in the non-rotating medium can
be written as [3]:
( λ + µ ) u j,ji + µ u i,jj = ρ ui, i , j = 1,2,3
(1.1)
where ρ is the density, u i is the displacement vector and
λ, µ are Lami’s constants. For a plane wave we assume
the solution of the form:
=
ui Aexp[ik(xn
j j − ct)]pi
(1.2)
where ni and pi are respectively the propagation and
polarization vectors of the wave. A is the amplitude of
the wave, c is the wave speed and k is wave number.
The secular equation is
0
( λ + µ ) ni n j p j + ( µ − ρc 2 ) pi =
(1.3)
It is known that in isotropic medium longitudinal
and transverse wave does propagate in all directions
with speeds [3]:
cL =
λ + 2µ
µ
and c T =
ρ
ρ
respectively.
Wave propagation in the rotating medium
The equation of motion in the absence of body
forces in the rotating medium can be written as
follows [1]:
σ ij,j = ρ (
ui + Ω ju jΩi − Ω u i + 2ε ijkΩj uk )
2
(1.4)
Assume that the body is rotating about
x3 -axis. Then Ω i = Ω(0,0,1), the secular equation
becomes
Corresponding Author: A. Khan, Department of Mathematics, COMSATS Institute of Information Technology, Islamabad,
Pakistan
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World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 1.1: Longitudinal and transverse wave speed in a rotating isotropic medium
Direction of propagation along
Speed of longitudinal wave m/s
Speed of transverse wave m/s
x 1 -axis
Nil
µ
ρ
x 2 -axis
Nil
µ
ρ
2
λ +2 µ
2
−Γ
ρ
x 3 -axis
µ
µ
µ
2
2 
2 
 + Γ  ±  +Γ  − − Γ 
ρ

ρ
 ρ

x 1 x 2 -plane
cL2
µ
ρ
x 2 x 3 -plane
Nil
Nil
x 1 x 3 -plane
Nil
Nil
{( λ + µ) k n n
2
i
+ ρ ΩiΩ j} p j + {µk − k ρc −ρΩ } p i − 2ikρ c εijk Ω jpk =
0
2
j
2
2
2
2
(1.5)
Speed of the longitudinal and transverse waves is given in Table 1.1.
where
Ω
= Γ . Rotational effect allows the longitudinal waves to move only along the axis of rotation subject to
k
λ + 2µ
≥ Γ 2 and perpendicular to the plane of it, while the speed of the transverse wave, propagate along the axis of
ρ
rotation, depend upon Γ and does not propagate in the planes containing the axis of rotation. The results of Table 1.1
are agreed with Schoenberg and Censor [1].
TRANSVERSELY ISOTROPIC MEDIUM
Basic equations and solution of the problem: It is known that the generalized Hooke’s law for transversely
isotropic medium, taking x3 -axis as the axis of the symmetry of the medium is:
c
 σ11   11
σ   c 21
 22   c
 σ33   31
  = 0
 σ23  
σ   0
 13  
 σ12   0

c12
c13
0
0
0
0
c11
c13
c13
c33
0
0
0
0
0
0
c 44 0
0 c 44
0
0
0
0
0

  ε11 
  ε 22 


  ε33 
0
  2ε 23 


0
  2ε13 

1

( c11 − c12 )   2ε12 

2
0
0
(2.1)
Thus secular equation in component form is:

1
1

2
2
2
2
0
c11n1 + ( c11 − c12 ) n2 + c 44 n3 − ρ c  p1 +  ( c11 + c12 ) n1n 2  p 2 + {( c13 + c 44 ) n1n 3 }p 3 =
2
2





 1

1
2
2
2
2
0
 ( c11 +c 12 ) n 1n 2  p1 +  ( c11 − c12 ) n1 + c11 n2 + c 44 n3 − ρ c  p2 + {( c13 + c 44 ) n 2 n 3 }p3 =

2

 2

{(c13 + c44 ) n1 n3 } p1 + {(c13 + c44 ) n 2n 3}p 2 + {c 44n12 + c 44 n22 + c 33 n 23 − ρc 2 } p3 = 0


(2.2)
The speed of the longitudinal and transverse waves are given in Table 2.1.
It is observed that two waves propagate along the plane perpendicular to the axis of symmetry where as three
waves propagate in the other directions. It is also noted that along the co-ordinate axes one longitudinal and two
transverse waves exist where as in the co-ordinate planes two longitudinal and one transverse wave exist except the
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World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 2.1: Longitudinal and transverse wave speed in transversely isotropic medium
Direction of propagation along
Speed of longitudinal wave m/s
=
c T1
c11 − c 22
=
,c T2
2ρ
c 44
ρ
c11
ρ
=
c T1
c11 − c 22
=
,c T2
2ρ
c 44
ρ
cL =
c33
ρ
=
c T1
cL =
c11
ρ
x 1 -axis
c11
ρ
x 2 -axis
cL =
x 3 -axis
x 1 x 2 -plane
x 2 x 3 -plane
x 1 x 3 -plane
Speed of transverse wave m/s
cT =
c L1
c n 2 + (c 13 + 2c 44 ) n 23
c n 2 + (c 13 + 2c44 ) n 22
= 11 2
, c L2 = 33 3
ρ
ρ
c L1
c n 2 + (c 13 + 2c 44 ) n 23
c n 2 + (c 13 + 2c44 ) n 21
= 11 1
, c L2 = 33 3
ρ
ρ
cT =
cT =
c 44
=
, c T2
ρ
c33
ρ
c 44
ρ
c11 − c 12 2
n2 + c 44 n32
2
ρ
c11 − c 12 2
n1 + c 44 n32
2
ρ
Table 2.2: Longitudinal and transverse wave speed in rotating transversely isotropic medium
Direction of propagation along
Speed of longitudinal wave m/s
x 1 -axis
Nil
c 44
ρ
x 2 -axis
Nil
c 44
ρ
x 3 -axis
x 1 x 2 -plane
x 2 x 3 -plane
c 33
ρ
νT
c 44
ρ
Nil
c 33 n 23
ρ
Nil
c11n1 + (c 13 + 2c 44 ) n3
2
x 1 x 3 -plane
Speed of transverse wave m/s
2
ρ
−
2
Ω
,
2
k
c 33 n 3 + (c 13 + 2c44 ) n 1
2
2
ρ
Nil
plane perpendicular to the axis of symmetry. The results of Table 2.1 are in good agreement with the results
produced by Chadwick [2].
Elastic wave propagation in a rotating transversely isotropic medium: Transversely isotropic medium rotating
with a uniform angular velocity Ω is considered. By using equation (1.4) and by choosing same Ω i , the equation of
motion for the rotating medium becomes,
 
1
 1

 2
2
2
2
2 2
2
2
0
 c11 n1 + ( c11 − c12 ) n 2+ c44 n3 − ρ c  k −ρΩ p 1 +  ( c11 + c12 ) n1 n 2  k + 2ikcρΩ  p2 + {( c13 + c 44 ) n1 n3 k } p 3 =
2
2








 1

 1
 2
2
2
2
2 2
2
2
 (c11 + c12 ) n1 n 2  k − 2ikcρΩ  p1 +  ( c11 − c12 ) n1 + c11 n2 + c 44 n3 − ρ c  k +ρΩ  p 2 + {( c13 + c 44 ) n 2n 3 k } p3 = 0 (2.3)
2
2





 

2
2
2
2
{( c13 + c 44 ) n1n 3 } p1 + {( c13 + c 44 ) n 2 n3 } p2 + { c 44 n1 + c 44 n 2 + c33 n3 − ρ c } p3 = 0

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World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 3.1: Speed of longitudinal and transverse waves in orthotropic medium
Direction of propagation along
Speed of Longitudinal waves m/c
Speed of Transverse waves m/s
x 1 -axis
c11
ρ
c 66 c 55
,
ρ
ρ
x 2 -axis
c 22
ρ
c 66
c
, 44
ρ
ρ
x 3 -axis
c 33
ρ
c 55
c
, 44
ρ
ρ
x 1 x 2 -plane
ν 1 and ν 2
x 2 x 3 -plane
ν 3 and ν 4
x 1 x 3 -plane
ν 5 and ν 6
c 55n 21 + c 44 n22
ρ
c 66 n 22 + c 55 n 23
ρ
c 66 n 12 + c 44 n 23
ρ
The speed of the longitudinal and transverse waves along the co-ordinate axes and planes are shown in
Table 2.2.
Where,
2
2c 44 + 2ρ
vT =
2
2
4
2

 2

Ω
Ω 
Ω
2Ω
2c
2
4
c
2
c 44 
±
+
ρ
−
+
ρ
−
ρ
44
44



2
4
2
k2
k
k
k




2ρ
Rotational effect allows the longitudinal wave to move along the axis of symmetry/rotation and planes
containing it whereas transverse waves don’t propagate in the planes containing axis of rotation.
ORTHOTROPIC MEDIUM
Basic equations and solution of the problem for the non-rotating material: It is known that the generalized
Hooke’s law for orthotropic medium is [3]:
0
0   ε11 
 σ11  c11 c12 c13 0
σ  c
c
c
0
0
0   ε 22 
 22   12 22 23
 σ33   c13 c23 c33 0
0
0   ε33 

  =

σ
0
0
0
c
0
0  2ε23 
44
 23  
σ   0 0 0 0 c
0  2ε13 
55
 13  


 σ12   0 0 0 0 0 c 66  2ε12 
(3.1)
The equation of motion in the absence of body forces in an elastic medium by using the constitutive equations
for the orthotropic material is
(ρ c − c11n1 − c66 n2 − c 55 n 3 )p1 − (c12 + c66 ) n1 n2p 2 − (c13 + c55 ) n1 n3 p3 =
0 

2
2
2
2
(c12 + c66 ) n1 n2 p1 − (ρ c − c 66 n1 − c22 n2 − c 44 n3 )p2 + (c 23 + c 44 ) n2 n3 p3 =
0

2
2
2
2
(c13 + c55 ) n1 n3p1 + (c23 + c44 )n 2n 3p 2 − (ρ c − c 55 n1 − c 44 n2 − c 33 n3 )p 3 =0 
2
2
2
2
Speed of longitudinal and transverse waves in the medium is presented in the Table 3.1.
1764
(3.2)
World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 3.2: Rotational effects on the longitudinal and transverse waves in orthotropic medium
Direction of propagation along
Speed of Longitudinal waves m/c
Speed of Transverse waves m/s
x 1 -axis
Nil
c 55
ρ
x 2 -axis
Nil
c 44
ρ
c 33
ρ
x 3 -axis
(
c44 + c55
+ Γ2 ) ±
2ρ
(
c44 − c55 2
c +c
) + 2( 44 55 ) Γ2
2ρ
ρ
c 55 n12 + c 44 n 22
x 1 x 2 -plane
Nil
x 2 x 3 -plane
Nil
Nil
x 1 x 3 -plane
Nil
Nil
ρ
where,
v=1
(c11 n12 + c66 + c 22 n 22 ) 2
c11 n12 c 66 c 22 n22 1
2
+
+
−
− 4(c11 c66 n14 − c12
n 21 n22 + c11 c 22n 21 n 22
2ρ
2ρ
2ρ
2ρ
− 2c12 c 66n 21 n 22 + c 22 c 66n 42)
v=
2
(c11 n21 + c6 6 + c22 n 22 ) 2
c1 1 n12 c 66 c2 2 n 22
1
4
2
2 2
2 2
+
+
+
− 4(c11 c66 n1 − c12 n 1 n 2 + c11 c 22 n 1 n 2
2ρ
2ρ
2ρ
2ρ
− 2 c12 c 6 6 n 21 n 22 + c 22 c 66 n 24 )
In a similar way ν3 , ν4 , ν5 and ν6 can be calculated.
From the Table 3.1 it is observed that along the axes one longitudinal and two transverse waves, where as in
coordinate planes one transverse and two longitudinal waves exist.
Rotational effects on the elastic wave propagation in orthotropic medium: The equation of motion in the
absence of body forces in rotating orthotropic medium can be written as follows

Ω2 
Ω


2
2
2
2
(ρ c − c11n1 − c66 n2 − c 55 n 3 ) + ρ 2  p1 −  2i ρc +( c12 + c66 ) n1 n 2  p2 − ( c13 + c55 ) n1 n 3p 3 =0
k 
k




2
Ω

Ω 


2
2
2
2
0
2iρ c − (c12 + c66 ) n1 n 2  p1 + ( ρ c − c66 n1 − c 22 n 2 − c 44 n 3 ) + ρ 2  p 2 − ( c 23 + c 44 ) n 2n 3 p3 =
k
k 




 c + c n n p + c + c n n p − ρ c2 − c n 2 − c n 2 − c n 2 p =
0
44 2
33 3 ) 3
55 1
( 13 55 ) 1 3 1 ( 23 44 ) 2 3 2 (

(3.3)
Speed of Longitudinal and transverse waves in rotating orthotropic medium are shown in the Table 3.2
It is noted that rotational effect restricts the longitudinal waves to move only along the axis of rotation, whereas
the transverse waves don’t exist in planes containing axis of rotation.
MONOCLINIC MEDIUM
Elastic wave propagation in non-rotating monoclinic medium: Generalized Hooke’s law for monoclinic
material, where x2 = 0 is the plane of symmetry, can be written as follows [4]:
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World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 4.1: Longitudinal and transverse wave speed in a stationary monoclinic medium
Direction of propagation along
Speed of longitudinal wave m/s
Speed of transverse wave m/s
cT =
x 1 -axis
Nil
x 2 -axis
cL =
x 3 -axis
Nil
cT =
x 1 x 2 -plane
x 2 x 3 -plane
x 1 x 3 -plane
Nil
Nil
ν L1 and ν L2
Nil
Nil
ν T3
σ11   c11
σ  c
 22   21
 σ33  c 31
  =
 σ23   0
 σ13  c 51
  
 σ12   0
c22
ρ
0
0
c 66
ρ
ν T1 and ν T2
c12
c22
c13
c23
c15
c 25
c32
0
c33 0 c 35
0 c 44 0
c52
0
c53 0 c 55
0 c 46 0
0   ε11 
0   ε 22 
0   ε 33 


c 46   2ε 23 
0   2ε13 


c 66   2ε12 
c44
ρ
(4.1)
The equation of motion in the component form can be written as follows
 {c11n12 + 2c15 n1n 3 + c66 n22 + c 55 n 32 − ρc 2 } + {( c12 +c 66 ) n1n 2 + (c 46 + c 25 ) n2 n 3 } p2

 +{( c13 +c 55 ) n1n 3 + c15 n12 + c 46 n 22 + c35 n32 }p 3 =
0

{( c12 + c66 ) n1 n 2 + ( c 46 + c 25 ) n 2 n3 } p1 + {2c 46 n1n 3 + c 66 n12 + c22 n 22 + c44 n 23 − ρc 2 } p 2


0
+ {(c 25 + c 46 ) n1 n 2 + ( c 23 + c 44 ) n 2 n3 } p3 =

2
2
2
{( c13 + c 55 ) n 1n 3 + c15 n1 + c46 n 2 + c35 n 3 } p1 + { (c 23 + c 44 ) n 2 n3 + ( c46 + c 25 ) n1n 2 } p2

2
2
2
2
0
+ {2c35 n1n 3 + c 55 n1 + c 44 n 2 + c 33 n 3 − ρ c }p 3 =
Table 4.1 shows the speed of the longitudinal and
transverse waves for monoclinic medium.
Where,
v=
c=
L1
L1
v=
c=
L2
L2
c11 n21 + (c 13 + 2c 55 ) n32 + 3c15n 1n 3 + c 35
ρ
c 33 n 32 + (c 13 + 2c 55 ) n12 + 3c35 n1n 3 + c15
v=
c=
T1
T1
v=
c=
T2
T2
ρ
(c44 + c66 ) +
( c44 −
n
n1
2
3
v=
c=
T3
SV
 {c11n12 + 2c15 n1n 3 + c66 n22 + c 55 n 23 − ρc 2 } k 2 −ρΩ2  p1


 + {( c + c ) n n + ( c + c ) n n } k 2 + 2ikcρΩ  p
46
25
2 3
 2
  12 66 1 2
 
2
2
2
2

0,
 + {( c13 + c 55 ) n1n 3 + c15 n1 + c 46 n 2 + c35 n3 } k  p3 =

2
 {( c12 + c66 ) n1 n 2 + ( c 46 + c 25 ) n 2 n3 } k − 2ikcρΩ  p1

2
2
2
2
2
2
+ {2c 46n 1 n3 + c 66 n1 + c 22 n 2 + c 44 n 3 − ρc } k − ρ Ω  p2 (4.3)

+ {(c 25 + c 46 ) n1 n 2 + ( c 23 + c 44 ) n 2 n3 } k 2  p3 =
0,

 {( c13 + c 55 ) n1n 3 + c15 n12 + c46 n 22 + c35 n23 } p1


+ {( c 23 + c 44 ) n 2 n3 + ( c46 + c 25 ) n1n 2 } p2


+ {2c35 n1n 3 + c55 n21 + c 44 n22 + c 33 n 23 − ρc 2 } p3 =
0.

n12
n3
c66 ) + 4c 246
2
( c 44 − c 66 )
2
c 66 n12 + c 44 n 23 + 2c 46 n1n 3
ρ
Elastic wave propagation in rotating monoclinic
medium: The equation of motion in the absence of
body forces in the rotating medium can be written in
component form as follows,
2ρ
(c 44 + c66 ) −
(4.2)
2
+ 4c 46
2ρ
and.
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World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 4.2: Longitudinal and transverse wave speed in a rotating monoclinic medium
Direction of propagation along
Speed of longitudinal wave m/s
Speed of transverse wave m/s
x 1 -axis
Nil
c
Ω2
c
c T1 = 66 − 2 ,c T 2 = 55
ρ k
ρ
x 2 -axis
x 3 -axis
x 1 x 2 -plane
x 2 x 3 -plane
x 1 x 3 -plane
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Table 5.1: Longitudinal and transverse wave speed in anisotropic medium
Direction of propagation along
Speed of longitudinal wave m/s
x 1 -axis
Nil
x 2 -axis
Nil
x 3 -axis
Nil
x 1 x 2 -plane
ν L1
ν L2
x 2 x 3 -plane
ν L3
x 1 x 3 -plane
Speed of transverse wave m/s
ν T1
ν T2
ν T3
Nil
Nil
Nil
Table 4.2 shows the speed of the longitudinal and transvers e waves in rotating monoclinic medium.
It is clear from Table 4.2 that longitudinal waves exist, neither along the coordinate axis nor in the coordinate
planes. Two transverse waves propagate when
constant speed
c55
Ω 2 c 66
≤
2
k
ρ
otherwise only one transverse wave propagates with
ρ . It is also noted that transverse wave neither exist along the axis of rotation nor along the axis
of symmetry.
ANISOTROPIC MEDIUM
Elastic wave propagation in general anisotropic medium: The constitutive equations for anisotropic medium are
as follows,
 σ11   c11
  
 σ22   c 21
 σ33   c 31
  =
 σ23   c 41
 σ13   c 51
  
 σ12   c 61
c12
c13
c22
c32
c23
c33
c42
c52
c43
c53
c62
c63
c14 c15 c16   ε11 


c24 c25 c 26   ε 22 
c34 c35 c 36   ε 33 


c44 c45 c 46  2ε 23 
c54 c55 c 56   2ε13 


c64 c65 c 66  2ε12 
(5.1)
The equation of motion in the component form can be written as,
{c11 n21 + c 66 n 22 + c 55 n 23 + 2c15 n1n 3 + 2c16n 1n 2 + 2c65n 2 n3 − ρc 2 } p1 +

{(c + c ) n n + (c + c ) n n + ( c + c ) n n + c n 2 + c n 2 + c n 2 } p +
46
25
2 3
56
14
1 3
16 1
62 2
54 3
2
 12 66 1 2
{(c13 + c55 ) n1n 3 + (c 14 + c 65 ) n 1n 2 + (c 63 + c 54 ) n 2 n3 + c15n 21 + c 46n 22 + c 35 n 23 } p3 =
0,

 ( c + c ) n n + (c + c ) n n + ( c + c ) n n + c n 2 + c n 2 + c n2 p +
66
1 2
46
25
2 3
56
14
1 3
16 1
62 2
54 3 } 1
{ 12

2
2
2
2
{c 66 n1 + c 22 n 2 + c 44 n 3 + 2c 46 n1n 3 + 2c 26n 1 n2 + 2c24n 2 n3 − ρc } p2 +

2
2
2
0,
{c 65 n1 + c 24 n 2 + c 43 n 3 + (c 25 + c46 ) n1 n 2 + (c 23 + c 44 ) n 2 n 3 + ( c 63 + c 45 ) n 1n 3} p3 =

2
2
2
 {( c13 + c55 ) n1 n3 + (c 14 + c 65 ) n1n 2 + (c 63 + c54 ) n 2 n 3 + c15 n1 + c46 n 2 + c3 5n 3 } p1 +

 {c 65 n12 + c24 n 22 + c 43 n32 + ( c 25 + c 46 ) n1 n2 + ( c 23 + c 44 ) n 2 n3 + (c 63 + c 45 ) n1 n3 } p2 +

0.
 {c55 n12 + c44 n 22 + c33n 32 + 2c35n 1n 3 + 2c54 n1n 2 + 2c 43n 2 n 3 − ρ c2 } p3 =

1767
(5.2)
World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 5.1 shows the speed of longitudinal and transverse waves in general anisotropic medium.
Where,
{( c
+ c 66 )
11
v L1 =
exist if
( c n 2 + c n2 + 2c n n )

66 2
16 1 2
2
2
2
 11 1


+
+
+

c
c
n
c
c
n
(
)
(
)
22
66
2
 11 66 1


2
2
n12 + (c 22 +c 66 ) n 22 + 2( c16 + c 62 ) n1n 2} ± 
 − 4 (c 66n1 + c22 n 2 + 2c26n 1n2 )

+2 ( c16 + c62 ) n1n 2



2
2
2
− ( c61n 1 + c 26n 2 + (c 66 + c21 ) n1n 2 ) 


2ρ
c n 2 + c n 2 + ( c 52 + c 46 ) n1 n2
p1
= − 56 12 42 22
holds, otherwise not.
p2
c 51 n1 + c46 n 2 + ( c 56 + c 41 ) n 1n 2
{( c
+ c 44 )
22
( c 22 + c 44 ) n22 + (c 44 +c
n22 + ( c44 + c33 ) n32 + 2 ( c24 + c43 ) n 2 n3} ± 
+2( c24 + c 43 ) n 2n 3
v L2 =
exist if
2ρ
c n 2 + c 53 n32 + ( c 63 + c 54 ) n 2n 3
p2
= − 46 22
holds, otherwise not.
p3
c62 n 2 + c54 n32 + ( c 52 + c 46 ) n 2 n3
{( c
11
+ c55 )
( c11 +c 55 ) n12 + (c 33 +c
n21 + ( c33 + c45 ) n32 + 2 ( c15 +c 35 ) n1n 3} ± 
+2 ( c15 +c 35 ) n1n 3
v L3 =
exist if
c 65 n12 + c 43 n32 + ( c 63 + c 54 ) n 1n 3
p1
= −
holds, otherwise not.
p3
c 61n12 + c54 n32 + ( c 65 + c 41 ) n1 n3
p2
p3
= −
c15
c16
2
− 4 ( c55c 66 − c 265 )
2ρ
(c 44 + c66 ) ± ( c 44 + c 66 )
2
2
− 4 ( c44 c 66 − c65
)
2ρ
p1
c
= − 24 holds, otherwise not.
p3
c 26
vT 3 =
exist if
(c 66 + c 55 ) ± ( c55 + c 66 )
holds, otherwise not.
vT 2 =
exist if
( c n2 + c n 2 + 2c n n )

55 3
15 1 3
 11 1

2 2

n


45 ) 3 
2
2
 − 4 (c 55 n1 + c 33 n3 + 2c53n 1n 3 )




2
2
2
−( c15n1 + c 53n 3 + (c13 + c55 ) n1n 3 ) 


2ρ
vT1 =
exist if
( c n 2 + c n 2 + 2c n n )

24 2 3
 22 2 44 3



2
2
 − 4 (c 44 n 2 + c 33n 3 + 2c 43n 2n 3 )




2
− (c 24 n22 + c 43n 23 + (c 23 + c44 ) n 2n 3 ) 


2
2

33 )n 3 
p1
p3
= −
c 34
c 35
( c 44 + c 55 ) ± (c 44 + c55 )
2ρ
holds, otherwise not.
1768
2
− 4 ( c44 c55 − c 245 )
World Appl. Sci. J., 15 (12): 1761-1769, 2011
Table 5.2: Longitudinal and transverse wave speed in rotating general anisotropic medium
Direction of propagation along
Speed of longitudinal wave m/s
x 1 -axis
x 2 -axis
x 3 -axis
x 1 x 2 -plane
x 2 x 3 -plane
x 1 x 3 -plane
Speed of transverse wave m/s
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Nil
Elastic wave propagation in rotating general anisotropic medium: The equation of motion in the absence of
body forces in a rotating elastic medium can be written in component form as follows,
{
}
{
 (s − ρc 2 ) k2 − ρ Ω2 p + { s k 2 + 2ikcρΩ} p + {s k 2} p =
0
1
5
2
6
3
 1

2
2
2
2
2
0
{s5 k − 2ikcρΩ} p1 + (s 2 − ρ c ) k −ρΩ p 2 + {s 4k } p 3 =

{s6 }p1 + {s4 } p2 + {s3 − ρc 2} p 3 = 0.


}
(5.3)
where
s1 = c11 n21 + c 66n 22 + c 55 n 23 + 2c15 n1n 3 + 2c16n 1n 2 + 2c65n 2n 3
s2 = c66 n12 + c22 n 22 + c44 n32 + 2c 46 n1n 3 + 2c 26n 1n 2 + 2c 24n 2n 3
s3 = c 55 n12 + c 44 n 22 + c33 n23 + 2c35n 1n 3 + 2c54n1n 2 + 2c 43n 2n 3
s4 = c 65 n12 + c 24 n 22 + c 43 n 23 + ( c 25 + c 46 ) n1 n2 + ( c 23 + c 44 ) n 2n 3 + (c 6 3 + c 45 ) n1 n3
s5 =(c 12 + c 66 ) n 1n 2 + (c 46 + c25 ) n 2 n 3 + ( c 56 + c14 ) n1n 3 + c16n12 + c 62n 22 + c 54 n 23
s 6 =(c 13 + c 55 ) n1n 3 + ( c14 + c65 ) n1 n2 + (c 63 + c54 ) n 2 n 3 + c15n12 + c46n 22 + c35 n23
It is observed that due to the rotation there do not
propagate any longit udinal and transverse wave in
general anisotropic material, neither along the
coordinate axes nor in the coordinate plane.
It is an interesting that in rotating general
anisotropic medium neither longitudinal nor transverse
wave propagate along co-ordinate axes and planes.
CONCLUSION
In isotropic medium longitudinal and transverse
waves propagates in all directions simultaneously
whereas in rotating medium the longitudinal wave
propagate along the axis of rotation and perpendicular
to the plane of the axis of rotation but the transverse
wave propagate everywhere except in the planes
containing the axes of rotation. It is further observed
that the results of Table 1.1 are in agreement with
Schoenberg and Censor [1].
In transversely isotropic mediums longitudinal
wave propagates only along the axis of rotation and
plane containing the axis of rotation whereas transverse
wave does not propagates in the plane containing the
axis of rotation. Rotational effect also reduces the
number of waves along the coordinate planes and axes.
It is further observed that the speed of the longitudinal
and transverse waves depends upon the frequency of
rotation of the medium as shown in Table 2.2.
In orthotropic medium it is observed that there is
only longitudinal wave propagate along the axis of
rotation whereas transverse wave does not propagates
in the plane containing the axis of rotation. Rotational
effect also effected the speed of the wave propagate
along the axis of rotation.
In monoclinic medium it is observed that the
rotational effect does not allow the longitudinal wave to
propagate along the co-ordinate axes and planes
whereas transverse wave propagate along an axis.
In general anisotropic medium rotational effect
does not allow the longitudinal and transverse waves to
propagate along the co-ordinate axes and planes.
REFERENCES
1.
2.
3.
4.
1769
Schoenberg, M. and D. Censor, 1973. Elastic
waves in rotating media. Quart. Appl. Math.,
31: 115-125.
Chadwick, P., 1989. Wave propagation in
transversely
isotropic
elastic
media
I.
Homogeneous plane waves. Proc. R. Soc. Lond.
A422: 23-66.
Dieulesaint, E. and D. Royer, 1974. Elastic waves
in solids. John Wiley and Sons Ltd., Paris.
Destrade, M., 2001. The explicit secular equation
for surface acoustic waves in monoclinic elastic
crystals. J. Acoust. Soc. Am., Vol: 109 (4).