Open Journal of Acoust i c s , 2011, 1, 15-26
doi:10.4236/oja.2011.12003 Published Online September 2011 (http://www.SciRP.org/journal/oja)
On Axially Symmetric Vibrations of Fluid Filled
Poroelastic Spherical Shells
Syed Ahmed Shah1, Mohammed Tajuddin2
1Department of Mathematics, Deccan College of Engineering and Technology, Hyderabad, India
2Department of Mathematics, Osmania University, Hyderabad, India
Received July 27, 2011; revised August 15, 2011; accepted August 19, 2011
Abstract
Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hol-
low poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially sym-
metric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical
shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness
to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervi-
ous and an impervious surface. Results of previous works are obtained as a particular case of the present
study.
Keywords: Biot’s Theory, Axially Symmetric Vibrations, Radial Vibrations, Rotatory Vibrations, Spherical
Shell, Elastic Fluid, Pervious Surface, Impervious Surface, Frequency
1. Introduction
Using exact three dimensional equations of linear elas-
ticity, Ram Kumar [1] studied the axially symmetric vi-
brations of fluid-filled spherical shells. Rand and Di-
Maggio [2] studied the vibrations of fluid filled elastic
spherical and spheroidal shells. Employing Biot’s [3]
theory, Paul [4] discussed the radial vibrations of poroe-
lastic spherical shells. Chao et al. [5] presented the theo-
retical and experimental results regarding wave propaga-
tion in porous formations. Ahmed Shah [6] investigated
the axially symmetric vibrations of fluid-filled poroelas-
tic circular cylindrical shells of various wall-thicknesses
in the absence of dissipation. Sharma and Sharma [7]
studied the three dimensional free vibrations of transra-
dially thermoelastic spheres. Ahmed Shah and Tajuddin
[8] studied torsional vibrations of poroelastic spheroidal
shells.
In the present analysis, wave propagation in fluid
filled poroelastic spherical shells is studied in absence of
dissipation. Frequency equation each for a pervious and
an impervious surface is obtained employing Biot’s [3]
theory of wave propagation in liquid saturated poroelas-
tic solid. Biot’s [3] model consists of an elastic matrix
permeated by a network of interconnected spaces satu-
rated with liquid. Frequency equation of axially symmet-
ric vibrations each for a pervious and impervious surface
is obtained for fluid filled and empty poroelastic spheri-
cal shells. The frequency equations of radial and rotatory
vibrations are obtained as a particular case. Non-
dimensional frequency as a function of ratio of thickness
to inner radius is computed for pervious and impervious
surfaces in each case, i.e., fluid filled and empty poroe-
lastic spherical shell in absence of dissipation. The re-
sults are displayed graphically for two types of poroelas-
tic materials and then discussed. Results of some previ-
ous works are shown as a particular case of the present
investigation. By ignoring the liquid effects, and after
some rearrangement of terms, results of purely elastic
solid are shown as a particular case considered by Ram
Kumar [1].
2. Governing Equations
The equations of motion of a homogeneous, isotropic
poroelastic solid [3] in presence of dissipation b are

2
22
12111122 12
2
ΦΦΦΦ ΦΦ,
t t
PQ ρρ b

 
 

2
22
12121222 12
2
ΦΦΦΦ ΦΦ,
t t
QR ρρ b





S. A. SHAH ET AL. 16

2
2
1111122 12
2
ΨΨΨΨΨ,
t t
Nρρ b

 



2
12 122212
2
0ΨρΨ ΨΨ,
t tρb



 (1)
where 1, 2, Ψ1, Ψ2 are potential functions of r, θ and
time t; P (=A + 2N), N, Q, R are all poroelastic constants
and 
ik (i, k = 1,2) are the mass coefficients following
Biot [3]. The poroelastic constants A, N correspond to
familiar Lame constants in purely elastic solid. The co-
efficient N represents the shear modulus of the solid. The
coefficient R is a measure of the pressure required on the
liquid to force a certain amount of the liquid into the ag-
gregate while total volume remains constant. The coeffi-
cient Q represents the coupling between the volume
change of the solid to that of liquid and
22
2
2222
12cot
.
θ
θ
rr θ
rr r
 
 

 (2)
The equation of motion for a homogeneous, isotropic,
inviscid elastic fluid is
2
2
22
f
1Φ
Φ,
Vt

  (3)
where  is displacement potential function and Vf is the
velocity of sound in the fluid.
The stresses 
ik and the liquid pressure s of the poroe-
lastic solid are
 
2, ,,,
ik ikik
σNeAe Qδik rθφ
,sQeR (4)
where ik is the well-known Kronecker delta function.
The fluid pressure σf is given by
2
2
Φ,
ff
σρ
t

  (5)
where 
f is the density of the fluid.
3. Solution of the Problem
Let (r,  , φ) be the spherical polar coordinates. Consider
a homogeneous, isotropic, poroelastic spherical shell
filled with inviscid elastic fluid. Let the inner and outer
radii be r1 and r2 respectively so that the thickness of
poroelastic spherical shell is h [=(r2r1) > 0]. Axially
sym- metric vibrations of the spherical shell is character-
ized by the displacement field of solid (,,0)uuv
and
liquid (,,0)UUV

where u, v, U and V are functions of
r, θ and time t. For axially symmetric vibrations of
poroelastic spherical shell, the displacement components
and solid and liquid, respectively are given potential
functions 1, 2, Ψ1, Ψ2 as
2
11 1
2
ΦΨ Ψ 1cot
,
θ
urrr θ
θ
 
 



2
11 1
ΦΨ Ψ
11
v,
rθrθrθ
 



2
22 2
2
ΦΨ Ψ
1cot
,
θ
Urrr θ
θ
 
 


2
22 2
ΦΨ Ψ 11
V.
rθrθrθ
 


 (6)
Similarly, the displacement vector of fluid is
(,,0)
fff
uuv


, where
Φ1Φ
, .
ff
uv
rrθ




 (7)
Solution of Equation (1) is


111112222
Φ()()()()(cos )e,
i ωt
nnn nn
Aj ξrByξrAjξrByξrP θ
22 22
2111122 222
Φ()()) (cos )e,
i ωt
n1n1n2n n
AδjξrBδyξrAδj(ξr)B δy(ξrP θ

 



133 33
Ψ))(cos )e,
i ωt
nnn
Aj(ξrBy(ξrP θ

12
23333
22
Ψ(()(cos )e.
i ωt
nnn
KAj ξr)B yξrP θ
K
  (8)
In Equation (8), jn, yn are Spherical Bessel functions of
first and second kind respectively, Pn is Legendre poly-
nomial of degree n (n is the order of spherical harmonic),
A1, B1, A2, B2, A3, B3 are constants, ω is circular fre-
quency. Here i is complex unity or i2 =1 and
1
k, 1,2,3 k
ξωVk

 (9)



 
1
222
11121222 , 1,2 kk
δRKQKVPR QRKQKk



 

(10)
11 1
11 1112122222
, , ,KρibωKρibωKρibω
 
 (11)
also 11
22
ππ
()(z), ()().
22
nn
nn
jzJyzY z
zz

 (12)

S. A. SHAH ET AL. 17
In Equation (9), V1, V2 are the dilatational wave ve-
locities of first and second kind, respectively and V3 is
shear wave velocity.
A bounded solution of Equation (3) is
Φ()(cos )e,
i ωt
fn fn
Aj ξrP θ (13)
where Af is constant and
1
f. f
ξωV

 (14)
Substituting Equation (13) into Equation (5), the fluid
pressure is
()(cos )e.
2i ωt
fffnfn
σAρω jξrP θ  (15)
Dilatations of solid and liquid respectively, are
22
12
eΦ, Φ. (16)
Now the required displacements, stresses and liquid
pressure are


1 111 122 132 143 153 16f17
()B()()()()()()(cos )e,
i ωt
f n
uuAMrMrAMrBMrAMrBMrAMrP θ    (17)


1 211 222 23224325326f27n
()()()()()()()P(cos )e,
i ωt
rr f
σsσAMrBMrAMrBMrAMrBMrAMr θ  (18)

1 311 322332343 353 36
d(cos )
()()()()()()sin e,
d
i ωt
n
rθ
Pθ
σAMrBMrAMrBMrAMrBMr θ
θ
 (19)


1411422 432 44f47
()()()()()(cos )e,
i ωt
f n
sσAMrBMrAMrBMrAMrP θ  (20)

1 411 422 43244f47
()()()()()(cos )e,
fi ωt
n
σ
sANrBNrA NrBNrA NrPθ
rr

 
 (21)
where the elements Mik(r) and Nik(r) are











111 1121113221422
, , , ,
nnn n
Mr ξjξrMrξyξrMrξjξrMrξyξr
 
 
   


 

153 16317
11
, , ,
nnfnf
nn nn
Mr jξrMr yξrMr ξjξr
rr









2222222
1
2111111 111 11
2
1
2
() δnn n
nn
ξ
Mr PQQ RδξjξrAQQδRδjξrAQQδRδjξr
rr

 
 








2222222 1
221 11n111n11 11
2
1
2,
n
nn
ξ
Mr PQQδRδξyξrAQQδRδyξrAQQδRδyξr
rr

 
 








2222222
2
23222222 n2222
2
1
2,
n n
nn
ξ
Mr PQQδRδξjξrAQQδRδjξrAQQδRδjξr
rr

 
 








222 2222 2
24222n222 2222
2
1
2
ξyξ
nn
nn
ξ
Mr PQQδRδξrAQQδRδyrAQQδRδyξr
rr

 
 
3
25n 33
2
2( 1)2( 1)
()() (),
n
Nn nξNn n
Mr jξrjξr
rr






32
2633 27
2
21 21
()()(), ()(),
nnfnf
Nn n+ξNn n
Mr yξryξrMr ρwj ξr
rr



  
11
3111 3211
22
22 22
, ,
nnnn
NξNξ NN
Mr jξrjξrM ryξryξr
rr rr


  
22
3322 3422
22
22 22
(), (),
nnn n
NξNξ NN
Mr jξrjξrMryξryξr
rr rr


 

  



2 2
353n 33363n 33
2 2
12 12
N
r, ,
n n
NnnNNnnN
Mr ξjξjξrMrNξyξryξr
rr
 
 





22 1
411 1111
2
1
2
() nn n
nn
ξ
Mr QRδξjξrjξrjξr
rr



 
 





S. A. SHAH ET AL. 18

 


22 1
421 1n1n11
2
1
2
() n
nn
ξ
Mr QRδξyξryξryξr
rr



 
 






 


22 2
4322 n222
2
1
2
nn
nn
ξ
Mr QRδξjξrjξrjξr
rr



 
 






 


22 2
442 2222
2
1
2,
nn n
nn
ξ
Mr QRδξyξryξryξr
rr

 
 


2
4546 47
()0, ()0, ()(),
fnf
Mr Mr Mrρω jξr


 


  
2
21
23 1
411 1n1n1n11
23
2ξ21
2
ξjn
nn nn
ξ
Nr QRξjrjξrξrjξr
rrr


 
  
 



 


  
2
21
23 1
42111n1n 11
23
2ξ21
2
() y
n n
nn nn
ξ
Nr QRδξyξrξryξryξr
rrr

 
  





 


  
2
22
23 2
432 22222
23
221
2r
nnn n
nn ξnn
ξ
Nr QRδξjξrjξrjξjξr
rrr

 
  




 


  
2
22
23 2
442 22222
23
221
2
() δξ
r
n
nn n
nn ξnn
ξ
Nr QRξyξryξryryξr
rr

 
  
 


2
45 46 47
()0, ()0, ()().
ffnf
NrNrNr ρωξ jξr

 (22)
In Equation (22), a prime over jn or yn represents dif-
ferentiation with respect to r.
4. Boundary Conditions-Frequency
Equation
The outer surface of the poroelastic spherical shell is
assumed to be free from traction. Continuity of radial
displacement and fluid pressure is assumed at the inter-
face of solid and fluid. Thus the boundary conditions for
a fluid-filled poroelastic spherical shell, in case of a per-
vious surface are
0, 0, 0, 0
frrfrθf
uu σsσσsσ
at r = r1,
0, 0, 0,
rr rθ
σsσs at r = r2. (23)
Similarly, the boundary conditions for a fluid-filled
poroelastic spherical shell, in case of an impervious sur-
face are
0, s0, 0, 0,
f
frr frθ
σ
s
uu σσσ rr




at r = r1,0, 0, 0,
rr rθ
s
σsσr


 at r = r2. .(24)
Substitution of Equations (17)-(20) into the boundary
conditions (23) result in a system of seven homogeneous
algebraic equations in seven constants A1, B1, A2, B2, A3,
B3 and Af. For a non-trivial solution, the determinant of
the coefficients must vanish. By eliminating these con-
stants, the frequency equation of axially symmetric vi-
brations of a fluid-filled poroelastic spherical shell in
case of a pervious surface is
0, ,1,2,3,7.
ik
Aik (25)
In Equation (25), the elements Aik are
1
(); 1,2,3,4 and 1,2,7,
ik ik
AMr ik


522632742
(), (), (), 1,2,6,
kk kk kk
A MrAMrA Mrk


37 57 67 77
0, 0, 0, 0.AAAA

 (26)
In Equation (26), the elements Mik(r) are defined in
Equation (22).
Arguing on similar lines, Equations (17)-(19), (21)
together with the Equation (24) yield the frequency equa-
tion of axially symmetric vibrations of a fluid-filled po-
roelastic spherical shell in case of an impervious surface
to be
0, ,1,2,7,
ik
Bik (27)
where the elements Bik are
, 1,2,3,5,6 and 1,2,7,
ik ik
BAi k



S. A. SHAH ET AL. 19
441
(), 1,2,7,
kk
BNrk
7k4k 277
N
(), 1,2,6, and 0,BrkB  (28)
where Mik(r) and Nik(r) are defined in Equation (22).
By eliminating liquid effects from frequency equation
of a pervious surface (25), that is, setting b0, 
120,

220, (AQ2/R) , N
 , Q0, R0 and after re-
arrangement of terms, the results of purely elastic solid
are obtained as a particular case considered by Ram
Kumar [1]. The frequency equation of an impervious
surface (27) has no counterpart in purely elastic solid.
The order of spherical harmonic n takes different integer
values. When n = 0, radial vibrations are obtained
whereas n = 1 results in rotatory vibrations. For n = 2
spheroidal vibrations are obtained in which the sphere is
distorted into an ellipsoid of revolution becoming prolate
or oblate according to the phase of motion.
4.1. Frequency Equation for an Empty
Poroelastic Spherical Shell
When the fluid density is zero, that is, 
f = 0 the fluid-
filled poroelastic spherical shell will become an empty
poroelastic spherical shell. Thus, the frequency equation
of pervious surface (25) under suitable boundary condi-
tions reduce to
0, 2,3,4,5,6,7 and 1,2,3,4,5,6,
ik
Ai k (29)
where the elements Aik are defined in Equation (26) are
now evaluated for 
f = 0.
Equation (29) is the frequency equation of axially
symmetric vibrations of an empty poroelastic spherical
shell in case of a pervious surface.
Similarly, the frequency equation of axially symmetric
vibrations of an empty poroelastic spherical shell for an
impervious surface under suitable boundary conditions is
0, 2,3,4,5,6,7 and 1,2,3,4,5,6,
ik
Bi k (30)
where the elements Bik are defined in Equation (28) are
now evaluated for 
f = 0.
4.2. Frequency Equation of Radial Vibrations
When the order of spherical harmonic n = 0, the fre-
quency equation of axially symmetric vibrations of fluid
filled spherical shell Equation (25) reduce to the fre-
quency equation of radial vibrations of fluid filled
spherical shell in case of a pervious surface under suit-
able boundary conditions. Radial vibrations involve u
and uf as the only non-zero displacement components.
Thus the frequency equation of radial vibrations of fluid
filled poroelastic spherical shell for a pervious surface is
1112 13 14 17
2122 23 24 27
4142 43 44 47
51 5253 54
71 7273 74
0,
0
0
AAAAA
AA AAA
AA AAA
AA AA
AAAA
 (31)
where the elements Aik are defined in Equation (26) are
now evaluated for n = 0.
Similarly, the frequency equation of radial vibrations
of fluid-filled poroelastic spherical shell in case of an
impervious surface is
11 1213 14 17
21 22232427
41 42434447
51 5253 54
71 727374
B0,
0
0
BBBBB
BBBBB
BBBB
BBBB
BBBB
 (32)
where the elements Bik are defined in Equation (28) are
now evaluated for n = 0.
When the fluid density is zero in Equation (31), it be-
comes the frequency equation of radial vibration of an
empty poroelastic spherical shell under suitable bound-
ary conditions as
21 2223 24
41 4243 44
51 5253 54
71 7273 74
0,
AAAA
AAAA
AAA A
AAAA
 (33)
where the elements Aik are defined in Equation (26) are
now evaluated for

f = 0 and n = 0.
In a similar way, the frequency equation of radial vi-
brations of an empty poroelastic spherical shell for an
impervious surface is
21 222324
41 424344
51 525354
71 727374
0,
BBBB
BBBB
BBBB
BBBB
 (34)
where the elements Bik are defined in Equation (28) are
now evaluated for

f = 0 and n = 0.
The frequency equation of radial vibrations of an
empty poroelastic spherical shell Equation (33) of a per-
vious surface was also studied by Paul [4] with the
boundary conditions 0, 0,
rr
σs

 at r = r1 and r2.
4.3. Frequency Equation of Rotatory Vibrations
When the order of spherical harmonic n = 1, the fre-
quency equation of axially symmetric vibrations of fluid
filled spherical shell Equation (25) reduce to the fre-

S. A. SHAH ET AL. 20
quency equation of rotatory vibrations of fluid filled
spherical shell in case of a pervious surface. Thus the
frequency equation of rotatory vibrations of fluid filled
poroelastic spherical shell for a pervious surface is
0, ,1,2,3,7.
ik
Aik (35)
The elements Aik appearing in Equation (35) are de-
fined in Equation (26) are now evaluated for n = 1.
Similarly, the frequency equation of rotatory vibra-
tions of a fluid filled poroelastic spherical shell in case of
an impervious surface is
0, ,1,2,3,7,
ik
Bik (36)
where elements Bik appearing in Equation (36) are de-
fined in Equation (28) are now evaluated for n = 1.
Setting n = 1 in Equations (29) and (30), the frequency
equations of rotatory vibrations of empty poroelastic
spherical shells of a pervious and an impervious surface
respectively, are obtained.
5. Non-Dimensionalisation of Frequency
Equation
To analyze the frequency equations of radial and rotatory
vibrations of fluid-filled and empty poroelastic spherical
shells in the cases of pervious and impervious surfaces, it
is convenient to introduce the following non-dimensional
variables:
1111
12 34
, , , ,aPH aQH aRH aNH

 11 11
11 1112122222
ρ, , , ,
f
mρmρρ mρρ tρ
ρ


 
1212 1211
010 20330
(), (VV), (), , Ω,
f
xVV yzVVmVVωhC

 
  (37)
where  is non-dimensional frequency, H = P + 2Q + R,
 = 
11 + 2

12
+

22, C0 and V0 are the reference veloci-
ties (C0
2 = N/, V0 2 = H/), h is the thickness of the
poroelastic spherical shell. Let
2
112
r1
, so that 1, .
r
hhg
gg
rrg


(38)
Employing the non-dimensional parameters defined in
Equations (37), (38) and using the relations given in A-
bramowitz and Stegun [9]
01
2
sinsin cos
(), (),
zzz
jz jz
zz z

232
31 3
()sin cos,
z
jzz z
zz

 


01
2
232
coscos sin
(z), (),
31 3
()cos sin,
z
zzz
yyz
zz
z
yzz z
z
z
 

 


the frequency equation of radial vibrations of a fluid
filled spherical shell in case of a pervious surface Equa-
tion (31) reduce to
0; ,1,2,3,4,5,
ik
Cik (39)
where elements Cik appearing in Equation (39) are
11111211
11
11
sin()cos(), Ccos()sin(),CTT TT
TT

1322 14221544
22 4
11 1
sin()cos(), cos()sin(), sin()cos(),CTTCTTCTT
TT T


22 4
21122 13 11141
1
4δδ sin( )4cos( ), a
CaaaaTTaT
T





22 4
22122 13 11141
1
4
δδ cos( )4 sin( ),
a
CaaaaT TaT
T

 



22 4
23122 23 22242
2
4δδsin()4cos( ), a
CaaaaTTaT
T





2
22 44
24122 23 22242254
2
24
4Ω
cos()4sin(), Csin(),
(1)
ata
CaaaδaδTTaT T
TgT

 
 

22
313 12113223 111
()sin(), ()cos(),CaδaTT Ca aδTT
2
22
4
33322223423 222354
2
4
Ω
(δ)sin(), ()cos(), sin(),
(1)
ta
CaaTTCaaδTTC T
gT
 


S. A. SHAH ET AL. 21
C41, C42, C51, C52 = Similar expressions as C21, C22, C31,
C32 with T1 replaced by T5,
C43, C44, C53, C54 = Similar expressions as C23, C24, C33,
C34 with T2 replaced by T6,
C45 = 0, C55 = 0. (40)
Similarly, the frequency equation of radial vibrations
of a fluid filled poroelastic spherical shell for an imper-
vious surface in non-dimensional form is
0; ,1,2,3,4,5,
ik
Dik (41)
Where ; 1,2,4, 1,2,3,4,5,
ik ik
DCi k
222
3123 1113 1211
()sin()()cos(),DaaδTT aδaT T 
222
32312113 1211
()cos()()sin(),DaδaT T aδaT T 
222
3323 2223 2222
()sin()()cos(),DaaδTT aδaT T 
222
343222232222
()cos()()sin(),DaδaT T aδaTT
22
44
354 4
22
4
ΩΩ
sin()cos(),
(1) (1)
ta ta
DTT
gT g


D51, D52 = Similar expressions as D31, D32 with T1 re-
placed by T5,
D53, D54 = Similar expressions as D33, D34 with T2 re-
placed by T6,
D55 = 0. (42)
The frequency equation of rotatory vibrations of a
fluid filled poroelastic spherical shell for a pervious sur-
face Equation (35) in non-dimensional form reduce to
0, ,1,2,3,7,
ik
Eik (43)
where the elements Eik are
1111 1211
22
11 11
222 2
1sin()cos(), 1cos()sin(),ETTE TT
TT
TT
 

 

1322 1422
22
22 22
222 2
1sin()cos(), 1cos()sin(),ETTE TT
TT
TT
 

 

1533 1633
2 2
33 33
22 22
cos()sin(), sin()cos(),ETTET T
TT TT

174 4
2
4
4
22
1sin( )cos(),ETT
T T

 



22 22 4 4
214122 13 11122 13 111
2
1
1
12 12
4sin() cos(),
aa
EaaaaδaδTaaaδaδTT
T T
 
 





22 22 44
22412213111 2 213111
2
1
1
12 12
4cos()sin(),
aa
EaaaaδaδTaaaδaδTT
T T
 
 





22 22
4 4
234122 23 22122 23 222
2
2 2
12 12
4δδsin( )δδ cos( ),
aa
EaaaaaTaaaaTT
T
T


 


 

22 22
44
244122 23 22122 23 222
2
2
2
12 12
4δδ cos( )(δδ)sin(),
aa
EaaaaaTaaaaT T
T T
 
 




44 44
25433 26433
22
33 33
121212 12
4sin()cos(), 4cos()sin(),
aa aa
EaTTEaTT
TT TT
 
 
 

22
44
27 44
222
44
ΩΩ
cos( )sin( ),
(1)(1)
ta ta
ETT
gT gT


44 44
31411 32411
22
11 11
66 66
E2sin(T )cos(), E2cos(T )sin(),
aa aa
aTa T
TT
TT
 
 
 

44 44
33422 34422
22
22 22
66 66
2sin()cos(), 2cos()sin(),
T
aa aa
EaTTEaTT
T
TT
 
 
 


S. A. SHAH ET AL. 22
4444
3543433364343337
2 2
3 3 3 3
6666
3sin()cos(), 3cos()sin(), 0,
aaaa
EaTaTTEaTaTTE
TT TT
   
 
   
 
 












22 22
4131 213121141231131 211
δsinδTcos, Eδcosδsin,Eaa TaaTaaTaaTT 













22 22
433222 32222442322 32222
δVsin δcos, δcos δsin,E aTa aTTE aaTa aTT 
 
22
44
45 46 4744
22
2
44
ΩΩ
0, 0, cossin,
11
ta ta
EEET T
gT gT
 

E51, E52, E61, E62, E71, E72 = Similar expressions as E21,
E22, E31, E32, E41, E42 with T1 replaced by T5,
E53, E54, E63, E64, E73, E74 = Similar expressions as E23,
E24, E33, E34, E43, E44 with T2 replaced by T6,
E55, E56, E65, E66 = Similar expressions as E25, E26, E35,
E36 with T3 replaced by T7,
57 67 75 76 77
0, 0, 0, 0, 0.EEEEE (44)
Similarly, the frequency equation of rotatory vibra-
tions of a fluid filled poroelastic spherical shell for an
impervious surface Equation (36) in non-dimensional
form reduce to
0, ,1,2,37,
ik
Fik (45)
where
; 1,2,3,5,6, 1,2,3,4,5,6,7,
ik ik
FEi k 
22 2
413 12113 1211
( )(2)sin()2()cos(),FaδaTT aδaT T 
22 2
42312113 1211
( )(2)cos()2( )sin(),FaδaT TaδaT T 
222
433 22223 2222
()(2)sin( )2()cos(),FaδaTT aδaT T 
22 2
44322223 2222
()(2)cos() 2()sin( ),FaδaT TaδaT T
45 46
22
44
474 4
22 2
44
0, 0,
Ω2Ω
21sin()cos( ),
(1) (1)
FF
ta ta
F
TT
gT gT






F71, F72 = Similar expressions as E41, E42 with T1 re-
placed by T5,
F73, F74 = Similar expressions as E43, E44 with T2 re-
placed by T6,
F75 = 0, F76 = 0, F77 = 0. (46)
In Equations (40), (42), (44) and (46) the quantities T1,
T2, T3, T4, T5, T6, T7 are
11 11
22 22
444 4
123 4
Ω() Ω()Ω()Ω()
,, , ,
(1) (1)(1)(1)
ax ayazaz
TTT T
gg gmg
 
 
 
111
222
44 4
567
Ω() Ω() Ω()
, , ,
(1) (1) (1)
ax gay gaz g
TTT
ggg


 (47)
and 22
12
, δδ
in non-dimensional form are

1
22
13 112121323 12222
()()x ,δamamaaaamam 





1
22
23 112121323 12222
()()y .δamamaa aamam





(48)
6. Numerical Results and Discussion
Two types of poroelastic materials are considered to find
the frequency as a function of ratio of thickness of
spherical shell to inner radius. One poroelastic material is
sandstone saturated with kerosene designated as Mate-
rial-I [10]. The other poroelastic material is sandstone
saturated with water, Material-II [11]. The non-dimen-
sional physical parameters of Material-I and II are given
in Table 1.
Frequency Equations (39), (41), (43), (45) and the cor-
responding frequency equations of empty poroelastic
spherical shells each for a pervious and impervious sur-
face constitute a relation between non-dimensional fre-
quency Ω and ratio of thickness of the poroelastic
spherical shell to inner radius h/r1. When the values of
h/r1 are small it represents thin poroelastic spherical shell.
Thickness of the poroelastic spherical shell increases
with the increase of h/r1. As h/r1→∞ or r1→0, empty
poroelastic spherical shell becomes a poroelastic solid
sphere. To compute the frequency of fluid filled and
empty poroelastic spherical shells values of h/r1 are taken
in the interval [0.5,7]. These values of h/r1 represents
transition of spherical shell from a thin spherical shell to
moderately thick shell and then to thick spherical shell.
For fluid-filled poroelastic spherical shells, the values of
m and t are taken as m = 1.5 and t = 0.4.
Table 1. Material Parameters.
Material/Parameter a1 a2 a3 a4 m11 m12 m22
x
 y
 z

I 0.843 0.065 0.028 0.234 0.901 –0.0010.101 0.999 4.763 3.851
II 0.960 0.006 0.028 0.412 0.877 0 0.123 0.913 4.347 2.129

S. A. SHAH ET AL. 23
Non-dimensional frequency of radial vibrations (n = 0)
as a function of ratio of thickness to inner radius of a
fluid-filled and an empty poroelastic spherical shell is
presented in Figure 1 for Material-I each for a pervious
and an impervious surface.
From Figure 1 it is clear that frequency of a fluid
filled shell for pervious surface decreases in case of a
thin spherical shell. As the thickness of the spherical
shell increases the frequency increases gradually for
moderately thick spherical shell. Still with the increase of
thickness of the shell, the frequency remains constant.
The frequency of fluid filled spherical shell decreases for
thin shell in case of an impervious surface also and it is
slightly higher than the frequency of a pervious surface.
Then, as the thickness of the spherical shell increases the
frequency increases rapidly and for moderately thick
shell it remains constant. With the increase of thickness
of the shell, the frequency decreases and then remains
constant. The frequency of an impervious surface is al-
ways higher than that of a pervious surface in case of a
fluid filled spherical shell. The frequency of an empty
spherical shell slowly increases with the increase of the
thickness and for thick spherical shell it is constant. The
frequency of an empty spherical thick shell is higher than
that of a fluid filled thick shell. This is not true in case of
a thin spherical shell. Thus it is seen that the presence of
fluid is decreasing the frequency of thick spherical shell.
The frequency of an empty spherical shell for an imper-
vious surface is almost same as that of fluid filled mod-
erately thick shell. The frequency of fluid filled and
empty thick shells is same. The frequency of an imper-
vious surface is higher than that of a pervious surface in
case of empty spherical shell also. Thus the frequency of
an impervious surface is higher than that of a pervious
surface for an empty and a fluid filled spherical shell.
The variation of frequency of radial vibrations of a
fluid filled and an empty poroelastic spherical shell each
for a pervious an impervious surface is presented in Fig-
ure 2 for Material-II. The variation of frequency of fluid
filled shell is similar as discussed in Figure 1. But the
frequency of empty spherical shell for a pervious an im-
pervious surface is nearly same for thin and moderately
thick spherical shells. The frequency of an impervious
surface is higher than that of a pervious surface in case of
thick empty spherical shells. Presence of mass-coupling
parameter has no significant effect on the frequency of
fluid filled spherical shell in case of a pervious surface.
The absence of mass-coupling parameter in Mate-
rial-II is increasing the frequency of an impervious sur-
face for fluid filled spherical shells. This phenomenon is
reversed in case of empty poroelastic spherical shells
where the presence of mass-coupling parameter is in-
creasing the frequency of an impervious surface. Also it
is seen that the presence of fluid is decreasing the fre-
quency in case of a pervious surface while it is increas-
ing the frequency in case of an impervious surface for
both the materials.
Frequency of rotatory vibrations (n=1) of fluid filled
and empty spherical shells each for a pervious and an
impervious surface is presented in Figures 3 and 4 for
Material-I and Material-II, respectively. From Figure 3 it
is clear that the frequency of a fluid filled spherical shell
0
1
2
3
4
5
6
7
0.5 1.5 2.53.5 4.5 5.56.5
Ratio of thickness to i nner ra dius
Frequency
_____________Perv ious Surface
- - - - - - - - - - -Impervious Surface
Fluid-fill ed shel l
Fluid-fill ed shell
Empty shell
Empty shell
Figure 1. Frequency as a function of ratio of thickness to inner radius Radial vibrations of Fluid-filled and empty spherical
shells (Material-Ⅰ, n = 0).

S. A. SHAH ET AL. 24
0
1
2
3
4
5
6
7
8
0.5 1.5 2.5 3.5 4.5 5.5 6.5
Ratio of thickness to inner radius
Frequency
____________Pervious Surface
- - - - - - - - - - -Im pervious
Flui d-filled shell
Fluid-filled shell
Empty shell
Empty shell
Figure 2. Frequency as a function of ratio of thickness to inner radius Radial vibrations of Fluid-filled and empty spherical
shells (Material-Ⅱ, n = 0).
0
1
2
3
4
5
6
0.5 1.5 2.5 3.5 4.5 5.5 6.5
Ratio of thi ckness to inner radius
Frequency
____________Perv ious Surface
- - - - - - - - - - -Im pervious
Fluid-filled shellFluid-filled shell
Empt y shell
Empt y shell
Figure 3. Frequency as a function of ratio of thickness to inner radius Rotatory vibrations of Fluid-filled and empty sphe rical
shells (Material-Ⅰ, n = 1).
varies in a staggered way when the thickness of the
spherical shell is small. With the increase of thickness it
remains almost constant. Frequency of thin shell is
higher than that of the frequency of moderately thick
shell and thick shell in case of a fluid filled shell. The
frequency of an impervious surface varies in staggered
form for small thickness of the fluid filled shell. But here
with the increase of thickness the frequency increases.
Thus the frequency of thick shell is higher than the fre-
quency of thin shell in case of fluid filled spherical shell.

S. A. SHAH ET AL. 25
0
2
4
6
8
10
12
0.5 1.5 2.5 3.5 4.5 5.5 6.5
Ratio of thickness to inner radius
Frequency
___________ _P er vio us Sur face
- - - - - - - - - - -I mpervious
Fluid-fi lled shel l
Flui d-filled shell
Em pty shel l
Empty shell
Figure 4. Frequency as a function of ratio of thickness to inner radius Rotatory vibrations of Fluid-filled and empty sphe rical
shells (Material-Ⅱ, n = 1).
Obviously, the frequency of an impervious surface is
higher than the frequency of pervious surface for thick
fluid filled spherical shells. Therefore, the presence of
fluid is increasing the frequency of an impervious sur-
face for thick shell. The frequency of an empty spherical
shell increases with the increase of thickness for a per-
vious surface. Frequency of an empty shell is higher than
the frequency of fluid filled shell. Thus the presence of
fluid is decreasing the frequency of a fluid filled shell for
a pervious surface. Frequency of an impervious surface
of empty spherical shell is lower than the frequency of a
fluid filled shell. Thus the absence of fluid in the spheri-
cal shell is decreasing the frequency of an empty shell in
case of an impervious surface for thick shell.
Figure 4 show the frequency of fluid filled and empty
spherical shell each for a pervious an impervious surface
in case of Material-II. Frequency of pervious and imper-
vious surface is almost same for thin fluid filled shell.
With the increase of thickness, the frequency of an
impervious surface is higher than the frequency of a per-
vious surface. In case of empty thin spherical shells,
again the frequency of pervious and impervious surface
is same. With the increase of thickness, the frequency of
a pervious surface is higher than the frequency of an im-
pervious surface. It is seen that the presence of mass-
coupling parameter has no significant effect in case of
thick fluid filled shell for a pervious surface. In case of
an impervious surface, the presence of mass- coupling
parameter is increasing the frequency of fluid filled thick
spherical shells. This phenomenon is reversed for empty
spherical shells where the absence of mass-coupling pa-
rameter is increasing the frequency of a pervious surface
while it has no effect on an impervious surface for thick
empty shells.
7. Concluding Remarks
The study of radial and rotatory vibrations of fluid-filled
and empty poroelastic spherical shells has lead to fol-
lowing conclusions:
1) The frequency of an impervious surface in case of
radial vibrations is higher than that of a pervious surface
in case of a fluid filled and empty spherical shells.
2) Mass-coupling parameter has no significant effect
on the frequency of fluid filled spherical shell for a per-
vious surface in case of radial vibrations.
3) The absence of mass-coupling parameter in Mate-
rial-II is increasing the frequency of an impervious sur-
face of fluid filled spherical shells in case of radial vibra-
tions.
4) Presence of fluid is increasing the frequency of an
impervious surface of thick spherical shell in case of
rotatory vibrations.
5) Presence of fluid is decreasing the frequency of a
pervious surface for rotatory vibrations.
6) Absence of fluid in the thick spherical shell is de-
creasing the frequency of an impervious surface for ro-
tatory vibrations.

S. A. SHAH ET AL. 26
7) Mass-coupling parameter is increasing the fre-
quency of fluid filled thick spherical shells of an imper-
vious surface in case of rotatory vibrations.
8. Acknowledgements
The authors are thankful to the editor, reviewers and the
editorial assistant for their suggestions and kind coopera-
tion in improving the quality of this paper.
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