Part-2
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM REGION
OF WATER
2.1
INTRODUCTION :
The study of blast wave propagation is now-a-days the target of scientist and medical
doctors who are using shock wave treatment as tools. Due to its medical and technical
applications, many techniques have been used to study the propagation of shock waves in
water/liquids by researchers. The first compendium of shock wave profiles was published
by Los Alamos National Laboratory (1982). A large effort has been made to the understanding
of this area in the recent past. Among possible experimental tools for studying these
processes, shock tubes are very convenient ones, since they allow for easier diagnostic
than any other experimental facility (Vetter and Sturtevant –1995, Houses and Chemouni
1996, Jaurden et al.-1997). Picone and Baris (1988), Yang et al.(1994), Yadav et
al.(1996), Grasso and Pirozzoli (2000), Vuong et al.(1999), Bates and Montogomery
(1999) and many others shared several features ,including the theoretical and computational
studies.
The motion of converging cylindrical shock waves in a perfect gas with constant
ratio specific heats has been studied by Stanyukovich (1960) using similar solutions. The
propagation of spherically converging shock in various metals has been studied by Yadav
and Singh (1982) by Whitham method. The shock wave propagation in water has been
studied by Bhatnager et al.(1969), Singh(1972), Rango rao and Ramanna (1973), Singh
et al.(1980), Singh and Shrivastava (1985) without taking gravity of earth. Viswakarma
et al.(1988) considered earth’s gravitation and time dependant energy release. The analysis
are completely analytical and are used for diverging shock only. The diverging shock
propagation in uniform and non-uniform gas has been studied by Yadav (1992), Yadav and
Tripathi (1995) considering the effect of overtaking disturbances.These results are in
good agreement with experimental results.
Chisnell (1998) described analytically the motion of converging spherical and
cylindrical shock waves in an ideal gas. Vuong et al.(1999) has studied strong spherical
converging shock by considering sonoluminiscence phenomenon using compressible
Navier-shocks equations. Recently, application of spherical micro-shock waves in
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
19
Biological science has been discussed by Jagdeesh and Takayama(2002). Recently, Yadav
and Gangwar (2003) have used Chester (1954)- Chisnell (1953)- Whitham (1958)
method to study the propagation of strong spherical shock in non-uniform medium
neglecting the effect of overtaking disturbances. Being an important phenomenon, it is
essential to take into account the effect of overtaking disturbances in the motion of shock
waves.The effect of overtaking disturbanes,on the propagation of shock waves in water has
been studied by Yadav and coworkers (2004).
Thomas(1957) used ‘Energy hypothesis’ for spherical blast waves. This hypothesis
was successfully applied by Bhutani(1966) to cylindrical blast waves in
Magnetogasdynamics. A theoretical study of explosion in water (without considering the
effect of gravity) is carried out by Singh and vola (1976) using Energy hypothesis.
Experimental verification of Energy hypothesis is given by Singh et al. (1980). Singh
(1982) used energy hypothesis to explosion problem in seawater considering the effect
of gravity was used by Vishvakarma et al. (1988). This hypothesis has been successfully
applied for the determination of entropy due to the propagation of blast wave in seawater
and in the propagation of blast wave in the air Yadav et al.(2005, 2006). Very recently
Sharma at al. (2008) studied the flow variables of seawater due to the propagation of blast
wave using the energy hypothesis of Thomas.
The aim of present part is to study the exploding cylindrical shock in uniform
region of water for two cases viz (I) when shock wave are strong and (II) when it is
weak.Assuming region where explosion occours, is of uniform density. Analytical relations
for mach number are obtained for freely propagation of shock using Chester - ChisnellWhitham in sectiosn 2.4. The effect of overtaking disturbances is included by Yadav co
worker approach in section 2.4.
It is found that shock strength decreases with propagation distance whereas it
increases with refractive index.The effect of overtaking disturbances is to enhance the
shock strength.Again it is found that the pressure and the particle velocity decreases with
propagation distance as well as with refractive index.The effect of overtaking disturbances
to reduce the pressure very much.where as it decreases with refractive index.The effect of
overtaking is to enhanace the shock strength.Similar results are reported by Sharma et al.
(2008).
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
20
2.2
BASIC EQUATIONS AND BOUNDRY CONDITIONS :
The basic equations for cylindrical shocks are -
u  u u  1 P  0
t
r
 r
{t  u r }  {ur   ur }  0
{ t  u r } c P  h  0

(2.1)
(2.2)
(2.3)
b g
where, u(r, t), P(r, t) and  r, t , are the particle velocity, the pressure and the density at
distance r from the origin at time t , respectively and  =1 for cylindrical shock.
2.2.1 BOUNDRY CONDITIONS :
Let p0 and  0 denotes the unperturbed values of pressure and density infront of
the shock and P and  be the values of respective quantities at any point immediately after
passes of shock, then the Rankine-Hugoniet conditions will permit us to express P ,  and
u in terms of unperturbed values of the quantities by means of the following equations,
LM M  1 OP
b gN MQ
L M  bn 1g OP 1
P   a M2
N bn 1g nbn 1gQ
b n  1g M
  
b n  1g M  2
2a0
n 1
u
2
2
0
(2.4a)
0
(2.4b)
2
0
a0 
2
b p  1gn
0
(2.4c)
(2.4d)
0
U
Where, M (= a ) is Mach number, U is the shock velocity, a0 is the sound velocity
0
in unperturbed medium and n is a constant
The boundry conditions for two cases are given belowCASE-I STRONG SHOCK :
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
21
For strong shock waves i.e. (M>>1) the condition (2.4a-2.4d) reduces to-
u
2a0 M
n1
P
2a02 M 2

n 1 0
  n 1 0
n 1
(2.5)
e j
a  Ma0 n  1
n 1
Where a0 
M
where
b p  1gn
0
0
U
a0 is called the Mach number.  0 , a0 , P ,  , n and U denote the
undisturbed value of density, local sound velocity behind the shock, pressure,the density,the
refractive index and shock velocity respectively.
CASE-II WEAK SHOCK :
We take Mach no. M as M  1 
Where (  <<1) for weak shock.
Thus, for weak shock, boundary conditions (2.4) reduce tou
P
b g
 b1  2 g
2a 0
1 
n 1
2a0
2
0
(2.6)
n 1
b g
a  a0  0 1  
Where a0 
2.3
b p  1gn
0
0
CHARACTERISTIC EQUATION :
The characteristic form of flow equation gives the following linear equation.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
22
e
j
p
 u  u u   P  P  u  a2 u   u  0
t
r
r r r
r
r
Where a is local sound Velocity immediately.
Linear combination of these equation [ from equations (2.1) to (2.3)] can be written
as-
b
g
c
h
P  P u     u a 2  u   u   a 2u  0
t r
r
t
r
(2.7)
The conditions that this combination involves the derivative in only one direction
(r, t) plane are given by-
b
P  P u  
t r
b
or rt  u  
g
(2.8a)
g
c
 u   u a 2  u
t
r
(2.8b)
h
(2.9a)
r
 ua
t
2
or  rt  a  u
(2.9b)
From equations (2.8) and (2.9)
b g
 u    a 2  u
2  a 2 and   a
For C+ disturbance   a
Equ. (2.8b) becomes
r
 ua
t
Now equ. (2.7) becomes
b g
c
h
P   u  a   u a 2  ua  a u   a 2u  0
t r
r
t
r
Using total derivatives for P and u ,we get the characterstic form of flow of diverging
shocks as
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
23
dP  adu  a 2 u dr  0
ua r
This equation is used to determine the stregnth of exploding cylindrical shock
waves.
For C- disturbance   a
(2.10)
Equ. (2.8b) and (2.9) becomesr  u  a
t
Now equ. (2.7) become-
b g c h
P  P bu  ag   u ca  auh  a u  a  u  0
t r
r
t
r
P  P u  a   u a2  au  a u  a2 u  0
t r
r
t
r
2
2
Writing total derivatives for P and u in this equationdP  adu  a 2
u dr  0
ua r
(2.11)
This equation is used to determine the stregnth of overtaking disturbances.
2.4.
STRONG EXPLODING SHOCKS :
2.4.1 FOR FREELY PROPAGATION OF CYLINDRICAL SHOCK WAVES :
Applying the boundary conditions given by the equation (2.5) to equ. (2.10) we
get for c+ disturbance  20 a02 M 2    2a
M 0
2 a0
n

1
n

1
n  1  M dr  0
4 a0
dM 
   0 a0 M
dM 

n 1
n 1 0
n 1
n  1 0 2a0
r
M   0 a0 M
n 1
Divided by
2a0  0
we get
n 1
n  1  2 M 3a
0
0
n

1
dr  0
2 MdM 
 0 MdM  n  1
n 1
a0 M 2   0 r
n 1
bn  1g  
dM  n  1
dr  0
M
2   bn  1g r
2
2  n  1 0
n 1
2
0
0
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
24
bn  1g  
dM 
n 1
dr  0
M L 2bn  1g  bn  1g O
r
MN n  1  PQ 2   bn  1g
bn  1g  
A
Let
2bn  1g  bn  1g 2   bn  1g
2
2
0
0
0
2
2
0
0
0
Then above equation takes the formdM  A dr  0
M
r
Integrating we get
log M  A log r  log K  Mr A  K
(2.12)
M  Kr  A
This equation represents the shock strength of exploding strong cylindrical shock
propagating in uniform region of water.
Now Shock Velocity
U  Ma0  Ka0r  A
(2.13)
The Particle Velocity just behind the shock
u
2 a0
Kr  A
n 1
(2.14)
and the Pressure
P
2a02 M 2  0 2a02 K 2r 2 A

0
n 1
n 1
(2.15)
2.4.2 OVERTAKING DISTURBANCES :
From equation(2.14) we get the particle velocity for C disturbances-
u 
2a0
Kr  A
n 1
b g
b g b g
(2.16)
2
n  1  20
A

where
2 n  1  n  1 0 2  0 n  1
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
b g
25
Again putting the value of boundry conditions given by the equation (2.5) on the
characteristic equation(2.11) we get
 20 a02 M 2    2a
M 0
2a0
n

1
n

1
n  1  M dr  0
4a0
dM 
   0a 0 M
dM 

n 1
n 1 0
n 1
n  1 0 2a0
r
M   0 a0 M
n 1
Divided by
2a0  0
we get
n 1
n  1  2 M 3a
0
0
n

1
dr  0
2 MdM 
 0 MdM  n  1
n 1
2
a0 M
 0 r
n 1
bn  1g  
n

1
dM
dr  0
2

 n 1
n 1
M
2   bn  1g r
bn  1g  
dM 
n 1
dr  0
M L 2bn  1g  bn  1g O
r

2


n

1
b
g
MN n  1
PQ
bn  1g  
B
2bn  1g  b n  1g  2   b n  1g
2
2
0
0
0
2
2
0
0
2
Let
0
2
0
0
0
Then above equation takes the formdM  B dr  0
M
r
Integrating we get
log M  B log r  log K '
Mr B  K '
M  K' r B
(2.17)
Now Shock Velocity
U  Ma0  K ' a0r  B
(2.18)
The particle velocityu
2a0
K' r B
n 1
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
(2.19)
26
Pressure
2a02 M 2  0 2a02 K '2 r 2 B

0
n 1
n 1
P
(2.20)
Now from equation(2.19)
u 
2a0
Kr  B
n 1
(2.21)
In presence of overtaking disturbances the resultant particle velocity is given by-
2
dU *
n1
du  du  du* 
(2.23)
Where U* is the resultant shock velocity in presence of overtaking disturbances.
For this we finding the values of du+ and du- as below
b g
2a

Kb Bgr
n 1
2a0
K  A r  A1dr
n 1
du 
du
0
(2.24)
 B 1
dr
(2.25)
from the equation (2.21),(2.24) and (2.25) we get
du  du  
a0 K
LM A
Nr
A 1
LM
N
OP
Q
2a0
B
2 dU *
K A
A1  B1 dr  du 
n 1 r
n 1
r
OP
Q
 B
dr  dU *
B 1
r
Integrating
LM A  B OP
N r  A r  B Q
 a KL A  B O
NM r r QP
U *   a0 K
U*
0
A
A
B
B
( 2.26)
U* is the shock velocity in presence of overtaking disturbances i.e. shock velocity modified
by overtaking velocity. Therefore modified shock strength will be
LM
N
*
M *  U  K 1A  1B
a0
r
r
OP
Q
(2.27)
The particle velocity in presence of overtaking disturbancesu* 
LM
N
2 a0
K 1A  1B
r
n 1 r
OP
Q
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
(2.28)
27
And the pressure just behind the shock in presence of overtaking disturbanceP* 
2.5
LM
N
2a02 M *2
2a 2
 0  0  0 K 2 1A  1B
n 1
n 1
r
r
OP
Q
(2.29)
WEAK EXPLODING SHOCK WAVES :
2.5.1 FOR FREELY PROPAGATION :
Appling the boundary conditions given by the equation. (2.5)to the eq.(210) we get-
b g
b g
b g
b g
2a 1 
2
 20 a02 1  
2a02  20
4a02  0
n  1 dr  0
1  d 
1  d   0 n  1
n 1
n 1
n  1 2a0
1    0a0 1 
n 1
b g
b g
e j
b g
Divided by 2a02  0 1  we have
LM
N
e jb g
b g
n  1  b1 g dr  0
n  1 2   bn  1g r
2
n  1 1  a0
0
n 1
dr  0
 2 d   20 n  1
n 1 n 1
2
a0 1 
 0 r
n 1
OP
Q
LM   2 OPd  
N n  1 n  1Q
0
2
0
0
LM   2 OP d    nn  11  dr  0
N n  1 n  1Q b1 g 2   bn  1g r
 bn  1g
d 
dr  0
1



b g bn  1g 2   bn  1g LM  2 OP r
N n  1 n  1Q
 bn  1g 
Let A  2   n  1  n  1  2n  1
b g
2
0
0
0
2
0
0
0
(2.30)
2
2
0
0
0
0
Equation (2.30) simplfy to
d   A dr  0
1  0 r
Integrating
b g
b1 gr  K
b1 g  K r
log 1   A0 log r  log K0
A0
0
 A0
0
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
28
  K0r  A0  1
Now shock strengthM  1  K0r
(2.31)
 A0
Shock VelocityU  K 0 a0 r
(2.32)
 A0
The particle velocityA
u  2U  2 K0a0r 0
n 1 n 1
(2.33)
PressureP
2U 2  0 2  0 2 2 2 A0

K a r
n 1
n 1 0 0
(2.34)
2.4.2 FOR OVERTAKING DISTURBANCES :
From equation(2.33) we get the particle velocity for C disturbancesu 
2
Ka0 r  A0
n 1
  K0r  A0  1
2
 20 n  1 
where A0  2   n  1  n  1  2n  1
0
0
b g
b g
Again putting the value of boundry conditions given by the equation (2.5) on the
characteri stic equation(2.11) w e get f or C - disturbances-
b g
b g
b g
b g
2a 1 
2
 20a02 1  
2a02  20
4a02  0
n  1 dr  0
1  d 
1  d   0 n  1
n 1
n 1
n  1 2a 0
1    0a0 1 
n 1
b g
b g
e j
e j b g dr  0
r
b g
b g dr  0
b g r
2
n  1 1  a0
0
n 1
 2 d   20 n  1
n 1 n 1
2
a0 1 
 0
n 1
 1 
0
 2 d   20 n  1
n 1 n 1
n  1 2  0 n  1
LM
OP
N
Q
LM
OP
N
Q
LM   2 OP d    nn  11  dr  0
N n  1 n  1Q b1 g 2   bn  1g r
b g
0
2
0
0
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
29
b g
dr
b g bn  1g 2   bn  1g LM   2 OP r  0
N n  1 n  1Q
 bn  1g 
B 
2   bn  1g  bn  1g  2bn  1g
 20 n  1 
d 
1 
0
0
2
2
0
0
0
(2.35)
o
Then above equation takes the formd   B dr  0
1  0 r
Integrating we get
b g
b1 gr  K '
log 1   B0 log r  log K '
B0
(2.36)
M  1 K' rB0
where M is called the shock strength of exploding strong cylindrical shockNow Shock Velocity
U Kar
' 0 B0
(2.37)
Particle Velocity
B
u  2U  2 K ' a0 r 0
n 1 n 1
(2.38)
Pressure
2U 2  0 2  0 2 2 2 B0
P

K a r
n 1
n 1 0 0
(2.39)
For overtaking disturbances, we use the conditions
du  du 
2
dU *
n 1
(2.40)
From eq.(2.33),eq.(2.38) and eq.(2.40)du  du 
K0a0
LM A
Nr
0
A0 1
LM
N
OP
Q
2 K0a0
A 1
A0r 0  B0r  B0 1 dr  2 dU *
n 1
n 1

B0
r
B0 1
OPdr  dU
Q
*
Integrating Integrating
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
30
LM
N
U *  a0 K0 1A0  1B0
r
r
OP
Q
(2.41)
U* is the shock velocity in presence of overtaking disturbances i.e. shock velocity
modified by overtaking velocity. Therefore modified shock strength will be
*
LM
MN
M *  U  K0 1A0  1
a0
B
r
r 0
OP
PQ
(2.42)
The particle velocity for overtaking disturbancesu* 
LM
MN
2a0
2 a0
M* 
K 1  1
n 1
n  1 0 r A0
B
r 0
OP
PQ
(2.43)
And the pressure for overtaking disturbances-
LM
N
2a02
2a02 2 2 1
*2
P 
M 0 
 K
 1
n 1
n  1 0 0 r A0 r B0
*
2.6
OP
Q
2
(2.44)
RESULTS AND DISCUSSION-
2.6.1 FOR STRONG SHOCK2.6.1.1 THE SHOCK STRENGTHEquation (2.12) represents the shock strength for freely propagation of exploding
shock waves, in uniform region of water.
r
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
M
24.64826
23.85423
23.14190
22.49786
21.91165
21.3749
20.88089
20.42411
20.00000
19.60477
19.23520
18.88857
18.56255
18.25513
Table 2.1 The variation of shock strength M with propagation distance r for exploding
strong cylindrical shock waves in uniform medium.
For the initial value of  0 =1,  =1, refractive index n=1.5 and a0 =1450, we have
calculated the constant of integration from equation (2.11).Taking this value of constant of
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
31
integration throughout, we have computed shock strength using equation (2.12) and
presented in table 2.1. Changing the value of r from 1.2 to 2.5, the corresponding shock
strength takes the values from 24.6482625 to 18.25513. It shows that as r increases shock
strength M decreases slowly, while its graphical representation confirms this statement
[fig 2.1].
26
24
22
M
20
18
1.2 1.4 1.6 1.8
2
2.2 2.4
r
Fig. 2.1. The variation of shock strength M with propagation distance r for
exploding strong cylindrical shock waves in uniform medium.
Again for the initial value of  0 =1,  =1, r=2 and changing the value of refractive
index n from 1.5 to 2.8 shock strength M increases very slowly from 20.17076 to 21.03295
shows very less effect of refractive index n on shock strength M as shown in table 2.2.Their
graphical representation is shown in fig.2.2 below.
n
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
M
20.17076
20.31244
20.43132
20.53207
20.61815
20.69226
20.75645
20.81239
20.86137
20.90445
20.9425
20.97622
21.00621
21.03295
Table 2.2. The variation of shock strength M with refractive index n for exploding
strong cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
32
21.2
21
20.8
M 20.6
20.4
20.2
20
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
Fig. 2.2 The variation of shock strength M with refractive index n for exploding strong
cylindrical shock waves in uniform medium.
Equation (2.27) represents the shock strength of exploding shock waves in presence
of overtaking disturbances. For the initial value of  0 =1,  =1 refractive index n=1.5 and
a0 =1450, the variation of shock strength of exploding cylindrical shock modified by
overtaking disturbances is computed using equation(2.27). Changing with the value of r
from 1.2 to 2.5 the corresponding to overtaking shock strength takes the value from 29.7915
to 18.26206. It shows that as r increases modified shock strength M* (strength of the
shock in presence of overtaking disturbances) decreases slowly as shown in table 2.3,
while the graphical representation is given in fig 2.3.
r
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
M*
29.79518
26.35854
24.42726
23.18867
22.2981
21.59884
21.01477
20.50641
20.05187
19.6382
19.25719
18.90331
18.5726
18.26209
Tab. 2.3 The variation of modified shock strength M* with propagation distance r for
exploding strong cylindrical shock waves in uniform medium
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
33
30
28
26
24
M*
22
20
18
1 .2
1 .4
1 .6
1 .8
2
2 .2
2 .4
r
Fig. 2.3 The variation of modified shock strength M* with propagation distance r for
exploding strong cylindrical shock waves in uniform medium
In presence of overtaking disturbances the modified shock strength is computed
,we conclude that the value of overtaking shock strength M* is increases from 20.2531 to
21.03295 as n increases from 1.5 to 2.8 as shown in table 2.4 while the graphical
representation in is shown fig 2.4.
n
M*
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
20.2531
20.41281
20.53427
20.62449
20.69176
20.74412
20.78823
20.82873
20.868
20.90636
20.94282
20.97624
21.00621
21.03295
Tab. 2.4 The variation of modified shock strength M* with refractive index n for
exploding strong cylindrical shock waves in uniform medium.
M*
21.2
21
20.8
20.6
20.4
20.2
20
1.5
1.7
1.9
2.1
2.3
2.5
2.7
n
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
34
Fig. 2.4 The variation of modified shock strength M* with refractive index n for
exploding strong cylindrical shock waves in uniform medium.
2.6.1.2 THE SHOCK VELOCITYEquation (2.13) represents the shock velocity U for freely propagation of exploding
shock waves in uniform region of water.
For the initial value of  0 =1,  =1, refractive index n=1.5 and a0 =1450, we calculate
the constant of integration from equation (2.12). Taking this value of constant of integration
throughout, we have computed shock velocity using equation (2.13) and presented in table
2.5. Changing with the value of r from 1.2 to 2.5 the corresponding shock velocity takes
the value from 8133.926 to 6024.194 respectively. It shows that as r increases shock
velocity U decreases slowly, while its graphical representation conforms this statment [fig
2.5.
r
U
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
8133.926
7871.897
7636.826
7424.295
7230.843
7053.717
6890.693
6739.955
6600
6469.573
6347.615
6233.228
6125.642
6024.194
Tab. 2.5 The variation of shock velocity U with propagation distance r for
strong cylindrical shock waves in uniform medium.
exploding
8500
8000
U
7500
7000
6500
6000
1 .2
1 .4 1 .6
1 .8
2
2 .2 2 .4
r
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
35
Fig. 2.5 The variation of shock velocity U with propagation distance r for
strong cylindrical shock waves in uniform medium.
exploding
Again for the initial value of  0 =1,  =1, r=2 and changing the value of refrative
index n from 1.5 to 2.8 .We conclude that the value of shock velocity U increasing fastly
from 6656.3521 to 6940.873 shows good effect of refrative index n . variation of shock
velocity U with refractive index n is shown in table 2.6. Their graphical representation is
shown in fig. 2.6.
n
U
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
6656.351
6703.104
6742.337
6775.582
6803.99
6828.444
6849.63
6868.088
6884.252
6898.469
6911.024
6922.153
6932.048
6940.873
Tab. 2.6 The variation of shock velocity U with refractive index n for exploding strong
cylindrical shock waves in uniform medium.
6950
6900
6850
U 6800
6750
6700
6650
1.5 1.6 1.7 1.8 1.9
2
2.1 2.2 2.3 2.4 2.5 2.6
n
Fig. 2.6 The variation of shock velocity U with refractive index n for exploding strong
cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
36
Equation (2.26) represents the shock velocity U* of exploding shock waves in
presence of overtaking disturbances. For the initial value of  0 =1,  =1 refractive index
n=1.5 and a0 =1450, the variation of shock velocity U* of exploding cylindrical shock
modified by overtaking disturbances is computed using equation(2.26). Changing with the
value of r from 1.2 to 2.5 the corresponding to overtaking shock velocity U* takes the
value from 9832.41to 6026.491. While the graphical representation is given in fig 2.7. It
shows that as r increases modified shock velocity U* (strength of the shock in presence of
overtaking diturbances) decreases very fastly as shown in table 2.7,while its graphical
representation is shown in fig 2.7.
r
U*
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
9832.41
8698.32
8060.997
7652.262
7358.373
7127.618
6934.875
6767.114
6617.117
6480.606
6354.874
6238.093
6128.959
6026.491
Tab. 2.7 The variation of shock velocity U* modified by overtaking disturbances with
propagation distance r for exploding strong cylindrical shock waves in uniform medium.
10000
9500
9000
8500
8000
U* 7 5 0 0
7000
6500
6000
1 .2
1 .4
1 .6
1 .8
2
2 .2
2 .4
r
Fig. 2.7 The variation of shock velocity U* modified by overtaking disturbances with
propagation distance r for exploding strong cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
37
In presence of overtaking disturbances the shock velocity U* is computed. We
conclude that the values of shock velocity U*, changing with the value of n from 1.5 to 2.8
the corresponding overtaking shock velocity U* takes the value from 66835.522 to
6940.873. We concluded that as n increases overtaking shock velocity U* is also increases
slowly as shown in table 2.8, while its graphical representation is given in fig 2.8.
n
U*
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
6683.522
6736.226
6776.311
6806.08
6828.282
6845.561
6860.115
6873.48
6886.441
6899.1
6911.131
6922.16
6932.048
6940.873
Tab. 2.8 The variation of shock velocity U* modified by overtaking disturbances with
refractive index n for exploding strong cylindrical shock waves in uniform medium.
6950
6900
6850
6800
U*
6750
6700
6650
6600
1.5
1.7
1.9
2.1
2.3
2.5
2.7
n
Fig.2.8 The variation of shock velocity U* modified by overtaking disturbances with
refractive index n for exploding strong cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
38
2.6.1.3 THE PARTICLE VELOCITY :
Equation (2.14) represents the particle velocity for freely propagation of exploding
shock waves, in uniform region of water.
For the initial value of  0 =1,  =1 refractive index n=1.5 and a0 =1450, we have
calculated the constant of integration from equation (2.12).Taking this value of constant of
integration throughout, we have computed particle velocity u using equation (2.12) and
presented in table 2.9.Changing with the value of r from 1.2 to 2.5 the corresponding
from particle velocity takes the value from 6778.272 to 5020.161 respectively. we concluded
that the particle velocity u decreases very sharply. While the graphical representation
conform this statement [fig 2.9].
r
u
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
6778.272
6559.914
6364.022
6186.913
6025.703
5878.097
5742.244
5616.629
5500
5391.31
5289.679
5194.357
5104.702
5020.161
Table 2.9 The variation of particle velocity u with propagation distance r for exploding
strong cylindrical shock waves in uniform medium.
7000
6500
u
6000
5500
5000
4500
1 .2
1 .4
1 .6
1 .8
2
2 .2
2 .4
r
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
39
Fig. 2.9 The variation of particle velocity u with propagation distance r for exploding
strong cylindrical shock waves in uniform medium.
Again for the initial value of  0 =1,  =1, r=2,we calculated the constant of
integration. With changing the value of refractive index n from 1.5 to 2.8 the value of
particle velocity u decreasing fastly from 5325.081 to3653.091 shows good effect of
refractive index n to the particle velocity u.Variation of particle velocity u with refractive
index n is shown in table 2.10.While its graphically representation is shown in fig. 2.10.
n
u
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
5325.081
5156.234
4994.324
4839.701
4692.407
4552.296
4419.116
4292.555
4172.274
4057.923
3949.157
3845.64
3747.053
3653.091
Tab. 2.10 The variation of particle velocity u with refractive index n for exploding
strong cylindrical shock waves in uniform medium.
u
5400
5200
5000
4800
4600
4400
4200
4000
3800
3600
1 .5
1 .7
1 .9
2 .1
2 .3
2 .5
2 .7
n
Fig 2.10 The variation of particle velocity u with refractive index n for exploding strong
cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
40
Equation (2.28) represents the particle velocity of exploding shock waves in presence
of overtaking disturbances. For the initial value of  0 =1,  =1 refractive index n=1.5 and
a0 =1450, the variation of particle velocity u of exploding cylindrical shock modified by
overtaking disturbances is computed using equation(2.28). Changing with the value of r
from 1.2 to 2.5 the corresponding to modified particle velocity u* (strength of the shock
in presence of overtaking disturbances) takes the value from 8193.675 to5022.076 it shows
that the modified particle velocity u* decreases very fastly as shown in table 2.11, while its
graphical representation is given in fig 2.11.
r
u*
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
8193.675
7248.6
6717.497
6376.885
6131.978
5939.682
5779.062
5639.262
5514.264
5400.505
5295.728
5198.411
5107.466
5022.076
Tab. 2.11 The variation of modified particle velocity u* with propagation distance r for
exploding strong cylindrical shock waves in uniform medium.
8300
7800
7300
6800
u *6 3 0 0
5800
5300
4800
1 .2
1 .4
1 .6
1 .8
2
2 .2
2 .4
r
Fig. 2.11 The variation of modified particle velocity u* with propagation distance r for
exploding strong cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
41
In presence of overtaking disturbances the particle velocity u* is computed. We
conclude that the value of overtaking particle velocity u* is increases from 5346.818 to
3653.091respectively. we concluded that as n increases from 1.5 to 2.8, overtaking particle
velocity u* is decreases slowly as shown in table 2.12,while its graphical representation
is shown in fig 2.12.
n
u*
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
5346.818
5181.712
5019.489
4861.486
4709.16
4563.707
4425.881
4295.925
4173.6
4058.294
3949.217
3845.644
3747.053
3653.091
Tab.2.12 The variation of modified particle velocity u* with refractive index n for
exploding strong cylindrical shock waves in uniform medium.
5500
5000
u*
4500
4000
3500
1.5
1.7
1.9
2.1
2.3
2.5
2.7
n
Fig. 2.12 The variation of modified particle velocity u* with refractive index n for
exploding strong cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
42
2.6.4 THE PRESSUREEquation (2.15) represents the pressure P for freely propagation of exploding shock
waves, in uniform region of water.
For the initial value of  0 =1,  =1 refractive index n=1.5 and a0 =1450, we have
calculated the constant of integration from equation (2.12).Taking this value of constant
of integration throughout, we have computed pressure P using equation (2.15) and presented
in table2.13. Changing the value of r from 1.2 to 2.5 the corresponding takes the value
from 55133967 to 30242425 respectively. It shows that as r increases pressure P decreases
very fastly, while its graphical representation confirms this statement [fig 2.13].
r
P
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
55133967
51638963
48600931
45933467
43570911
41462432
39568043
37855829
36300000
34879474
33576845
32377608
31269574
30242425
Table 2.13 The variation of pressure P with propagation distance r for exploding strong
cylindrical shock waves in uniform medium.
59000000
54000000
49000000
P
44000000
39000000
34000000
29000000
1.2 1.4 1.6 1.8
2
2.2 2.4
r
Fig 2.13 The variation of pressure P with propagation distance r for exploding strong
cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
43
Again for the initial value  0 =1, M=20,  =1, r=2, with changing the value of
refractive index n from 1.5 to 2.8 of pressure P decreases very fastly from 35445608 to
25355638. From above discussion we concluded that as far as the refractive index n
increases the pressure P takes the lower value as shown in table 2.14. Their graphical
representation is shown in fig.2.14.
n
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
P
35445608
34562768
33673411
32791791
31927092
31085101
30269310
29481648
28722982
27993454
27292720
26620111
25974752
25355638
Table 2.14 The variation of pressure P with refractive index n for exploding strong
cylindrical shock waves in uniform medium.
P
37000000
35000000
33000000
31000000
29000000
27000000
25000000
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
Fig 2.14 The variation of pressure P with refractive index n for exploding strong
cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
44
Equation (2.29) represents the pressure P* of exploding shock waves in presence
of overtaking disturbances. For the initial value of  0 =1,  =1, refractive index n=1.5 and
a0 =1450, the variation of pressure P* of exploding cylindrical shock modified by overtaking
disturbances is computed using equation(2.29). Changing with the value of r from 1.2 to
2.5 the corresponding to modified pressure P* (strength of the shock in presence of
overtaking disturbances) takes the value from 22.13081 to 22.13081,Shows that the
modifited pressure P* does not depend on the variation of propagation distance r overtaking
pressure P* as shown in table 2.15,while its graphical representation is given in fig 2.15.
r
P*
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
22.13081
Table 2.15 The variation of pressure P* modified by overtaking disturbances with
propagation distance r for exploding strong cylindrical shock waves in uniform medium.
P*
22.14
22.12
22.1
22.08
22.06
22.04
22.02
22
1.2
1.4
1.6
1.8
2
2.2
2.4
r
Fig. 2.15 The variation of pressure P* modified by overtaking disturbances with
propagation distance r for exploding strong cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
45
In presence of overtaking disturbances the pressure P* is computed . We conclude
that the value of overtaking shock pressure P* is increases from 21.24558 to 13.97736
when the refractive index n changing from 1.5 to 2.8 as shown in table 2.16. While its
graphically representation is given in the fig. 2.16.
n
P*
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
21.24558
20.42844
19.67183
18.96927
18.31516
17.70465
17.13353
16.59811
16.09514
15.62175
15.17542
14.75388
Table 2.16 The variation of pressure P* modified by overtaking disturbances with
refractive index n for exploding strong cylindrical shock waves in uniform medium.
P*
22
21
20
19
18
17
16
15
1.5
1.7
1.9
2.1
2.3
2.5
2.7
n
Fig 2.16 The variation of pressure P* modified by overtaking disturbances with
refractive index n for exploding strong cylindrical shock waves in uniform medium.
2.6.2 FOR WEAK SHOCK2.6.2.1-THE SHOCK STRENGTH- Equation (2.31) represents the shock strength for
freely propagation of exploding weak shock wave in uniform region of water.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
46
For weak shock , the initial value M=1.25,  0 =1,  =1, refractive index n=1.5 and
a0 =1450, we have calculated the constant of integration from equation (2.31).Taking this
value of constant of integration throughout, we have computed shock strength using equation
(2.36) and presented in table 2.17.Changing the value of r from 1.2 to 2.5 the corresponding
shock strength takes the value from 2.25 to 1.934154 respectively. It shows that as r
increases shock strength M decreases slowly. Its graphical representation confirm this
statement [fig 2.17].
r
M
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.25
2.21092
2.175828
2.144073
2.115145
2.088637
2.064223
2.041633
2.020645
2.001074
1.982764
1.96558
1.94941
1.934154
Table 2.17 The variation of shock strength M with propagation distance r for
ploding weak cylindrical shock waves in uniform medium.
ex-
2.3
2.2
M
2.1
2
1.9
1.2
1.4
1.6
1.8
2
2.2
2.4
r
Fig. 2.17 The variation of shock strength M with propagation distance r for exploding
weak cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
47
Again for the initial value  0 =1,  =1, r=2, and changing the value of refractive
index n from 1.5 to 2.8 of shock strength M increases very slowly from 2.020645 to
2.064272 respectively. It shows that as r increases shock strength M decreases slowly as
shown in table 2.18. Their graphical representation is shown in table and fig 2.18.
n
M
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.020645
2.027814
2.03383
2.038927
2.043283
2.047033
2.050281
2.053112
2.05559
2.05777
2.059695
2.061402
2.062919
2.064272
Table 2.18 The variation of shock strength M with refractive index n for exploding
weak cylindrical shock waves in uniform medium.
M
2.07
2.06
2.05
2.04
2.03
2.02
2.01
1.5
1.7
1.9
2.1
2.3
2.5
2.7
n
Fig. 2.18 The variation of shock strength M with refractive index n for exploding weak cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
48
Equation (2.42) represents the shock strength of exploding shock waves in presence
of overtaking disturbances. For the initial value of  0 =1,  =1 refractive index n=1.5 and
a0 =1450, the variation of shock strength of exploding cylindrical shock modified by
overtaking disturbances is computed using equation (2.27). Changing the value of r from
1.2 to 2.5 the corresponding to modified shock strength takes the value from 1.544095 to
0.934803 respectively. It shows that as r increases modified shock strength M* (strength
of the shock in presence of overtaking disturbances) decreases slowly as shown in table
2.19, while its graphical representation is given in graph 2.19..
r
M*
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
1.54409485
1.36185981
1.25722304
1.18987686
1.14189497
1.10477791
1.07424717
1.04802106
1.02481147
1.00384878
0.98464656
0.96688016
0.95032151
0.9348028
Table 2.19 The variation of modified shock strength M* with propagation distance r for
exploding weak cylindrical shock waves in uniform medium
M*
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
1.2
1.4
1.6
1.8
2
2.2
2.4
r
Fig. 2.19 The variation of modified shock strength M* with propagation distance r for
exploding weak cylindrical shock waves in uniform medium
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
49
In presence of overtaking disturbances the modified shock strength is computed.
We conclude that the value of overtaking shock strength M* is increases from 1.02481147
to 1.06427202 as shown in table 2.20. While its graphical representation is shown in [fig
2.20].
n
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
M*
1.02481147
1.0328927
1.03903906
1.04360379
1.04700804
1.04965753
1.05188914
1.0539384
1.05592575
1.05786681
1.05971156
1.06140272
1.06291894
1.06427202
Tab. 2.20 The variation of modified shock strength M* with refractive index n for
exploding weak cylindrical shock waves in uniform medium.
1.07
1.06
1.05
M*
1.04
1.03
1.02
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
.
Fig. 2.20 The variation of modified shock strength M* with refractive index n for
exploding weak cylindrical shock waves in uniform medium.
2.6.2.2 THE SHOCK VELOCITY U
Equation (2.32) represents the shock velocity U for freely propagation of exploding
weak shock wave in uniform region of water.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
50
For weak shock ,initially we take M=1.25,  0 =1,  =1 refractive index n=1.5 and
a0 =1450. Using these initial constant values first of all, we calculated the value of the
constant of integration. After that changing with the value of r from 1.2 to 2.5 the
corresponding shock velocity U takes the value from 3262.5 to 2804.523 respectively. It
shows that as r increases shock velocity U decreases slowly as shown in table 2.21, while
its graphical representation confirms this statement [fig 2.21].
r
U
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
3262.5
3205.834
3154.951
3108.906
3066.96
3028.524
2993.123
2960.368
2929.936
2901.558
2875.007
2850.091
2826.644
2804.523
Tab. 2.21 The variation of propagation distance r for shock velocity U exploding weak
cylindrical shock waves in uniform medium.
3300
3200
3100
U
3000
2900
2800
1.2 1.4 1.6 1.8 2 2.2 2.4
r
Fig. 2.21 The variation of propagation distance r for shock velocity U exploding weak
cylindrical shock waves in uniform medium.
For weak shock ,initially we take M=1.25,  0 =1,  =1 propagation distance r=2
and a0 =1450.with changing the value of n from 1.5 to 2.8 the corresponding shock velocity
U takes the value from 2929.936 to 2993.194 respectively. It shows that as n increases
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
51
shock velocity U increases slowly as shown in table 2.22. Their graphical representation
is shown in table and fig 2.22.
n
U
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2929.936
2940.33
2949.053
2956.445
2962.761
2968.198
2972.908
2977.012
2980.606
2983.767
2986.558
2989.032
2991.232
2993.194
Tab. 2.22 The variation of shock velocity U with refractive index n for exploding weak
cylindrical shock waves in uniform medium.
2995
2985
2975
2965
U
2955
2945
2935
2925
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
Fig. 2.22 The variation of shock velocity U with refractive index n for exploding weak
cylindrical shock waves in uniform medium.
Equation (2.41) represents the shock velocity U* of exploding cylindrical shock
waves in presence of overtaking disturbances. For the initial value of M=1.25,  0 =1,  =1
refractive index n=1.5 and a0 =1450, the variation of shock velocity U* (strength of the
shock in presence of overtaking disturbances) of exploding cylindrical shock modified by
overtaking disturbances is computed using equation (2.41). Changing with the value of r
from 1.2 to 2.5 the corresponding to modified shock velocity U* takes the value from
2238.93754to 1355.46406 as shown in table 2.23, while its graphical representation is
shown in fig 2.23.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
52
r
U*
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2238.93754
1974.69673
1822.9734
1725.32144
1655.7477
1601.92797
1557.6584
1519.63053
1485.97663
1455.58073
1427.73752
1401.97624
1377.96619
1355.46406
Tab. 2.23The variation of shock velocity U* modified by overtaking disturbances with
propagation distance r for exploding weak cylindrical shock waves in uniform medium.
2400
2200
2000
1800
u*
1600
1400
1200
1000
1.2 1.4 1.6 1.8
2
2.2 2.4
r
Fig. 2.23 The variation of shock velocity U* modified by overtaking disturbances with
propagation distance r for exploding weak cylindrical shock waves in uniform medium.
In presence of overtaking disturbances the shock velocity U* is computed. We
conclude that the value of overtaking shock velocity U* Changing with the value of n from
1.5 to 2.8 the corresponding overtaking shock velocity U* takes the value from 1485.97663
to 1543.19443 with the fixed value of propagation constant r=2. It can be shown in table
2.24. Their graphical representation is shown in figure 224.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
53
n
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
U*
1485.97663
1497.69441
1506.60664
1513.22549
1518.16166
1522.00341
1525.23925
1528.21068
1531.09233
1533.90687
1536.58177
1539.03395
1541.23246
1543.19443
Tab. 2.24 The variation of shock velocity U* modified by overtaking disturbances with
refractive index n for exploding weak cylindrical shock waves in uniform medium.
1560
1540
1520
U*
1500
1480
1460
1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7
n
Fig. 2.24 The variation of shock velocity U* modified by overtaking disturbances with
refractive index n for exploding weak cylindrical shock waves in uniform medium.
2.6.3 THE PARTICLE VELOCITY :
Equation (2.33) represents the particle velocity u for freely propagation of exploding
weak shock wave in uniform region of water.
For weak shock ,initially we take M=1.25,  0 =1,  =1, refractive index n=1.5 and
a0 =1450,we have calculated the constant of integration, from equation (2.31). Taking this
value of constant of integration throughout, we have computed particle velocity u using
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
54
equation (2.33) and presented in table 2.25.Changing with the value of r from 1.2 to 2.5,
the corresponding from particle velocity takes the values from 2610.0 to 2243.619
respectively. It shows that as r increases the particle velocity u decreases fastly. Their
graphical representation is shown in fig 2.25.
r
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
u
2610
2564.667
2523.961
2487.124
2453.568
2422.819
2394.498
2368.294
2343.948
2321.246
2300.006
2280.073
2261.315
2243.619
Tab. 2.25 The variation of particle velocity u with propagation distance r for exploding
weak cylindrical shock waves in uniform medium.
2640
2540
u
2440
2340
2240
1.2 1.4 1.6 1.8
2
2.2 2.4
r
Fig. 2.25 The variation of particle velocity u with propagation distance r for exploding
weak cylindrical shock waves in uniform medium.
For weak shock, initially we take M=1.25,  0 =1,  =1, propagation distance r=2
and a0 =1450, changing the value of n from 1.5 to 2.8 the corresponding shock particle
velocity u takes the value from 2343.948 to 1575.365. It shows that as n increases, particle
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
55
velocity u decreases very fastly as shown in table 2.26. Their graphical representation is
shown in fig 2.26.
n
u
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2343.948
2261.792
2184.484
2111.746
2043.283
1978.799
1918.005
1860.632
1806.428
1755.157
1706.605
1660.574
1616.882
1575.365
Tab. 2.26 The variation of particle velocity u with refractive index n for exploding weak
cylindrical shock waves in uniform medium.
2500
2300
u
2100
1900
1700
1500
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
Fig. 2.26 The variation of particle velocity u with refractive index n for exploding weak
cylindrical shock waves in uniform medium.
Equation (2.43) represents the particle velocity of exploding weak shock waves in
presence of overtaking disturbances. For the initial value of M=1.25,  0 =1,  =1 refractive
index n=1.5 and a0 =1450, the variation of particle velocity u of exploding cylindrical
shock modified by overtaking disturbances is computed using equation (2.43). Changing
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
56
with the value of r from 1.2 to 2.5 the corresponding to modified particle velocity u*
(strength of the shock in presence of overtaking disturbances) takes the value from u* is
also decreases from 1791.15003to 1084.37125 as shown in table 2.27. Their graphical
representation is shown in table and fig 2.27.
r
u*
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
1791.15003
1579.75738
1458.37872
1380.25715
1324.59816
1281.54237
1246.12672
1215.70443
1188.7813
1164.46458
1142.19001
1121.58099
1102.37295
1084.37125
Tab.2.27 The variation of particle velocity u with refractive index n for exploding weak
cylindrical shock waves in uniform medium.
1800
1600
u* 1400
1200
1000
1.2 1.4 1.6 1.8
2
2.2 2.4
r
Fig. 2.27 The variation of particle velocity u with refractive index n for exploding weak
cylindrical shock waves in uniform medium.
In presence of overtaking disturbances the particle velocity u* is computed. We
conclude that the value of particle velocity u* is decreases from 1188.7813 to 812.207594
as n increases from 1.5 to 2.8 ,with the fixed value of propagation constant r=2 as shown
in table 2.28. Their graphical representation is shown in fig 2.28.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
57
n
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
u*
1188.7813
1152.07262
1116.00492
1080.87535
1047.00804
1014.66894
984.025322
955.131675
927.934747
902.29816
878.046725
855.018862
833.098628
812.207594
Tab. 2.28 The variation of modified particle velocity u* with propagation distance r for
exploding weak cylindrical shock waves in uniform medium.
1200
1100
u* 1000
900
800
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
Fig. 2.28 The variation of modified particle velocity u* with propagation distance r for
exploding weak cylindrical shock waves in uniform medium.
2.6.4 THE PRESSURE P :
Equation (2.39) represents the pressure P of freely propagation of exploding weak
cylindrical shock waves in uniform region of water.
For weak shock, initially we take M=1.25,  0 =1,  =1, refractive index n=1.5 and
a0 =1450. Using these initial constant values first of all, we calculated the value of the
constant of integration.Changing the value of r from 1.2 to 2.5, the corresponding the
pressure P takes the values from 8515125 to 6292280 respectively. We concluded that as
r increases the pressure P decreases fastly as shown in tab.2.29, while its graphical
representation confirms this statement [fig 2.29].
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
58
r
P
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
8515125
8221899
7962971
7732235
7524993
7337566
7167028
7011021
6867618
6735230
6612534
6498416
6391933
6292280
Tab. 2.29. The variation of pressure P with propagation distance r for exploding weak
cylindrical shock waves in uniform medium.
8700000
8200000
P
7700000
7200000
6700000
6200000
1.2
1.4 1.6
1.8
2
2.2
2.4
r
Fig. 2.29 The variation of pressure P with propagation distance r for exploding weak
cylindrical shock waves in uniform medium.
For weak shock ,initially we take M=1.25,  0 =1,  =1, propagation distance r=2
and a0 =1450. Changing the value of n from 1.5 to 2.8, the corresponding shock pressure
P takes the value from 6867618to 4715375. It shows that as n increases, pressure P
decreases very fastly as shown in table 2.30. While their graphical representation is shown
in fig 2.30.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
59
n
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
P
6867618
6650417
6442159
6243260
6053760
5873465
5702053
5539125
5384248
5236978
5096874
4963508
4836471
4715375
Tab. 2.30 The variation of pressure P with refractive index n for exploding weak cylindrical shock waves in uniform medium.
7000000
6500000
6000000
P
5500000
5000000
4500000
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
Fig. 2.30 The variation of pressure P with refractive index n for exploding weak
cylindrical shock waves in uniform medium.
Equation (2.44) represents the pressure P* of exploding weak shock waves in
presence of overtaking disturbances. For the initial value of M=1.25,  0 =1,  =1, refractive
index n=1.5 and a0 =1450, the variation of pressure P* of exploding cylindrical shock
modified by overtaking disturbances is computed using equation (2.44). Changing with the
value of r from 1.2 to 2.5 the corresponding to modified pressure P* (strength of the
shock in presence of overtaking disturbances) takes the value from 4010273.04 to 69826.25
as in table 2.31.Their graphical representation is shown in fig. 2.31.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
60
r
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
P*
4010273.04
3119541.73
2658585.63
2381387.26
2193200.37
2052938.57
1941039.75
1847421.56
1766501.23
1694972.21
1630747.54
1572429.9
1519032.66
1469826.25
Tab. 2.31 The variation of pressure P* modified by overtaking disturbances with propagation distance r for exploding weak cylindrical shock waves in uniform medium.
4500000
4000000
3500000
3000000
P* 2500000
2000000
1500000
1000000
1.2 1.4 1.6 1.8
2
2.2 2.4
r
Fig. 2.31 The variation of pressure P* modified by overtaking disturbances with
propagation distance r for exploding weak cylindrical shock waves in uniform medium.
In presence of overtaking disturbances the pressure P* is computed. We conclude
that the value of overtaking shock pressure P* is increases from 1766501.23 to1253394.23
when the refractive index n changing from 1.5 to 2.8 with the fixed value of propagation
constant r=2 as shown in table 2.32. Their graphhical representation is shown in fig. 2.32.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
61
n
P*
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
1766501.23
1725452.72
1681380.42
1635608.13
1589527.48
1544329.59
1500874.04
1459642.43
1420753.78
1384041.35
1349190.59
1315903.06
1283998.65
1253394.23
Tab. 2.32 The variation of pressure P* modified by overtaking disturbances with refractive index n for exploding weak cylindrical shock waves in uniform medium.
1800000
1700000
1600000
P* 1500000
1400000
1300000
1200000
1.5 1.7 1.9 2.1 2.3 2.5 2.7
n
Fig. 2.32 The variation of pressure P* modified by overtaking disturbances with refractive index n for exploding weak cylindrical shock waves in uniform medium.
EXPLODING CYLINDRICAL SHOCKS IN UNIFORM
REGION OF WATER
62