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coefficient can be expressed by the ratio C
t
=C
l
of the transverse and longitudinal
wave velocities:
1
2C
t
=C
l
2
1
C
t
=C
l
D
:
(7.103)
Substituting Eq. (
7.100
)for
u
r
into Eq. (
7.102
)gives
s
rr
D
m
R
0
.1
/@
t
f
C
R
0
.1
2/
@
t
f
C
R
0
f
;
2C
l
C
l
(7.104)
where f
D
f .t
1
/. Equating Eq. (
7.42
)fors
rr
and Eq. (
7.104
) taken at r
D
R
0
yields
d
t
f
C
R
0
f
2C
l
.1
2/
R
0
.1
/
C
l
P
0
C
.P
P
0
/ exp .
t=t
r
/
m
R
0
.1
/
d
t
f
C
D
; (7.105)
where f
D
f .t/. The initial condition for Eq. (
7.105
)isasfollows:f .0/
D
0 and
d
t
f .0/
D
0. The solution of this problem can be written as
f .t/
D
a
1
C
a
2
exp
"
.a
1
C
a
2
/ cos
2
2
C
t
t
R
0
t
t
r
#
exp
sin
2
2
C
t
t
R
0
: (7.106)
.2
1
.a
1
C
a
2
/
a
2
a
3
/
2
1=2
1
2
1
C
l
t
R
0
C
2
Here the following designations are introduced
P
0
.1
1
/
2
1
m
C
l
2.P
P
0
/.1
1
/
m
C
l
a
3
4
1
a
3
C
4
1
a
1
D
;a
2
D
;
C
t
C
l
2
R
0
C
l
t
r
;
2
D
.1
1
/
1=2
:
a
3
D
;
1
D
(7.107)
It should be noted that we have corrected some errors in the coefficients a
1
and a
2
which were made in the work by Surkov (
1989b
).
The normalized potential (
7.106
) versus time is illustrated in Fig.
7.7
with lines
1 and 2, which correspond to t
r
D
0:3 and 0:03 s, respectively. In making these
plots the following parameters have been used: R
0
D
100 m, P
D
5
10
8
Pa,
P
0
D
1:5
10
8
Pa, C
l
D
5 km/s ,
m
C
l
D
5
10
10
Pa, and
1
D
0:2.
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