Geoscience Reference
In-Depth Information
Now we will establish that the function Eq. (
7.95
) is a solution of not only
Eq. (
7.91
) but also Eq. (
7.90
). Taking the time-derivative of the both sides of
Eq. (
7.95
) and substituting Eq. (
7.89
)for@
t
g
1
into the integral (
7.95
), we obtain
Z
m
@
r
0
g
1
C
2
@
r
0
V
C
2
V
r
0
dr
0
: (7.96)
r
1
2r
3
2@
r
0
g
1
r
0
r
03
@
r
0
@
t
g
2
D
0
Integrating Eq. (
7.96
) by parts and taking into account that V and @
r
V are equal
to zero at r
D
0, we come to Eq. (
7.90
). This completes the proof of the above
statement.
The sought functions ıB
r
and ıB
can be expressed through the functions g
1
and g
2
B
0
3
.g
1
C
2g
2
/ cos and ıB
D
B
0
3
.g
2
g
1
/ sin :
ıB
r
D
(7.97)
Substituting Eq. (
7.92
)forg
1
and Eq. (
7.95
)forg
2
into Eq. (
7.97
), taking inte-
gral (
7.95
) by parts and rearranging we come to Eqs. (
7.35
) and (
7.36
).
The electric field can be found from Eq. (
7.84
) and (
7.88
) that leads to Eq. (
7.37
).
The general solution of the problem given by these equations and Eq. (
7.92
) can
be applied to an arbitrary function V
D
V .r;t/.
Normalized Potential of Elastic Displacement
In this section we deal with a spherically symmetric acoustic wave which results
in the radial displacement of elastic medium. Since all the functions are dependent
only on radius r, the acoustic wave equation (
7.39
) is reduced to the form:
@
t
u
r
D
C
l
@
r
1
r
2
@
r
r
2
u
r
;
(7.98)
where
u
r
D
u
r
.r;t/ denotes the radial displacement of the medium and C
l
is
longitudinal wave velocity.
Equation (
7.98
) should be supplemented by the proper boundary and initial
conditions. Let R
0
be the radius of the effective acoustic source and a given function
u
r
.R
0
;t/
D
u
.t/ be the radial displacement of the medium at the radius r
D
R
0
.
This function must satisfy the following conditions:
u
.0/
D
0 and @
t
u
.0/
D
0.
At the initial moment the medium is at rest, that is,
u
r
.r;0/
D
0. Since the radial
displacement has to be continuous at the front of acoustic wave, it is necessary that
u
r
.r
l
;t/
D
0, where r
l
D
R
0
C
C
l
t is the radius of acoustic wave front.
It is convenient to introduce a new auxiliary unknown function f .r;t/ instead of
the function
u
r
.r;t/ as follows:
u
r
D
R
0
@
r
.f=r/. Substituting the last expression
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