Civil Engineering Reference
In-Depth Information
In this case we have:
(
)
(
)
(
)
(
)
0
0
0
0
Y=Y+Y
i
exp
i kX
+ωτ+ωτ =Y+Y −ωτ×
i
i
exp
*
**
**
*
**
**
(
)
(
)
cos
kX
+ ωτ +
i
sin
kX
+ ωτ
{ }
(
)
(
)
0
Re
Y =
exp
−ω τ Y
sin
ϕ+
kX
+ωτ
**
(
)
0
02
0 2
0
0
Y = Y +Y
,
ϕ =
arctg
−Y Y
/
*
**
*
**
Here
Thus, the decision of the first type is a sinusoidal coordinate and
ω
>
0
decaying
*
exponentially in time perturbation, which is called
k
--wave:
( )
Œ
Ž
2
π +
Xvk
τ
'
'
( )
( )
(11)
Φ
Y = Y
k
0
exp
−ω
k
τ
sin
ϕ+
'
“
( )
**
λ
k
'
'
”
•
( )
( )
/
Where
,
( )
),)
ϕ
--initial phase.
vk
=ω
kk
λ =
k
2π
/
k
Φ
( )
vk
Φ
--phase velocity or the velocity of phase fluctuations
( )
Here,
λ
--wave-
( )
ω
--damping the oscillations in time. In other words,
k
--waves - waves
have uniform length, but time-varying amplitude. These waves are analogue of free
oscillations.
Type II
. Decisions, or wave, the second type, when
k
length)
ω =
*
ω
--a real positive num-
ber
(
)
ω
> ω
0
=
0
. In this case we have:
*
(
)
(
)
(
)
(
)
0
0
0
0
Y= Y + Y
i
exp
i kX
+ωτ+
ik
z
= Y + Y
i
exp
−
k
X
×
*
**
**
*
**
**
(
)
(
)
cos
kX
+ ωτ +
i
sin
kX
+ ωτ
(
)
{ }
(
)
Y == − Y
ϕ + + ωτ
Thus, the solution of the second type is a sinusoidal oscillation in time (excited, for
example, any stationary source of external monochromatic vibrations at)
Re
exp
k
X
0
sin
kX
**
=
X
, de-
caying exponentially along the length of the amplitude. Such disturbances, which are
analogous to a wave of forced oscillations, called ω--waves:
0
( )
Œ
Ž
2
π +
Xv
ω τ
'
'
(
)
( )
( )
( )
(12)
Φ
0
Y ω = Y ω
exp
− ω
k
X
sin
ϕ+
'
“
( )
**
λω
'
'
”
•
( )
( )
( )
( )
v
Φ
ω=ω ω λω=π ω
/
k
,
2/
k
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