Geology Reference
In-Depth Information
2.5.3 Example 2-7. String with variable properties.
We consider weakly damped transverse vibrations of a string with
constant
lineal mass density
l
and tension T. In Chapter 1, we examined
l
2
u/ t
2
+
u/ t - T
2
u/ x
2
= 0 and determined that
l
(
r2
-
i2
)= Tk
2
+
i
and
i
= -
/ 2
l
. For small dissipations satisfying |
i
/
r
| << 1, the first is approximated
by
l
r2
= Tk
2
. We also indicated that this dispersion relation holds to leading
order for slow variations when
l
=
l
(x) and T = T(t), provided the dependences
on x and t are weak. This fact arises formally using asymptotic WKB methods
(Nayfeh, 1973). For our purposes,
r
(k,x,t) = {T(t)/
l
(x)}
1/2
k (2.106)
i
(k,x,t) = - /2
l
(x) (2.107)
To evaluate the right sides of Equations 2.104 and 2.105, we take the
derivative
r
t
(k,x,t) = 1/2 {T(t)/
l
(x)}
-1/2
{ T(t)/ t/
l
(x)}k, and then,
r
x
(k,x,t)
= 1/2 {T(t)/
l
(x)}
-1/2
{T(t)}
1/2
(-1/2){
l
(x)}
-3/2
{
l
(x)/ x}k. The group velocity
is simply
r
x
(k,x,t) = {T(t)/
l
(x)}
1/2
. Physically, Equation 2.104 indicates that
energy changes to the system are expected on a guitar string that is being
stretched or relieved in tension in time. Equation 2.105 indicates that
momentum changes are expected whenever variations in spatial mass density
distribution exist.
2.5.4 Computational solution.
Integration of Equations 2.104 and 2.105, in general, must proceed
numerically. For E(x,t), we have the sequence of steps,
r
k
(k,x,t)E)/ x = E{2
i
+
r
t
(k,x,t)/
r
}
E/ t +
(
(2.108)
E/ t +
r
k
(k,x,t) E/ x + E
r
k
(k,x,t)/ x
= E{ 2
i
+
r
t
(k,x,t)/
r
}
(2.109)
E/ t +
r
k
(k,x,t) E/ x + E
r
kk
(k,x,t) k/ x + E
r
kx
(k,x,t)
= E{ 2
i
+
r
t
(k,x,t)/
r
}
(2.110)
E/ t +
r
k
(k,x,t) E/ x
= E{ 2
i
+
r
t
(k,x,t)/
r
-
r
kx
(k,x,t) -
r
kk
(k,x,t) k/ x} (2.111)
Drawing upon earlier work, we immediately write the required
characteristic equations for wavenumber and position,
dk/dt = -
r
x
(k,x,t)
(2.112)
along the ray path
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