Geology Reference
In-Depth Information
+
+
{1/(2 )}
e
-i
t
K(
) U(
) d
k(t-x) u(x) dx
(1.81)
- -
The transform functions e
ic t
and e
-ic t
are associated with the
delta
functions
(see next section) shown in Equations 1.82 and 1.83,
e
i
x
(x-ct) dx = e
ic
t
(1.82)
e
i
x
(x+ct) dx = e
-ic
t
(1.83)
Thus, the solution in Equation 1.80 can be written as
+
+
u(x,t) = 1/2
(x - - ct) f( ) d + 1/2
(x - + ct) f( ) d
(1.84)
-
-
which is easily evaluated to give
u(x,t) = 1/2 f(x-ct) + 1/2 f(x+ct) (1.85)
This agrees with the general solution in Equation 1.4. Equation 1.85 shows that,
at each point x, half of the initial u(x,0) = f(x) disturbance propagates to the
right, while the remaining half propagates to the left.
1.6.2 Example 1-12. Directional properties, special wave operators.
In constructive solutions to
2
u(x,t)/ t
2
- c
2 2
u/ x
2
= 0, the independent
variable combinations “x-ct” and “x+ct” arise all too frequently. This suggests
that
= x-ct
(1.86)
and
= x+ct
(1.87)
may be more natural than x and t. This is, in fact, true: and are called
“natural,” “characteristic,” or “canonical variables.” Let us write u(x,t) =
U( , ), and determine the equation satisfied by U in the coordinates ( , ).
Application of the chain rule of calculus shows that
u
t
(x,t) = U
t
+ U
t
= - c U + cU
(1.88)
u
tt
(x,t) = -c(U
t
+ U
t
) + c(U
t
+ U
t
)
= c
2
U
- 2c
2
U
+ c
2
U
(1.89)
u
x
(x,t) = U
x
+ U
x
= U + U
(1.90)
u
xx
(x,t) = U
x
+ U
x
+ U
x
+ U
x
= U
+ 2U
+ U
(1.91)
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