Geology Reference
In-Depth Information
where t
ref
is a constant with dimensions of time. The A {(t +x/c)/t
ref
} term is
the incoming wave, while - A {(t -x/c)/t
ref
} represents the reflected outgoing
wave, had we assumed rigid displacement boundary conditions. Two time
scales (ac)
-1
and (bc)
-1
are introduced by the “mixed boundary condition” in
Equation 1.56. Our solution shows that they are responsible for a propagating
exponential distortion {ae
a(ct-x)
- be
b(ct-x)
} having nonzero width.
1.5.3.3 Example 1-10c. Incident sinusoidal wavetrain.
We consider the incident displacement wave A sin
(ct+x)/c, where
is
the
frequency
. The exact solution is
u(x,t) = A sin {
(ct+x)/c} - A sin {
(ct-x)/c}
(1.65)
+ 2 ATb{ b s i n ( (ct-x)/c)
+ ( /c) cos (
c
2
(a-b)(b
2
+
2
/c
2
)}
(ct-x)/c)}/{
- 2ATa{a sin (
(ct-x)/c)
c
2
(a-b)(a
2
+
2
/c
2
)}
+ (
/c) cos (
(ct-x)/c)}/{
+{ 2 AT
/[
c
3
(a-b)]}{ae
a(ct-x)
/(a
2
+
2
/c
2
)
- be
b(ct-x)
/(b
2
+
2
/c
2
)}
The first line represents the incoming wave and its pinned end reflection,
where A is the wave amplitude. The second and third lines represent the
phase
change
due to the support, while the fourth shows that a distortion with
exponential smearing
emerges that shifts the mean DC level.
1.6 Fourier Transforms
We have seen how Laplace transforms can be used to solve wave problems
with nontrivial boundary conditions. Another popular transformation is the
“Fourier transform” (not to be confused with the “Fast Fourier Transform” or
“FFT” algorithm) useful in solving problems on infinite domains. Consider a
function u(x) defined on -
< x <
. Its Fourier transform U(
) is defined by
+
U(
) =
e
i
x
u(x) dx
(1.66)
-
where is a parameter. If the transform U( ) were known, then the original
function u(x) can be recovered by the complementary integral
+
u(x) = {1/(2 )} e
-i
x
U(
) d
(1.67)
-
Excellent classic references to Fourier transforms exist, e.g., Carrier, Krook and
Pearson (1966), Churchill (1941) and Churchill (1958). The first provides solid
mathematical developments while the latter two illustrate transform techniques
with important engineering examples.
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