Geology Reference
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U"(x) + {(
2
- i
)/c
2
} U(x) = 0
(1.40)
for a complex U(x). Exponential rather than trigonometric functions are again
used to satisfy boundary conditions, and we attempt solutions of the form U(x) =
e
x
. Substitution in Equation 1.40 leads to
2
= -
2
/c
2
+ i
/c
2
(1.41)
When = 0, the solutions to Equation 1.41 are
= i
/c and - i
/c. The
assumption u(x,t)
e
i
t
U(x) produces the solutions e
i(
t+
x/c)
and e
i(
t-
x/c)
as
before, the first of which is eliminated since it is left-going.
In general, we can express any complex number X + i Y in a “polar form”
X + i Y = R(cos + i s i n ), where R = (X
2
+ Y
2
) and = tan
-1
{Y/X}.
From De Moivre's Theorem in complex variables, it follows that {X + i Y}
1/2
=
{R(cos + i s i n )}
1/2
= R
1/2
{cos ( +2 k )/2 + i sin ( +2 k )/2}, k = 0, 1.
For dissipative problems, the solution to Equation 1-41 is required. Then,
setting X = -
2
/c
2
and Y =
/c
2
, the two square roots of
obtained are
1
= (
4
/c
4
+
2
2
/c
4
)
1/4
(1.42a)
x [cos{
½
arctan(- /
)} + i sin{
½
arctan(- /
)}]
2
= (
4
/c
4
+
2
2
/c
4
)
1/4
(1.42b)
x [cos{
½
arctan(- /
) + } + i sin{
½
arctan(- /
) + }]
Following Example 1-7, only the root corresponding to the right-going
wave is physically meaningful. In the zero damping limit when = 0, our
dissipative solution should reduce to the earlier one. We choose as the
“principal branch” of the tan
-1
function, the one yielding tan
-1
(-
/
) = when
/c) [cos{
½
)} + i sin{
½
}] = i
= 0, so that
1
= (
/c and
2
= (
/c)
[cos{
½
+ } + i sin{
½
+ }] = - i
/c as before.
Substitution in u(x,t)
e
i
t
U(x) = e
i
t
e
x
clearly shows that the
1
solution “e
i
t+x/c
” is left-going, while the
2
solution “e
i
t-x/c
” is the desired
right-going wave. Only the
2
root is relevant, with
2
/c
4
)
1/4
cos{
½
arctan(- /
4
/c
4
+
2
2
= (
) + }
2
/c
4
)
1/4
sin{
½
arctan(- /
+ i (
4
/c
4
+
2
) + }
(1.42c)
If Equation 1.42c is introduced into u(x,t) e
i t
U(x) = e
i t
e
x
, with a
factor A to handle the boundary condition, we obtain the complex function
2
/c
4
)
1/4
cos{
½
arctan(- /
4
/c
4
+
2
) + }}
u(x,t) = Aexp { (
2
/c
4
)
1/4
sin{
½
arctan(- /
X exp i{
t + (
4
/c
4
+
2
) + }x }
(1.43)
whose real part is
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