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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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