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852 Answers to Selected Problems

and

2 2
2
2
2
2
r n = 4(BL + n π c ) − A L .
15. (a) With the forcing term, the solution is
∞

y f (x,t) = d n sin(nπx/4)cos(3nπt/4)
n=1
1
+ (cos(πx) − 1),
9π 2
where
#
−32(1−(−1) n )(288−17n 2 ) for n
= 4,
d n = 9n 3 π 3 (n 2 −16)
0 for n = 4.
(b) Without the forcing term, the solution is
∞
128
y(x,t) = sin((2n − 1)πx/3)cos(3(2n − 1)πt/4).
3
π (2n − 1) 3
n=1
Section 16.3 Wave Motion in an Infinite Medium
10

∞
1. y(x,t) = cos(ωx)cos(12ωt)dω
0 π(25 + ω )
2
1 sin(πω)
∞

3. y(x,t) = sin(ωx)sin(4ωt)dω
0 2πω 1 − ω 2
e 2cos(ω) − ω sin(ω)

−2
∞

5. y(x,t) = cos(ωx)
0 3πω 4 + ω 2
−2
e ω cos(ω) + 2sin(ω)
+ sin(ωx) sin(3ωt)dω
3πω 4 + ω 2
Section 16.4 Wave Motion in a Semi-Infinite Medium
∞ 2 2 − ω sin(ω) − 2cos(ω)

1. y(x,t) = sin(ωx)cos(3ωt)dω
0 π ω 3
∞ 1 sin(πω/2) − sin(5πω/2)

3. y(x,t) = sin(ωx)sin(2ωt)dω
0 πω ω − 1
2
5.
∞ 3

y(x,t) = d ω sin(ωx)sin(14ωt)dω,
0 7πω 5
where
2
d ω = 2sin(3ω) − 4ω cos(3ω) − 3ω sin(3ω) − 2ω
Section 16.5 Laplace Transform Techniques
x K x x 1

2
1. y(x,t) = f t − − t − H t − + Kt 2
c 2 c c 2
A x 3 x A
3. y(x,t) = t − H t − − t 3
6 c c 6
x x 1
5. y(x,t) = f t − H t − − Axt 4
c c 6
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October 14, 2010 17:50 THM/NEIL Page-852 27410_25_Ans_p801-866