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16.2 Wave Motion on an Interval 577

2

1

0
0 5 . 0 1 1.5 2 2.5 3
x
–1

–2

FIGURE 16.8 Wave profiles in Example 16.7.

∞
2AL 3
(−1) n
= sin(nπx/L)cos(nπt/L)
π 3 n 3
n=1
∞
2L
1 − (−1) n
+ sin(nπx/L)sin(nπt/L).
π 2 n 2
n=1
The solution of the original problem is
1 2 2
y(x,t) = Y(x,t) + Ax(L − x ).
6
Figure 16.8 shows wave profiles for c =1 and L =π at times t =0.03,0.2, 0.5,0.9, 1.4, and
2.2. The waves move upward as t increases over these times.

SECTION 16.2 PROBLEMS

2
2
∂ y ∂ y
In each of Problems 1 through 8, solve the initial- 2. = 9 for 0 < x < 4,t > 0
∂t 2 ∂x 2
boundary value problem using separation of variables.
y(0,t)= y(4,t) = 0for t ≥ 0
Graph the fiftieth partial sum of the solution for some
values of t, with c = 1if c is unspecified in the y(x,0)= 2sin(πx), ∂y (x,0) = 0for 0 ≤ x ≤ 4
problem. ∂t
2
2
∂ y ∂ y
3. = 4 for 0 < x < 3,t > 0
∂t 2 ∂x 2
y(0,t)= y(3,t) = 0for t ≥ 0
2
2
∂ y ∂ y ∂y
1. = c 2 for 0 < x < 2,t > 0 y(x,0)= 0, (x,0) = x(3 − x) for 0 ≤ x ≤ 3
∂t 2 ∂x 2 ∂t
2
2
y(0,t)= y(2,t) = 0for t ≥ 0 4. ∂ y = 9 ∂ y for 0 < x <π,t > 0
∂t 2 ∂x 2
∂y
y(x,0)= 0, (x,0) = g(x) for 0 ≤ x ≤ 2 y(0,t)= y(π,t) = 0for t ≥ 0
∂t
∂y
y(x,0)= sin(x), (x,0) = 1for 0 ≤ x ≤ π
2x for 0 ≤ x ≤ 1 ∂t
where g(x) =
0 for 1 < x ≤ 2.
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October 14, 2010 15:23 THM/NEIL Page-577 27410_16_ch16_p563-610