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16.3 Wave Motion in an Infinite Medium 579

15. (a) Write a series solution for 16. (a) Write a series solution for
2
2
2
2
∂ y ∂ y ∂ y ∂ y
= 9 − e −x for 0 < x < 4,t > 0 = 9 + cos(πx) for 0 < x < 4,t > 0
∂t 2 ∂x 2 ∂t 2 ∂x 2
y(0,t) = y(4,t) = 0for t ≥ 0 y(0,t) = y(4,t) = 0for t ≥ 0
∂y ∂y
y(x,0) = sin(πx), (x,0) = 0for0 ≤ x ≤ 4. y(x,0) = x(4 − x), (x,0) = 0for 0 ≤ x ≤ 4.
∂t ∂t
(b) Write a series solution when the forcing term e −x (b) Write a series solution when the forcing term
is removed. cos(πx) is removed.
(c) In order to gauge the effect of the forcing term (c) In order to gauge the effect of the forcing term on
on the motion, graph the fortieth partial sum of the motion, graph the fortieth partial sum of the
the solutions in parts (a) and (b) on the same set solutions in (a) and (b) on the same set of axes
of axes when t = 0.4 seconds. Repeat this for when t = 0.6 seconds, then for t = 1,1.4, 2,3, 5
t = 0.8,1.4, 2,2.5, 3, and 4. and 7.

16.3 Wave Motion in an Infinite Medium

If great distances are involved (as with sound waves through the ocean or cosmic background
radiation across the universe), wave motion is often modeled by the wave equation on −∞ <
x < ∞. In this case, there is no boundary, hence no boundary condition. However, we seek
bounded solutions.
∞
The analysis is similar to that for solutions on a closed interval, except that ···dω
−∞
∞
replaces . As we did on a bounded interval, we will consider separately the cases of zero
n=1
initial velocity and no initial displacement.
Zero Initial Velocity The initial-boundary value problem is
2
2
∂ y ∂ y
= c 2 for −∞ < x < ∞,t > 0,
∂t 2 ∂x 2
∂y
y(x,0) = f (x), (x,0) = 0for −∞ < x < ∞.
∂t
Separate variables by putting y(x,t) = X(x)T (t). Exactly as with wave motion on a bounded
interval, we obtain
2

X + λX = 0, T + λc T = 0.
There are three cases on λ.
Case 1: If λ = 0 then X = ax + b
This is bounded if a = 0, so 0 is an eigenvalue with constant eigenfunctions.

2
Case 2: If λ< 0, write λ =−ω with ω> 0
Then X(x) = c 1 e ωx + c 2 e −ωx , and this function is unbounded on the entire real line if either
constant is nonzero. This problem has no negative eigenvalue.

2
Case 3: If λ> 0, write λ = ω with ω> 0
Now X(x) = c 1 cos(ωx) + c 2 sin(ωx), a bounded function for any choices of positive ω.Every
2
positive number λ = ω is an eigenvalue, with eigenfunctions of the form X ω = c 1 cos(ωx) +
c 2 sin(ωx).

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October 14, 2010 15:23 THM/NEIL Page-579 27410_16_ch16_p563-610