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Solved Problems

Exercise 114. Consider a semi-infinite string, x > 0. For all time the end of
the string is displaced according to u(0, t) = f(t). Find the motion of the string,
u(x, t) with the method of characteristics. The wave speed is c.

Solution: The problem is

2
u − c u xx = 0, x > 0, −∞ < t < ∞,
tt
u(0, t) = f(t).

Because the left end of the string is being displaced, there will only be right-moving

waves. The solution has the form

u(x, t) = F(x − ct).

We substitute this into the boundary condition.

F(−ct) = f(t)

ξ
F(ξ) = f −
c
u(x, t) = f(t − x/c)

Exercise 115. Sketch the solution to the wave equation:

Z x+ct
1 1
u(x, t) = (u(x + ct, 0) + u(x − ct, 0)) + u (τ, 0) dτ,
t
2 2c x−ct
for various values of t corresponding to the initial conditions:

(a) u(x, 0) = 0, u (x, 0) = sin ωx where ω is a constant,
t

1 for 0 < x < 1

(b) u(x, 0) = 0, u (x, 0) = −1 for − 1 < x < 0
t

0 for |x| > 1.

Solution:

(a)

Z x+ct
1 1
u(x, t) = (u(x + ct, 0) + u(x − ct, 0)) + u (τ, 0) dτ
t
2 2c x−ct
1 Z x+ct
u(x, t) = sin(ωτ) dτ
2c x−ct
sin(ωx) sin(ωct)
u(x, t) =
ωc

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