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594 CHAPTER 16 The Wave Equation

SECTION 16.5 PROBLEMS

∂y
1. Use the Laplace transform to write the solution (in
y(x,0) = (x,0) = 0,
terms of f (t)) of the boundary value problem ∂t
y(0,t) = f (t), and lim y(x,t) = 0,
2
2
∂ y ∂ y x→∞
= c 2 + K for x > 0,t > 0 in which A is a positive constant.
∂t 2 ∂x 2
4. Use the Laplace transform to find the solution
∂y
y(x,0) = (x,0) = 0for x > 0, 1
∂t y(x,t) = ( f o (x + ct) + f o (x − ct))
2
y(0,t) = f (t), lim y(x,t) = 0for t ≥ 0.
x→∞ of the problem
2
2
2. Use the Laplace transform to write the solution (in ∂ y 2 ∂ y
= c for x > 0,t > 0
terms of f (t)) of the boundary value problem ∂t 2 ∂x 2
∂y
2
2
∂ y ∂ y ∂ y y(x,0) = ∂t (x,0) = 0for x > 0,
2
9 + − 6 = 0for x > 0,t < 0
∂t 2 ∂x 2 ∂x∂t
y(0,t) = f (t), lim y(x,t) = 0for t ≥ 0.
x→∞
∂y
y(x,0) = (x,0) = 0for x > 0, y(2,t) = f (t), 5. Use the Laplace transform to solve:
∂t
∂y ∂y
y(0,t) = 0, lim y(x,t) = 0for t ≥ 0. = c 2 − Axt for x > 0,t > 0,
∂t ∂x
x→∞ 2 2
3. Use the Laplace transform to solve ∂y
y(x,0) = (x,0) = 0for x > 0,
∂t
2
2
−t
∂ y ∂ y y(0,t) = e , lim y(x,t) = 0for t > 0.
= c 2 − At for x > 0,t > 0, x→∞
∂t 2 ∂x 2
16.6 Characteristics and d’Alembert’s Solution
In this section, we will derive d’Alembert’s solution of a wave problem on the real line. We will
denote partial derivatives by subscripts, ∂u/∂t =u t and ∂u/∂x =u x . The problem we will solve is
2
u tt = c u xx for −∞ < x < ∞,t > 0
and
u(x,0) = f (x),u t (x,0) = g(x) for −∞ < x < ∞.
We are using u(x,t) for the position function of the wave. A graph of the wave’s profile at time
t is the graph of y = u(x,t) in the x, y-plane for that value of t.

This initial-boundary value problem is called the Cauchy problem for the wave equation.
The lines x −ct =k 1 and x +ct =k 2 in the x,t-plane are called characteristics of the wave
equation. These are straight lines of slope 1/c and −1/c in the x,t-plane. Exploiting these
characteristics, make the change of variables
ξ = x − ct,η = x + ct.

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October 14, 2010 15:23 THM/NEIL Page-594 27410_16_ch16_p563-610