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Chapter 6

Wave Equation

6.1 The Method of Characteristics and the Wave

Equation

Consider a homogeneous string of length l and density ρ = ρ(x). Assume the string is
moving in the transverse direction, but not in the longitudinal direction. Let u(x, t) denote
the displacement of the string from equilibrium at time t and position x. Therefore, the
slope of the string at time t, position x is given by u x (x, t). Let T(x, t) be the magnitude
of the tension (force) tangential to the string at time t and position x.
We will assume that both T and ρ are independent along the string, and there are
no external forces F. The wave equation, which describes the dynamics of the amplitude
u(x, t) of the point at position x on the string at time t, has the following form

2
u tt = c u xx , x ∈ R (6.1)
r
T
where c = represents the wave speed.
ρ
The reason for this solution becomes obvious when we consider the physics of the
problem: The wave equation describes waves that propagate with the speed c (the speed
of sound, or light, or whatever). Thus any perturbation to the one dimensional medium
will propagate either right- or leftwards with such a speed. This means that we would
expect the solutions to propagate along the characteristics x ± ct = constant.

Theorem 6.1.1 The general solution of equation (6.1) is given by

u(x, t) = f(x + ct) + g(x − ct), (6.2)

for arbitrary (smooth) functions f and g.
In particular, f(x + ct) is a wave moving to the left with speed c,

while g(x − ct) is a wave moving to the right with speed c.

Proof 6.1.1 As we saw in the last lecture, the wave equation has the

second canonical form for hyperbolic equations. By passing to the

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