
Chapter 6
Second-Order
Hyperbolic Equations
with One Space Variable
6.1 Constant Coefficient Equations
6.1.1 Wave Equation
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
This equation is also known as the equation of vibration of a string. It is often encountered
in elasticity, aerodynamics, acoustics, and electrodynamics.
◮ General solution. Some formulas.
1
◦
. General solution:
w(x, t) = ϕ(x + at) + ψ(x − at),
where ϕ(x) and ψ(x) are arbitrary functions.
Physical interpretation: The solution represents two traveling waves that propagate,
respectively, to the left and right along the x-axis at a constant speed a.
2
◦
. Fundamental solution:
E (x, t) =
1
2a
ϑ
at − |x|
, ϑ(z) =
(
0 for z ≤ 0,
1