The Wave Equation and its Connections with the Laplace and Heat Equations

Abstract

The wave equation is the PDE

$$ \frac{{\partial ^2 }} {{\partial t^2 }}u(x,t) - \Delta u(x,t) = 0 for x \in \Omega \subset \mathbb{R}^d , t \in (0,\infty ) or t \in \mathbb{R}. $$

((6.1.1))

As with the heat equation, we consider t as time and x as a spatial variable. For illustration, we first consider the case where the spatial variable x is one-dimensional. We then write the wave equation as

$$ u_{tt} (x,t) - u_{xx} (x,t) = 0. $$

((6.1.2))

Let ϕ, ψ ∈ C²(ℝ). Then

$$ u(x,t) = \phi (x + t) + \psi (x - t) $$

((6.1.3))

obviously solves (6.1.2).