The Wave Equation

Abstract

This chapter treats the equation

$$\frac{{{\partial ^2}u}}{{\partial {t^2}}}\left( {x,t} \right) - {\omega ^2}\Delta u\left( {x,t} \right) = f\left( {x,t} \right),x \in \Omega ,t \in \left( {0,T} \right)$$

(1)

which is called the wave equation and plays an important role in mathematical physics. In particular, this equation is satisfied by the waves in homogeneous elastic media and by the electromagnetic waves. The constant w is just the speed of propagation. The boundary value problem and the Cauchy problem in R ⁿ are the main topics studied here. The functional framework developed for the heat equation and in particular, the Fourier method and the semigroup approach are applicable in this case too. However, we shall see that there are some significant differences between the wave and the heat equations, the most important being perhaps the finite speed propagation and the conservation of energy.