The Wave Equation

Abstract

According to (1.24), the wave equation in R ₁ is of the form

$$ {u_{xx}} - \frac{1} {{{\gamma |2}}}{u_{tt}} = 0,\quad where\quad \gamma = const > 0 $$

(2.1)

The symbol x denotes a real variable here. Introduction of a new time scale t̃ = γt shows that we may restrict ourselves to the case γ = 1. Therefore we base (2.1) on the form

$$ {u_{xx}} - {u_{tt}} = 0,\quad or\quad {u_{\bar x\bar t}} = 0 $$

(2.2)

where the last equation results from the transformation x = x + t, t̄ = x - t. By integration we find all solutions of (2.2):

$$ u\left( {x,t} \right) = {w_1}\left( {\bar x} \right) + {w_2}\left( {\bar t} \right) = {w_1}\left( {x + t} \right) + {w_2}\left( {x - t} \right) $$

((2.3))

where the w _i ε C ²(i = 1, 2) are arbitrary functions.