\lambda
= t \,$ in the previous relation (1.2.6), the particular point $\, y \,=\, x(t) \,=\,
x-at \,\,$ on the axis of abscissa is strictly positive then the initial condition $\, u_0(y) \,$ is well defined. The solution $\,\, u({\scriptstyle \bullet} ,\,
{\scriptstyle \bullet}) \,\, $ is constant on the characteristic line (see Proposition 1.1) that contains this particular point. Then relation (1.2.4) is established.

$\bullet \qquad \,\,\, $ Second case : $\,\, x-at \,< \,0 \,. \, $ We consider the particular value $\,\,
\lambda = {{x}\over{a}} \,$ inside the expression (1.2.6). Then the corresponding foot of the characteristic belongs to the time axis : $\,\, \theta \,=\, t-\lambda =
t- {{x}\over{a}} \,$ and $\,\, \theta > 0 \,\,$ due to the inequalities $\, x < a t
\,$ and $\, a > 0 \,.\,$ The solution is constant along the characteristic line going through this point and the relation (1.2.5) is established. $ \hfill \square
\kern0.1mm $

$\bullet \qquad \,\,\, $ In the particular case where datum $\,\, u_0(x) \,\,$ is identically equal to zero, [*i.e.*]{}

(1.2.7) $\qquad \displaystyle 
u_0(x) \,\,= \,\, 0 \,\,,\qquad \qquad \,\, x > 0 \,,\,$

and if the boundary condition $\, \, 