change the sign of celerity $\, a ,\,$ [*i.e.*]{} if we suppose now

(1.3.4) $\qquad \displaystyle 
a < 0  \,\,, \,$

the above analysis remains unchanged, but the algebraic relations (1.2.4) and (1.2.5) have to be modified (see Figure 1.6). We still start from relation (1.1.5) that expresses that the solution of the advection equation (1.1.1) is constant along the characteristics lines. The foot of the characteristic line that contains the particular point $\,\, (x,\,t) \,\,$ in space-time is either the point $\,\,(y=x-at,\,0) \,\,$ if $\,\, x-at < L \,,\,$ either the point $\,\, \bigl(L,\,
\theta=t-{{1}\over{a}}(x-L)\bigr) \,\,$ if $\,\, x-at > L \,.\,$ In the first case, we have $\,\, y>0 \,\,$ and $\,\, \theta <0 \,\,$ then the initial condition (1.3.2) is advected inside the domain (1.3.1) and we have :

(1.3.5) $\qquad \displaystyle 
u(x, \,t) \,\,\, = \,\, u_0(x - at) \,\,, \qquad  \qquad   \,\,\,\,\,  \, x-at \,< \, L
\,. \,$

$\bullet \qquad \,\,\, $ On the contrary, if $\,\, x-at > L,\,$ we have $\,\, y>L \,\,$ and $\,\, \theta > 0
\,\,$ then the boundary condition at $\,\,x=L \,\,$ that takes now the expression

(1.3.6) $\qquad \displaystyle 
u(L, \,t) \,\,\, = \,\, w_{L}(t) \,,\qquad \quad 