v_0(t) \,\,$ is sinusoïdal for time positive to fix the ideas,

(1.2.8) $\qquad \displaystyle 
v_0(t) \,\,\,= \,\, {\rm sin}(\omega t)  \,\,,\qquad t > 0 \,,\,$

the solution of the advection equation in the domain $\,\, x > 0 \,,\,\, t>0 \,\,$ via the relations (1.2.4) and (1.2.5) can be considered with the two following view points.

[**(i)** ]{} We take a snap shot of the solution $\,\, u({\scriptstyle \bullet} ,\,
{\scriptstyle \bullet}) \,\,$ at a fixed time $\, T>0 .\,$ We consider the partial function $\,\, [0,\,+\infty[ \, \, \ni x \longmapsto u(x,\,T) \in \R \,\,$ and taking into account the relations (1.2.4), (1.2.5), (1.2.7) and (1.2.8), we have

21= 22= 30= 31= 44=

$ \box44 $

and this function is illustrated on Figure 1.3.

[**(ii)** ]{} We fix a particular position $\,\, X \,\,$ in space and we look, as time is increasing, to the solution $\,\, u({\scriptstyle \bullet} ,\, {\scriptstyle
\bullet}) \,\,$ at this particular point. We show on Figure 1.4 the function $\,\, 
[0,\,+\infty[ \, \ni t \longmapsto u(X,\,t) \in \R \,\,$ and taking into account the relations (1.2.4), (1.2.5), (1.2.7) and (1.2.8), we have

21= 22= 30= 31= 44=

$ \box44 $

[**Figure 1.3.**]{} Snap shot 