f[x_]=Piecewise[{{1+x,-1<x<0},{1-x,0<x<1}}]

Plot[f[x],{x,-4,4},AspectRatio->Automatic,Ticks->{Automatic,{1}},ImageSize->Large,PlotRange->All]

u1[x_,t_,c_]=f[x-c t]/2+f[x+c t]/2

Plot[u1[x,2,1],{x,-4,4},PlotRange->{0,1},AspectRatio->1/5,Ticks->{Automatic,{1}},ImageSize->Large]

Animate[Plot[u1[x,t,1],{x,-10,10},PlotRange->{0,1},AspectRatio->1/8,
Ticks->{Automatic,{1,2}},ImageSize->Large,PlotStyle->Thick],{t,0,10},AnimationRunning->False]

g[x_]=Piecewise[{{1+x,-1<x<0},{1-x,0<x<1}}]

u2[x_,t_,c_]=Integrate[g[y],{y,x-c t,x+c t},Assumptions->{c>0,t>0,x∈Reals}]/2c

Plot[u2[x,2,1],{x,-10,10},PlotRange->{0,1},PlotStyle->Thick,AspectRatio->1/8,ImageSize->Large,Ticks->{Automatic,{1,2}}]

Animate[Plot[u2[x,t,1],{x,-10,10},PlotRange->{0,1},AspectRatio->1/8,
Ticks->{Automatic,{1,2}},ImageSize->Large,PlotStyle->Thick],{t,0,10},AnimationRunning->False]

Clear[v2,c]

v2[x_,t_,c_]=D[u2[x,t,c],t]

Animate[Plot[v2[x,t,1],{x,-10,10},PlotRange->{0,1},AspectRatio->1/8,
Ticks->{Automatic,{1,2}},ImageSize->Large,PlotStyle->Thick],{t,0,10},AnimationRunning->False]

Animate[Plot[{u2[x,t,1],v2[x,t,1]},{x,-10,10},PlotRange->{0,1},AspectRatio->1/8,
Ticks->{Automatic,{1,2}},ImageSize->Large,PlotStyle->Thick],{t,0,10},AnimationRunning->False]