Clear[r, f]
ρ = 1
c1 = 1
c2 = 1
μ = 1
f[x_] := 
      x - 3 /; 3 ≤ x ≤ 4
f[x_] := 
      5 - x /; 4 ≤ x ≤ 5
f[x_] := 0 /; (x < 3 || x > 5)
r[x_] := ((ρ*c1)/(ρ*c1 + ρ*c2 + μ)) 
        f[c1*x]
u1[x_] := 
      If[(x ≥ 0 && (x - c1*t) ≥ 
              0), (1/2) (f[x + c1*t] + f[x - c1*t]), 
        0]
u2[x_] := 
      If[(x ≥ 0 && (x - c1*t) ≤ 
              0), (1/2) (f[x + c1*t] - f[-x + c1*t]) + 
          r[((-x + c1*t)/c1)], 0]
u3[x_] := 
      If[(x ≤ 0 && (x + c2*t) ≥ 0), 
        r[((x + c2*t)/c2)], 0]
g[x_] := u1[x] + u2[x] + u3[x]
Table[
      Plot[g[x], {x, -10, 10}, 
        PlotRange -> {{-10, 10}, {-1, 1}}, 
        PlotStyle -> {RGBColor[1, 0.7, 0], Thickness[0.015]}], {t, 0, 
        10, 0.1}]