I would first define the odd-extension into the interval which below I'm using Piecewise in Mathematica:

f_0[x_] := Piecewise[{{0, -2 <= x <= 2}, {2 (x - 2),

2 <= x <= 3}, {-2 (x - 3) + 2,

3 < x < 4}, {2 (x + 3) - 2, -3 <= x <= -2}, {-2 (x + 4), -4 <=

x <= -3}, {0, Abs[x] > 4}}];

You read that as: zero in the interval (-2,2), 2(x-2) in the interval (2,3), and so on.

and then convert it to a problem on the infinite string in which the solution is then:

.

You can then draw this piecewise function at various time periods or use Manipulate in Mathematica:

Manipulate[

Plot[1/2 (f_0[x + t] + f_0[x - t]), {x, -10, 10},

PlotRange -> {{-10, 10}, {-2, 2}}, PlotPoints -> 50], {t, 0, 10}]