Plucked string

f // ClearAll;  w // ClearAll;

w[x_, t_]  :=  ((f[x  -  t]  +  f[x  +  t])/2)

At midpoint

f[x_]  := (mx  =  Mod[x - 1/2, 2];  
          Which[ mx <  1, 1 /2 - mx, True, mx - 3 /2 ]);

\(\(\(Plot[w[x, #], {x, 0, 1}, 
            PlotRange\  -> {{0, 1}, {\(- .6\),  .6}}, 
            PlotStyle\  -> \ {Thickness[ .004], 
                Hue[ .65]}] &\)\  /@ 
\((	\ Range[0, 
            2,  .02])\);\)\)

At 2/3 of length

f[x_]  := (mx  =  Mod[x - 2/3, 2];  
          Which[ mx < 2/3, 2 (1/3 - mx), True, 
            mx - 4/3 ]);

\(\(\(Plot[w[x, #], {x, 0, 1}, PlotRange\  ->  .7 {\(-1\), 1}, 
            PlotStyle\  -> \ {Thickness[ .004], 
                Hue[ .05]}] &\)\  /@ 
\((	\ Range[0, 
            2,  .02])\);\)\)