eq = D[f[t, x], {t, 2}] - D[f[t, x], {x, 2}] - D[f[t, x], {t, 2},{x, 2}]==0

shape = Cos[32*Pi*(-0.5 + x)]/E^(120.*(x-0.5)^2);
ics ={f[0, x] == shape, Derivative[1,0][f][0,x]==0};
Plot[shape,{x,0,1},Frame->True, PlotLabel->{"x","f[0,x]","initial shape"]}]

sol=NDSolveValue[{eq, ics} // Flatten, f, {t, 0, 0.25}, {x} [Element]region,
  Method -> {"FiniteElement", "MeshOptions" -> {"MaxCellMeasure" -> 0.005}}];

eq2 = D[f[t, x], {t, 2}] - D[f[t, x], {x, 2}]==0