n = 10; k0 = 1; m0 = 1;
k = SparseArray[{Band[{1, 1}] -> -2. k0, Band[{1, 2}] -> k0,
    Band[{2, 1}] -> k0}, {n, n}, 0.];
m = m0*IdentityMatrix[n];
xx = Table[x[i][t], {i, 1, n}];
eq = Join[
   Thread[(m.D[xx, t, t])[[2 ;; -2]] == (k.xx)[[2 ;; -2]]], {x[1][
      t] == 1., x[n][t] == n}, Thread[xx == Range[n] /. t -> 0],
   Thread[D[xx, t] == 0.25 Sin[Subdivide[0., 2. Pi, n - 1]] /.
     t -> 0]];
sol = NDSolve[eq, xx, {t, 0, 100}];
Plot[Evaluate[xx /. sol[[1]]], {t, 0, 100}]

A = 1;
q[j_, n_, k_] := A Sin[(n [Pi])/(k + 1) j]
Manipulate[
 ListPlot[Table[{i, q[l, i, 10]}, {i, 1, 10}]], {l, 1, 10, 1}]