Solution

Plot[BesselJ[2, x], {x, 0, 18}];

Consider the 4th zero

FindRoot[ BesselJ[2, x]==0, {x, 14}]

The (2, 4,) normal mode

Do[
CylindricalPlot3D[
         u[2, 14.796, r, theta, t, 10], {r, 0, 1},
                {theta, 0, 2 Pi},
        PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}},
        PlotPoints->50,
        Axes -> False, Boxed ->False],
               {t, 0, 2 Pi/10, Pi/ (7 10)}]
                       

Created by Mathematica  (October 29, 2006)