<< NumericalMath`BesselZeros`

mm  =  3;  nn  =  2;  k  =  6; dt  =   .005;

bj  =  Table[BesselJZeros[m, nn], {m, 0, mm}];

ci  =  {{1,  .3}, {1,  .2}, {-2,  .5}, {2,  .4}};

Do[v[r_, θ_]  =  
        Sum[ci[[m + 1, 
                n]] (E^(- bj[[m + 1, n]]^2 t)) BesselJ[m,  bj[[m + 1, n]] r] Cos[m (θ +  3.5)], {m, 0, 
            mm}, {n, 1, nn}]; 
zz = 
        ParametricPlot3D[{r Cos[θ], r Sin[θ], 
            v[r, θ]}, {r, 0, 1}, {θ, 0, 2  π}, 
          Boxed -> False, Axes -> False, 
          PlotPoints  -> {50, 50}, 
          PlotRange  ->  2 {-1, 1}], {t, 0,  .5,  dt}]