<< NumericalMath`BesselZeros`

h[x_, y_]  :=  
      Exp[-100 ((x -  .5)^2  +  y^2)]

ParametricPlot3D[{r Cos[θ], r Sin[θ], 
        h[r Cos[θ], r Sin[θ]]}, {r, 0, 
        1}, {θ, -π, π}, Boxed -> False, 
      Axes -> False, PlotPoints  -> {50, 50}, 
      PlotRange  ->  {-1, 1}]

mm  =  20; nn  =  10;

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

cf  =  
        Table[2  NIntegrate[
                h[x, y] BesselJ[m,  
                    bj[[m + 1, n]] (Sqrt[(x^2) +  (y^2)])] 
                  Cos[m  ArcTan[
                        y/x]], {x,  .3,  .7}, {y, - .2,  .2}]/( π BesselJ[m + 1, bj[[m + 1, n]]]^2), {m, 0, mm}, {n, 1, 
            nn}];

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