Bessel functions of order m

Plot[BesselJ[1,z],{z,0,40}]

Plot[BesselY[1,z],{z,0,40}]

GraphicsGrid[Table[{Plot[BesselJ[m,z],{z,0,40},PlotRange->All],Plot[BesselY[m,z],{z,0,40}]},{m,0,5}],
ImageSize->Medium]

Table[N[BesselJZero[m,n]],{m,0,5},{n,1,5}]//TableForm

General normal modes of a circular membrane

Plot[BesselJ[1,z],{z,0,30}]

α11=N[BesselJZero[1,1]]

t11=2 Pi/α11

RevolutionPlot3D[BesselJ[1,α11* r]Cos[θ],{r,0,1},{θ,0,2Pi}]

Animate[
RevolutionPlot3D[BesselJ[1,α11* r]*Cos[θ]*Cos[α11* t],{r,0,1},{θ,0,2Pi},PlotRange->{-1,1},Mesh->True],{t,0,t11},AnimationRunning->False]

α36=N[BesselJZero[3,6]]

t36=2Pi/α36

Animate[
RevolutionPlot3D[BesselJ[3,α36* r]*Cos[3θ]*Cos[α36* t],{r,0,1},{θ,0,2Pi},PlotRange->{-1,1},Mesh->True],{t,0,t36},AnimationRunning->False]