How to write a specific Bessel function in Mathematica

Question

I want to plot the following function on Mathematica, and I gave it a go on wolframalpha.

Bessel[n,z] is the usual form, but I am not sure how to use this to compute the following plot:

\begin{equation} u(r,t)=\frac{\alpha J_{4}(i\sqrt{2}r)}{J_{4}\big(\frac{100*2}{\alpha}\big)}e^{-16t^2} \end{equation}

I tried

BesselJ[4, I Sqrt[2] x]/BesselJ[4, 200]

But I don't know how to include the zeros defined by $\alpha$

Any help appreciated!

where $\alpha$ are the zeros of the Bessel function.

Answer 1

Clear["Global`*"]

u[x_, t_, α_] := α*BesselJ[4, I Sqrt[2] x]/BesselJ[4, 200/α]*
  E^(-16 t^2)

u[x, t, α] == -u[x, t, -α]

(* True *)

u[x, t, α] == u[x, -t, α]

(* True *)

u[x, t, α] == u[-x, t, α]

(* True *)

Manipulate[
 Plot3D[u[x, t, α],
  {x, -5, 5}, {t, -2, 2},
  AxesLabel -> Automatic,
  ClippingStyle -> None],
 {{α, 1}, 0.05, 5, 0.05, Appearance -> "Labeled"}]

EDIT: For α = BesselJZero[4, k]

Manipulate[
 α = BesselJZero[4, k];
 Plot3D[u[x, t, α], {x, -5, 5}, {t, -2, 2},
  AxesLabel -> Automatic,
  ClippingStyle -> None,
  PlotLabel -> StringForm["α = `` = ``", α, α // N],
  WorkingPrecision -> 15],
 {{k, 1}, Range[10], ControlType -> SetterBar}]