# Helmholtz Problem in Circular Disk

The following expression $$u_{nm}(r,\theta)=J_n(x_{nm}\frac{r}{a}) \begin{cases} \cos n \theta, & n=0,1,2 , m=1,2 \\ \sin n \theta, & \end{cases}$$ gives the solution for the Helmholtz problem. On the circular disc with center 0 and radius a. For a = 1, the plot in 3-dimensional graphics of the solutions on Mathematica for $$(n,m)=(2,2)$$ and $$(n,m)=(0,2)$$ and then calculate the eigenfunction

$$k_{nm}= \frac{x_{nm}}{a}$$

a = 1;
m = 2; (*Set the value of m*)
x[m_] := N[BesselJ[2, m]]
(*calculate the eigenfunction k[m]*)
k[m_] := x[m]/a
TableForm [Table[{m, k[m]}, {m, 2, 2}], TableSpacing -> {3, 5},
TableAlignments -> Center,
TableHeadings -> {None, {"m", "\!$$\*SubscriptBox[\(k$$, $$m$$]\)"}}]


My question is how could I define the function $$u$$ and create 3D graphics as the following picture shows

I was thinking to use this

u[r_, t_, m_] := (Cos[\[Omega][m]*t] + Sin[\[Omega][m]*t])*
BesselJ[0, k[m]*r]


Is my consideration right?

It could be better to separate functions with $$\sin (n\theta)$$ and $$\cos (n\theta)$$ as follows

a = 1;
mmax = nmax = 2; (*Set the value of m, n *)
b[m_, n_] := BesselJZero[n, m];
(*calculate the eigenfunction k[m,n]*)
k[m_, n_] := N[b[m, n]/a];
uc[r_, t_, m_, n_] := (Cos[n t]) BesselJ[n, k[m, n]*r];
us[r_, t_, m_, n_] := (Sin[n t]) BesselJ[n, k[m, n]*r];


Visualization

rule = {r -> Sqrt[x^2 + y^2], t -> ArcTan[x, y]}; region =
ImplicitRegion[x^2 + y^2 <= 1, {x, y}];
Table[Plot3D[Evaluate[uc[r, t, m, n] /. rule],
Element[{x, y}, region], PlotLabel -> {n, m},
PlotTheme -> "Marketing", ColorFunction -> "AuroraColors"], {n, 0,
nmax}, {m, 1, mmax}]

Table[Plot3D[Evaluate[us[r, t, m, n] /. rule],
Element[{x, y}, region], PlotLabel -> {n, m},
PlotTheme -> "Marketing", ColorFunction -> "AuroraColors"], {n, 1,
nmax}, {m, 1, mmax}]


• Thank you that was very helpful Apr 14, 2023 at 10:43
• @AthanasiosParaskevopoulos You are welcome! Apr 14, 2023 at 11:34