-SurfaceGraphics-
In[2]:=
Plot3D[Sin[2 Pi x] Sin[5 Pi y/2], {x,0,1},{y,0,2}]
Out[2]=
-SurfaceGraphics-
(Return to Chapter IX, Exercise IX.7.)
When we separate variables in polar coordinates, instead of sines and cosines, we
encounter products of trigonometric functions of theta with radial functions of a new sort.
The radial equation leads to Bessel functions. Here are some graphs to show what Bessel
In[3]:=
Plot[{BesselJ[0,r], BesselJ[1,r], BesselJ[2,r]},{r,0,4}]
Out[3]=
-Graphics-
In[4]:=
Plot[BesselJ[0,r],{r,0,40}]
Out[4]=
-Graphics-
Notice that the Bessel function oscillates qualitatively like a sine or cosine.
It drops off as
the variable increases, and the oscillations are not quite as regular. We can
calculate the
positions of the zeroes quite easily. They are called
In[5]:=
j0[n_] := x /. FindRoot[BesselJ[0,x] == 0, {x, n 2.5}]
In[6]:=
jvals = Array[j0, 5]
Out[6]=
{2.40483, 5.52008, 8.65373, 8.65373, 11.7915}
In[7]:=
j01 := 2.4048255577
j02 := 5.5200781103
j11 := 3.83171
j12 := 7.01559
The normal
modes of the disk problem are products of radial functions of the
form
BesselJ[m, jm[n] r] times Sin[m theta] or Cos[m theta] . Let's now look at some
of the
normal modes:
In[8]:=
Needs["Graphics`ParametricPlot3D`"]
In[9]:=
CylindricalPlot3D[BesselJ[0,j01 r] , {r, 0, 1}, {phi, 0, 2 Pi}]
Out[9]=
-Graphics3D-
In[10]:=
CylindricalPlot3D[BesselJ[0,j02 r] , {r, 0, 1}, {phi, 0, 2 Pi}]
-Graphics3D-
In[11]:=
CylindricalPlot3D[BesselJ[1,j11 r] Cos[phi], {r, 0, 1}, {phi, 0, 2 Pi}]<
/B>
Out[11]=
-Graphics3D-
In[12]:=
j21 := 5.13562
j22 := 8.41724
In[13]:=
CylindricalPlot3D[BesselJ[2,j21 r] Cos[2 phi], {r, 0, 1}, {phi, 0, 2 Pi}]
Out[13]=
-Graphics3D-
In[14]:=
CylindricalPlot3D[BesselJ[2,j22 r] Cos[2 phi], {r, 0, 1}, {phi, 0, 2 Pi}]
Out[14]=
-Graphics3D-
As a representative Disk problem, let us consider the heat equation on a disk,
with zero
DBC at r = 1, and initial conditions u[r,theta, t=0] = 1. We need to expand the
function
f[r] = 1 in a Fourier Bessel series. Since the function is radial (independent
of theta) only
m=0 modes contribute.
In[15]:=
Clear[j0]
j0[1] := 2.4048255577
j0[2] := 5.5200781103
j0[3] := 8.65373
j0[4] := 11.7915
In[16]:=
BesselCoeffs[n_] := (2/(BesselJ[1,j0[n]]^2)) \
Integrate[BesselJ[0,j0[n] x] x, {x,0,1}]
In[17]:=
Array[BesselCoeffs, 4]
General::intinit:
Loading integration packages -- please wait.
Out[17]=
{1.60197, -1.0648, 0.851399, -0.729645}
In[18]:=
Needs["Graphics`ParametricPlot3D`"]
In[19]:=
CylindricalPlot3D[Sum[BesselCoeffs[n] BesselJ[0,j0[n] r], {n,1,4}] , {r, 0, 1}
, {phi, 0,
2 Pi}]
Out[19]=
-Graphics3D-
This is meant to be the initial condition, which should be at constant height
1, but we have
kept only a few terms in the series. Notice that the boundary conditions are
verified