This is an evaluated

(c) Copyright 1994,1995 by Evans M. Harrell, II. All rights reserved

Let us begin with some representative normal modes for the vibrations of a rectangular

membrane, after which we will compare with those of a circular membrane, like a drum.

In the x-direction the rectangle has length 1 and in the y-direction it has length 2.

-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

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}]

*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