14.2 Problems in Polar and Cylindrical Coordinates:
Bessel Functions
Introduction In this section we are going to consider BVPs involving
forms of the heat and wave equation in polar coordinates and a form of
Laplace's equation in cylindrical coordinates. There is a commonality
throughout the examples and exercises–––each BVP in this section
possesses radial symmetry.
Radial Symmetry The two-dimensional heat and wave equations
2
  2u  2u  u
 2u   2u
2  u
and a  2  2   2
k  2  2  
x  t
y  t
 x
 x
expressed in polar coordinates are, in turn
2
1 u 1  2u   2u
  2u 1 u 1  2u  u
2  u
and a  2 
k 2 
 2
 2

  2 (1)
2
2

r
r

r
r



t

r
r

r
r





 t
where u  u (r ,  , t ) . However, we are going to consider the simpler, but
still important, problems that possess radial symmetry–––that is,
problems in which the unknown function u is independent of the angular
coordinate  . In this case the heat and wave equations in (1) take, in
turn, the forms
  2 u 1 u  u
  2u 1 u   2u
and a 2  2 
k 2 

  2 (2)

r
r

r

t

r
r

r



 t
where u  u(r , t ) . Vibrations described by the second equation in (2) are
said to be radial vibrations.
Example 1. (pp. 734-5) Radial Vibrations of a Circular Membrane
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Laplacian in Cylindrical Coordinates Recall that the relationship between
the cylindrical coordinates of a point in space and its rectangular
coordinates is given by the equations
x  r cos , y  r sin  , z  z.
It follows immediately from the derivation of the Laplacian in polar
coordinates (see Section 14.1) that the Laplacian of a function u in
cylindrical coordinates is
 2 u 1 u 1  2 u  2 u
2
 u 2 


.
r
r r r 2  2 z 2
Example 2. (pp. 737-8) Steady Temperatures in a Circular Cylinder
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14.3 Problems in Spherical Coordinates: Legendre Polynomials
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