In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(k) is the product of three terms, each depending on one coordinate alone. The ρ-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).

Definition

are the cylindrical coordinates, and n and k constants that differentiate the members of the set. As a result of the superposition principle applied to Laplace's equation, very general solutions to Laplace's equation can be obtained by linear combinations of these functions.

Since all surfaces with constant ρ, φ and z are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables and Laplace's equation, divided by V

The Z part of the equation is a function of z where k is, in general, a complex number. For a particular k, the Z(z) function has two linearly independent solutions. If k

If k

It can be seen that the Z(k,z) functions are the kernels of the Fourier transform or Laplace transform of the Z(z) function and so k may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions.

, we may now separate the P and Φ functions and introduce another constant (n

is periodic, we may take n to be a non-negative integer

is a form of Bessel's equation.

If k is zero, but n

If k is a real number are ordinary Bessel functions.

If k is an imaginary number are modified Bessel functions.

The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions allow the constants to be determined.

then:

In solving problems, the space may be divided into any number of pieces, as long as the values of the potential and its derivative match across a boundary which contains no sources.

Example: Point source inside a conducting cylindrical tube

.). Since the potential is bounded by the planes on the z axis, the Z(k,z), and it must be chosen so that one of its zeroes lands on the bounding cylinder. For the measurement point below the source point on the z axis, the potential will be:

is the r

.

Point source inside cylinder

Point source in open space

As the radius of the cylinder (a) approaches infinity, the sum over the zeroes of and R

Point source in open space at origin

See also

• Spherical harmonics

Notes

References