Cylindrical harmonics
Encyclopedia
In mathematics
, the cylindrical harmonics are a set of linearly independent 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 term cylindrical harmonics is also used to refer to the Bessel function
s (that are cylindrical harmonics in the sense described above).
of this basis consists of the product of three functions:
where
are the cylindrical coordinates, and n and k are constants which distinguish the members of the set from each other. 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 of the surfaces of constant ρ, φ and z are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables
, a separated solution to Laplace's equation may be written:
and Laplace's equation, divided by V, is written:
The Z part of the equation is a function of z alone, and must therefore be equal to a constant:
where k is, in general, a complex number
. For a particular k, the Z(z) function has two linearly independent solutions. If k is real they are:
or by their behavior at infinity:
If k is imaginary:
or:
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.
Substituting
for 
, Laplace's equation may now be written:
Multiplying by
, we may now separate the P and Φ functions and introduce another constant (n) to obtain:
Since
is periodic, we may take n to be a non-negative integer and accordingly, the 
the constants are subscripted. Real solutions for 
are
or, equivalently:
The differential equation for
is a form of Bessel's equation.
If k is zero, but n is not, the solutions are:
If both k and n are zero, the solutions are:
If k is a real number we may write a real solution as:
where
and 
are ordinary Bessel function
s. If k is an imaginary number, we may write a real solution as:
where
and 
are modified Bessel function
s. 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 of these solutions:
where the
are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary conditions of the problem. Note that the integral may be replaced by a sum for appropriate boundary conditions. The orthogonality of the 
is often very useful when finding a solution to a particular problem. The 
and 
functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions. When 
is simply 
, the orthogonality of 
, along with the orthogonality relationships of 
and 
allow the constants to be determined.
see smythe p 185 for more orthogonality
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.
inside a conducting cylindrical tube (e.g. an empty tin can) which is bounded above and below by the planes 
and 
and on the sides by the cylinder 
(Smythe, 1968). (In MKS units, we will assume 
). Since the potential is bounded by the planes on the z axis, the Z(k,z) function can be taken to be periodic. Since the potential must be zero at the origin, we take the 
function to be the ordinary Bessel function 
, 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:
where
is the r-th zero of 
and, from the orthogonality relationships for each of the functions:
Above the source point:
It is clear that when
or 
, the above function is zero. It can also be easily shown that the two functions match in value and in the value of their first derivatives at 
.
and R is the distance from the point source to the measurement point:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation
Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
,
, expressed in cylindrical coordinatesCylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...
, 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 term cylindrical harmonics is also used to refer to the Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s (that are cylindrical harmonics in the sense described above).
Definition
Each function of this basis consists of the product of three functions:where
are the cylindrical coordinates, and n and k are constants which distinguish the members of the set from each other. As a result of the superposition principleSuperposition principle
In physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...
applied to Laplace's equation, very general solutions to Laplace's equation can be obtained by linear combinations of these functions.
Since all of the surfaces of constant ρ, φ and z are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables
Separation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
, a separated solution to Laplace's equation may be written:
and Laplace's equation, divided by V, is written:
The Z part of the equation is a function of z alone, and must therefore be equal to a constant:
where k is, in general, a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
. For a particular k, the Z(z) function has two linearly independent solutions. If k is real they are:
or by their behavior at infinity:
If k is imaginary:
or:
It can be seen that the Z(k,z) functions are the kernels of the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
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.
Substituting
for , Laplace's equation may now be written:Multiplying by
, we may now separate the P and Φ functions and introduce another constant (n) to obtain:Since
is periodic, we may take n to be a non-negative integer and accordingly, the the constants are subscripted. Real solutions for areor, equivalently:
The differential equation for
is a form of Bessel's equation.If k is zero, but n is not, the solutions are:
If both k and n are zero, the solutions are:
If k is a real number we may write a real solution as:
where
and are ordinary Bessel functionBessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s. If k is an imaginary number, we may write a real solution as:
where
and are modified Bessel functionBessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s. 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 of these solutions:
where the
are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary conditions of the problem. Note that the integral may be replaced by a sum for appropriate boundary conditions. The orthogonality of the is often very useful when finding a solution to a particular problem. The and functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions. When is simply , the orthogonality of , along with the orthogonality relationships of and allow the constants to be determined.see smythe p 185 for more orthogonality
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
As an example, consider the problem of determining the potential of a unit source located at inside a conducting cylindrical tube (e.g. an empty tin can) which is bounded above and below by the planes and and on the sides by the cylinder (Smythe, 1968). (In MKS units, we will assume ). Since the potential is bounded by the planes on the z axis, the Z(k,z) function can be taken to be periodic. Since the potential must be zero at the origin, we take the function to be the ordinary Bessel function , 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:where
is the r-th zero of and, from the orthogonality relationships for each of the functions:Above the source point:
It is clear that when
or , the above function is zero. It can also be easily shown that the two functions match in value and in the value of their first derivatives at .Point source inside cylinder
Removing the plane ends (i.e. taking the limit as L approaches infinity) gives the field of the point source inside a conducting cylinder:Point source in open space
As the radius of the cylinder (a) approaches infinity, the sum over the zeroes of J_n(z) becomes an integral, and we have the field of a point source in infinite space:and R is the distance from the point source to the measurement point:
Point source in open space at origin
Finally, when the point source is at the origin,