In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, ${\displaystyle \nabla ^{2}V=0}$, expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function V_n(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

Each function ${\displaystyle V_{n}(k)}$ of this basis consists of the product of three functions:

$${\displaystyle V_{n}(k;\rho ,\varphi ,z)=P_{n}(k,\rho )\Phi _{n}(\varphi )Z(k,z)\,}$$

${\displaystyle (\rho ,\varphi ,z)}$

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, a separated solution to Laplace's equation can be expressed as:

$${\displaystyle V=P(\rho )\,\Phi (\varphi )\,Z(z)}$$

$${\displaystyle {\frac {\ddot {P)){P))+{\frac {1}{\rho ))\,{\frac {\dot {P)){P))+{\frac {1}{\rho ^{2))}\,{\frac {\ddot {\Phi )){\Phi ))+{\frac {\ddot {Z)){Z))=0}$$

The Z  part of the equation is a function of z alone, and must therefore be equal to a constant:

$${\displaystyle {\frac {\ddot {Z)){Z))=k^{2))$$

$${\displaystyle Z(k,z)=\cosh(kz)\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,\sinh(kz)\,}$$

$${\displaystyle Z(k,z)=e^{kz}\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,e^{-kz}\,}$$

If k is imaginary:

$${\displaystyle Z(k,z)=\cos(|k|z)\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,\sin(|k|z)\,}$$

$${\displaystyle Z(k,z)=e^{i|k|z}\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,e^{-i|k|z}\,}$$

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 ${\displaystyle k^{2))$ for ${\displaystyle {\ddot {Z))/Z}$ , Laplace's equation may now be written:

$${\displaystyle {\frac {\ddot {P)){P))+{\frac {1}{\rho ))\,{\frac {\dot {P)){P))+{\frac {1}{\rho ^{2))}{\frac {\ddot {\Phi )){\Phi ))+k^{2}=0}$$

Multiplying by ${\displaystyle \rho ^{2))$, we may now separate the P  and Φ functions and introduce another constant (n) to obtain:

$${\displaystyle {\frac {\ddot {\Phi )){\Phi ))=-n^{2))$$

$${\displaystyle \rho ^{2}{\frac {\ddot {P)){P))+\rho {\frac {\dot {P)){P))+k^{2}\rho ^{2}=n^{2))$$

Since ${\displaystyle \varphi }$ is periodic, we may take n to be a non-negative integer and accordingly, the ${\displaystyle \Phi (\varphi )}$ the constants are subscripted. Real solutions for ${\displaystyle \Phi (\varphi )}$ are

$${\displaystyle \Phi _{n}=\cos(n\varphi )\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,\sin(n\varphi )}$$

$${\displaystyle \Phi _{n}=e^{in\varphi }\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,e^{-in\varphi ))$$

The differential equation for ${\displaystyle \rho }$ is a form of Bessel's equation.

If k is zero, but n is not, the solutions are:

$${\displaystyle P_{n}(0,\rho )=\rho ^{n}\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,\rho ^{-n}\,}$$

If both k and n are zero, the solutions are:

$${\displaystyle P_{0}(0,\rho )=\ln \rho \,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,1\,}$$

If k is a real number we may write a real solution as:

$${\displaystyle P_{n}(k,\rho )=J_{n}(k\rho )\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,Y_{n}(k\rho )\,}$$

${\displaystyle J_{n}(z)}$

${\displaystyle Y_{n}(z)}$

If k is an imaginary number, we may write a real solution as:

$${\displaystyle P_{n}(k,\rho )=I_{n}(|k|\rho )\,\,\,\,\,\,\mathrm {or} \,\,\,\,\,\,K_{n}(|k|\rho )\,}$$

${\displaystyle I_{n}(z)}$

${\displaystyle K_{n}(z)}$

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:

$${\displaystyle V(\rho ,\varphi ,z)=\sum _{n}\int d\left|k\right|\,\,A_{n}(k)P_{n}(k,\rho )\Phi _{n}(\varphi )Z(k,z)\,}$$

${\displaystyle A_{n}(k)}$

${\displaystyle J_{n}(x)}$

${\displaystyle \Phi _{n}(\varphi )}$

${\displaystyle Z(k,z)}$

${\displaystyle P_{n}(k\rho )}$

${\displaystyle J_{n}(k\rho )}$

${\displaystyle J_{n))$

${\displaystyle \Phi _{n}(\varphi )}$

${\displaystyle Z(k,z)}$

If ${\displaystyle (x)_{k))$ is the sequence of the positive zeros of ${\displaystyle J_{n))$ then:^[2]

$${\displaystyle \int _{0}^{1}J_{n}(x_{k}\rho )J_{n}(x_{k}'\rho )\rho \,d\rho ={\frac {1}{2))J_{n+1}(x_{k})^{2}\delta _{kk'))$$

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 ${\displaystyle (\rho _{0},\varphi _{0},z_{0})}$ inside a conducting cylindrical tube (e.g. an empty tin can) which is bounded above and below by the planes ${\displaystyle z=-L}$ and ${\displaystyle z=L}$ and on the sides by the cylinder ${\displaystyle \rho =a}$.^[3] (In MKS units, we will assume ${\displaystyle q/4\pi \epsilon _{0}=1}$). 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 ${\displaystyle P_{n}(k\rho )}$ function to be the ordinary Bessel function ${\displaystyle J_{n}(k\rho )}$, 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:

$${\displaystyle V(\rho ,\varphi ,z)=\sum _{n=0}^{\infty }\sum _{r=0}^{\infty }\,A_{nr}J_{n}(k_{nr}\rho )\cos(n(\varphi -\varphi _{0}))\sinh(k_{nr}(L+z))\,\,\,\,\,z\leq z_{0))$$

${\displaystyle k_{nr}a}$

${\displaystyle J_{n}(z)}$

$${\displaystyle A_{nr}={\frac {4(2-\delta _{n0})}{a^{2))}\,\,{\frac {\sinh k_{nr}(L-z_{0})}{\sinh 2k_{nr}L))\,\,{\frac {J_{n}(k_{nr}\rho _{0})}{k_{nr}[J_{n+1}(k_{nr}a)]^{2))}\,}$$

Above the source point:

$${\displaystyle V(\rho ,\varphi ,z)=\sum _{n=0}^{\infty }\sum _{r=0}^{\infty }\,A_{nr}J_{n}(k_{nr}\rho )\cos(n(\varphi -\varphi _{0}))\sinh(k_{nr}(L-z))\,\,\,\,\,z\geq z_{0))$$

$${\displaystyle A_{nr}={\frac {4(2-\delta _{n0})}{a^{2))}\,\,{\frac {\sinh k_{nr}(L+z_{0})}{\sinh 2k_{nr}L))\,\,{\frac {J_{n}(k_{nr}\rho _{0})}{k_{nr}[J_{n+1}(k_{nr}a)]^{2))}.\,}$$

It is clear that when ${\displaystyle \rho =a}$ or ${\displaystyle |z|=L}$, 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 ${\displaystyle z=z_{0))$.

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:

$${\displaystyle V(\rho ,\varphi ,z)=\sum _{n=0}^{\infty }\sum _{r=0}^{\infty }\,A_{nr}J_{n}(k_{nr}\rho )\cos(n(\varphi -\varphi _{0}))e^{-k_{nr}|z-z_{0}|))$$

$${\displaystyle A_{nr}={\frac {2(2-\delta _{n0})}{a^{2))}\,\,{\frac {J_{n}(k_{nr}\rho _{0})}{k_{nr}[J_{n+1}(k_{nr}a)]^{2))}.\,}$$

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:

$${\displaystyle V(\rho ,\varphi ,z)={\frac {1}{R))=\sum _{n=0}^{\infty }\int _{0}^{\infty }d\left|k\right|\,A_{n}(k)J_{n}(k\rho )\cos(n(\varphi -\varphi _{0}))e^{-k|z-z_{0}|))$$

$${\displaystyle A_{n}(k)=(2-\delta _{n0})J_{n}(k\rho _{0})\,}$$

$${\displaystyle R={\sqrt {(z-z_{0})^{2}+\rho ^{2}+\rho _{0}^{2}-2\rho \rho _{0}\cos(\varphi -\varphi _{0}))).\,}$$

Point source in open space at origin

Finally, when the point source is at the origin, ${\displaystyle \rho _{0}=z_{0}=0}$

$${\displaystyle V(\rho ,\varphi ,z)={\frac {1}{\sqrt {\rho ^{2}+z^{2))))=\int _{0}^{\infty }J_{0}(k\rho )e^{-k|z|}\,dk.}$$