The Bessel function is a special function in mathematics. This function expresses inward or outward propagating waveforms, such as a circular membrane or line source. Moreover, the function also has a unique form describing the radial dependence of a spherical waveform. We will focus on an overview of the Bessel and Hankel functions.

The cylindrical form of the Helmholtz equation is given by

$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial P}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2P}{\partial\phi^2}+\frac{\partial^2p}{\partial z^2}+k^2P=0$

where k is the wavenumber. Applying separation of variables, we assume $P=R\left(r\right)\Phi\left(\phi\right)Z\left(z\right)$. Substituting and dividing by P yields

$\frac{1}{rR}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r}\right)+\frac{1}{r^2\Phi}\frac{\partial^2\ \Phi}{\partial\phi^2}+\frac{1}{Z}\frac{\partial^2Z}{\partial z^2}+k^2=0.$

The third addend is only dependent on z, meaning it must be a constant. This gives

$\frac{\partial^2Z}{\partial z^2}=-k_z^2$.

Defining the radial wavenumber as $k_r^2=k^2-k_z^2$ and multiplying by $r^2$ yields

$\frac{r}{R}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r}\right)+\frac{1}{\Phi}\frac{\partial^2\ \Phi}{\partial\phi^2}+{k_r}^2r^2=0.$

Observing that the second term is independent of r and z, we assume this term is also equal to a constant,

$\frac{1}{\Phi}\frac{\partial^2\ \Phi}{\partial \phi^2}=-n^2.$

After substitution, the expression is an ordinary differential equation dependent on r,

$r\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r}\right)+\left({k_r}^2r^2-n^2\right)R=r^2\frac{\partial^2}{\partial r^2}+r\frac{\partial R}{\partial r}+\left({k_r}^2r^2-n^2\right)R=0.$

The above differential equation is the Bessel equation. The general solution is given as

$R=\alpha J_n(kr)+\beta Y_n(kr),$

where $J_n$ is the Bessel function of the first kind of order n and $Y_n$is Bessel function of the second kind of order n. The Hankel functions are two linearly independent combination of the Bessel function:

$H_n^{\left(1\right)}=J_n(kr)+\text{j}Y_n(kr) \\ H_n^{\left(2\right)}=J_n(kr)-\text{j}Y_n(kr).$

The Bessel functions may also describe the propagating wave in a spherical sense. These functions are called spherical Bessel functions j. The relationship between Bessel and spherical Bessel functions is given as

$j_n(kr)=\sqrt(\frac{\pi}{2})J_{n+1/2},$

where the order n is a half-integer, n+1/2, instead of an integer.