Sommerfeld identity

{{short description|Result used in the theory of propagation of waves}}

The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves,

:$$

\frac{{e^{ik R} }}

{R} = \int\limits_0^\infty I_0(\lambda r) e^{ - \mu \left| z \right| } \frac{{\lambda d \lambda}}{{\mu}}

where

:$$

\mu =

\sqrt {\lambda ^2 - k^2 }

is to be taken with positive real part, to ensure the convergence of the integral and its vanishing in the limit $z\; \backslash rightarrow\; \backslash pm\; \backslash infty$ and

:$$

R^2=r^2+z^2

.

Here, $R$ is the distance from the origin while $r$ is the distance from the central axis of a cylinder as in the $(r,\backslash phi,z)$ cylindrical coordinate system. Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld. The function $I\_0(z)$ is the zeroth-order Bessel function of the first kind, better known by the notation $I\_0(z)=J\_0(iz)$ in English literature.

This identity is known as the Sommerfeld identity.{{sfn|Sommerfeld|1964|p=242}}

In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves:{{sfn|Chew|1990|p=66}}

:$$

\frac{{e^{ik_0 r} }}

{r} = i\int\limits_0^\infty {dk_\rho \frac{{k_\rho }}

{{k_z }}J_0 (k_\rho \rho )e^{ik_z \left| z \right|} }

Where

:$$

k_z=(k_0^2-k_\rho^2)^{1/2}

The notation used here is different form that above: $r$ is now the distance from the origin and $\backslash rho$ is the radial distance in a cylindrical coordinate system defined as $(\backslash rho,\backslash phi,z)$. The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in $\backslash rho$ direction, multiplied by a two-sided plane wave in the $z$ direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers $k\_\backslash rho$.

The Sommerfeld identity is closely related to the two-dimensional Fourier transform with cylindrical symmetry, i.e., the Hankel transform. It is found by transforming the spherical wave along the in-plane coordinates ($x$,$y$, or $\backslash rho$, $\backslash phi$) but not transforming along the height coordinate $z$. {{sfn|Chew|1990|p=65-66}}