Sommerfeld identity

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

${\displaystyle {\frac {e^{ikR}}{R}}=\int \limits _{0}^{\infty }I_{0}(\lambda r)e^{-\mu \left|z\right|}{\frac {\lambda d\lambda }{\mu }}}$

where

${\displaystyle \mu ={\sqrt {\lambda ^{2}-k^{2}}}}$

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

${\displaystyle R^{2}=r^{2}+z^{2}}$.

Here, ${\displaystyle R}$ is the distance from the origin while ${\displaystyle r}$ is the distance from the central axis of a cylinder as in the ${\displaystyle (r,\phi ,z)}$ 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 ${\displaystyle I_{0}(z)}$ is the zeroth-order Bessel function of the first kind, better known by the notation ${\displaystyle I_{0}(z)=J_{0}(iz)}$ in English literature. This identity is known as the Sommerfeld identity.^[1]

In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves:^[2]

${\displaystyle {\frac {e^{ik_{0}r}}{r}}=i\int \limits _{0}^{\infty }{dk_{\rho }{\frac {k_{\rho }}{k_{z}}}J_{0}(k_{\rho }\rho )e^{ik_{z}\left|z\right|}}}$

Where

${\displaystyle k_{z}=(k_{0}^{2}-k_{\rho }^{2})^{1/2}}$

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

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 (${\displaystyle x}$,${\displaystyle y}$, or ${\displaystyle \rho }$, ${\displaystyle \phi }$) but not transforming along the height coordinate ${\displaystyle z}$. ^[3]

Notes

References

• Sommerfeld, Arnold (1964). Partial Differential Equations in Physics. New York: Academic Press. ISBN 9780126546583.
• Chew, Weng Cho (1990). Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold. ISBN 9780780347496.