Sommerfeld identity

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

$\frac{e^{ikR}}{R}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{I}_0(\lambda r)e^{-\mu \left|z\right|}\frac{\lambda d\lambda }{\mu }$

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 $z\to \pm \mathrm{\infty }$ 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 $(r,\varphi ,z)$ 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 in radial-spectral form.Template:Sfn

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

$\frac{e^{i{k}_{0}r}}{r}=i\underset{0}{\overset{\mathrm{\infty }}{\int }}d{k}_{\rho }\frac{k_{\rho }}{k_z}{J}_0(k_{\rho }\rho )e^{i{k}_z\left|z\right|}$

where

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

The notation used here is different from that above: $r$ is now the distance from the origin and $\rho$ is the radial distance in a cylindrical coordinate system defined as $(\rho ,\varphi ,z)$. The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in $\rho$ 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 $k_{\rho }$.

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 $\rho$, $\varphi$) but not transforming along the height coordinate $z$. Template:Sfn

Axial-spectral representation of the Sommerfeld identity

The radial forms above integrate over the transverse spectral variable λ (equivalently $k_{\rho }$), which leaves the axial dependence to appear through a two-sided exponential in $z$. A complementary form rather integrates over the axial wavenumber $k_z$, keeping the axial phase explicit and expressing the transverse dependence through a cylindrical (Hankel) waveTemplate:Sfn:

$\frac{e^{i{k}_{0}R}}{R}=\frac{i}{2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{H}_0^{(1)}(k_{\rho }\rho )e^{i{k}_{z}z}d{k}_z,k_{\rho }=\sqrt{k_0^2-k_z^2},\text{Im}(k_{\rho })\ge 0,$

where $\rho$ is the radial distance in the cylindrical system $(\rho ,\varphi ,z)$, $R=\sqrt{{\rho }^2+z^2}$, and $H_0^{(1)}$ is the Hankel function of zeroth-order of the first kind. The branch of $k_{\rho }$ is chosen such that $H_0^{(1)}(k_{\rho }\rho )$ is representative of an outgoing wave for $k_z^2<k_0^2$ and is evanescent for $k_z^2>k_0^2$, which ensures convergence.Template:Sfn

This form is obtained via the same spherical wave by transforming along thew axial coordinate $z$ alone, instead of along the in-place coordinates. This representation is used in spectral-domain analyses of cylindrical structures and layered media, where the $e^{i{k}_{z}z}$ factor allows axial and transverse operations to be separate, and where the Hankel-function argument makes the two-centre translation of cylindrical waves tractable (via Bessel-family addition theorems, for example).

References

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