The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves,
e i k R R = ∫ 0 ∞ I 0 ( λ r ) e − μ | z | λ d λ μ {\displaystyle {\frac {e^{ikR}}{R}}=\int \limits _{0}^{\infty }I_{0}(\lambda r)e^{-\mu \left|z\right|}{\frac {\lambda d\lambda }{\mu }}}
where
μ = λ 2 − k 2 {\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 z → ± ∞ {\displaystyle z\rightarrow \pm \infty } and
R 2 = r 2 + z 2 {\displaystyle R^{2}=r^{2}+z^{2}} . Here, R {\displaystyle R} is the distance from the origin while r {\displaystyle r} is the distance from the central axis of a cylinder as in the ( r , ϕ , z ) {\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 I 0 ( z ) {\displaystyle I_{0}(z)} is the zeroth-order Bessel function of the first kind, better known by the notation I 0 ( z ) = J 0 ( i z ) {\displaystyle I_{0}(z)=J_{0}(iz)} in English literature. This identity is known as the Sommerfeld identity in radial-spectral form. In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves:
e i k 0 r r = i ∫ 0 ∞ d k ρ k ρ k z J 0 ( k ρ ρ ) e i k z | z | {\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
k z = ( k 0 2 − k ρ 2 ) 1 / 2 {\displaystyle k_{z}=(k_{0}^{2}-k_{\rho }^{2})^{1/2}}
The notation used here is different from that above: r {\displaystyle r} is now the distance from the origin and ρ {\displaystyle \rho } is the radial distance in a cylindrical coordinate system defined as ( ρ , ϕ , z ) {\displaystyle (\rho ,\phi ,z)} . The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in ρ {\displaystyle \rho } direction, multiplied by a two-sided plane wave in the z {\displaystyle z} direction; see the Jacobi-Anger expansion. The summation has to be taken over all the wavenumbers k ρ {\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 ( x {\displaystyle x} , y {\displaystyle y} , or ρ {\displaystyle \rho } , ϕ {\displaystyle \phi } ) but not transforming along the height coordinate z {\displaystyle z} .
Axial-spectral representation of the Sommerfeld identity The radial forms above integrate over the transverse spectral variable λ (equivalently k ρ {\displaystyle k_{\rho }} ), which leaves the axial dependence to appear through a two-sided exponential in z {\displaystyle z} . A complementary form rather integrates over the axial wavenumber k z {\displaystyle k_{z}} , keeping the axial phase explicit and expressing the transverse dependence through a cylindrical (Hankel) wave:
e i k 0 R R = i 2 ∫ − ∞ ∞ H 0 ( 1 ) ( k ρ ρ ) e i k z z d k z , k ρ = k 0 2 − k z 2 , Im ( k ρ ) ≥ 0 , {\displaystyle {\frac {e^{ik_{0}R}}{R}}={\frac {i}{2}}\int _{-\infty }^{\infty }H_{0}^{(1)}(k_{\rho }\rho )e^{ik_{z}z}dk_{z},\quad k_{\rho }={\sqrt {k_{0}^{2}-k_{z}^{2}}},\ {\text{Im}}(k_{\rho })\geq 0,}
where ρ {\displaystyle \rho } is the radial distance in the cylindrical system ( ρ , ϕ , z ) {\displaystyle (\rho ,\phi ,z)} , R = ρ 2 + z 2 {\displaystyle R={\sqrt {\rho ^{2}+z^{2}}}} , and H 0 ( 1 ) {\displaystyle H_{0}^{(1)}} is the Hankel function of zeroth-order of the first kind. The branch of k ρ {\displaystyle k_{\rho }} is chosen such that H 0 ( 1 ) ( k ρ ρ ) {\displaystyle H_{0}^{(1)}(k_{\rho }\rho )} is representative of an outgoing wave for k z 2 < k 0 2 {\displaystyle k_{z}^{2}<k_{0}^{2}} and is evanescent for k z 2 > k 0 2 {\displaystyle k_{z}^{2}>k_{0}^{2}} , which ensures convergence. This form is obtained via the same spherical wave by transforming along thew axial coordinate z {\displaystyle 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 {\displaystyle e^{ik_{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).
Notes
References Sommerfeld, Arnold (1964). Partial Differential Equations in Physics. New York: Academic Press. ISBN 9780126546583. {{cite book}}: ISBN / Date incompatibility (help) Chew, Weng Cho (1990). Waves and Fields in Inhomogeneous Media. New York: Van Nostrand Reinhold. ISBN 9780780347496.