• Source: Sommerfeld identity

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 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 form 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}

.




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.

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