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,

R=∫0∞I0(λr)e−μ|z|λdλμ

where

μ=λ2−k2

is to be taken with positive real part, to ensure the convergence of the integral and its vanishing in the limit z->±∞ and

R2=r2+z2.

Here, R is the distance from the origin while r is the distance from the central axis of a cylinder as in the (r,ϕ,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 I0(z) is the zeroth-order Bessel function of the first kind, better known by the notation I0(z)=J0(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]

r=i∫0∞dkρkρkzJ0(kρρ)eikz|z|

Where

kz=(k02−kρ2)1/2

The notation used here is different form that above: r is now the distance from the origin and ρ is the radial distance in a cylindrical coordinate system defined as (ρ,ϕ,z). The physical interpretation is that a spherical wave can be expanded into a summation of cylindrical waves in ρ 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 kρ.

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 ρ, ϕ) but not transforming along the height coordinate z. [3]

## Notes

1. ↑ Sommerfeld 1964, p. 242.
2. ↑ Chew 1990, p. 66.
3. ↑ Chew 1990, p. 65-66.

## 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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