Sommerfeld identity

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

[math]\displaystyle{ {R} = \int\limits_0^\infty I_0(\lambda r) e^{ - \mu \left| z \right| } \frac{{\lambda d \lambda}}{{\mu}} }[/math]

where

[math]\displaystyle{ \mu = \sqrt {\lambda ^2 - k^2 } }[/math]

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

[math]\displaystyle{ R^2=r^2+z^2 }[/math].

Here, [math]\displaystyle{ R }[/math] is the distance from the origin while [math]\displaystyle{ r }[/math] is the distance from the central axis of a cylinder as in the [math]\displaystyle{ (r,\phi,z) }[/math] 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 [math]\displaystyle{ I_0(z) }[/math] is the zeroth-order Bessel function of the first kind, better known by the notation [math]\displaystyle{ I_0(z)=J_0(iz) }[/math] 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]

[math]\displaystyle{ {r} = i\int\limits_0^\infty {dk_\rho \frac{{k_\rho }} {{k_z }}J_0 (k_\rho \rho )e^{ik_z \left| z \right|} } }[/math]

Where

[math]\displaystyle{ k_z=(k_0^2-k_\rho^2)^{1/2} }[/math]

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

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

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