Sommerfeld identity

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

\frac{{e^{ik R} }} {R} = \int\limits_0^\infty 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 \rightarrow \pm \infty 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. The function I0 is a Bessel function. Here the notation for Bessel functions follows the German convention, to be consistent with the original notation used by Sommerfeld. In English literature it is more common to use

In(ρ) = Jn(iρ).

This identity is known as the Sommerfeld Identity [Ref.1,Pg.242].

An alternative form is

\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_\rho^2)^{1/2}

[Ref.2,Pg.66]. The notation used here is different form that above: r is now the distance from the origin and ρ is the axial distance in a cylindrical 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 plane wave in the z direction. The summation has to be taken over all the wavenumbers kρ.

[edit] References

  1. Sommerfeld, A.,Partial Differential Equations in Physics,Academic Press,New York,1964
  2. Chew, W.C.,Waves and Fields in Inhomogenous Media,Van Nostrand Reinhold,New York,1990


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