Common pilot channel

Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as

"the sources must be sources, not sinks of energy. The energy which is radiated from the sources must scatter to infinity; no energy may be radiated from infinity into ... the field."

Mathematically, consider the inhomogeneous Helmholtz equation

$({\mathrm{\nabla }}^2+k^2)u=-f\text{ in }{ℝ}^n$

where $n=2,3$ is the dimension of the space, $f$ is a given function with compact support representing a bounded source of energy, and $k>0$ is a constant, called the wavenumber. A solution $u$ to this equation is called radiating if it satisfies the Sommerfeld radiation condition

$\underset{|x|\to \mathrm{\infty }}{lim}|x{|}^{\frac{n-1}{2}}(\frac{\partial }{\partial |x|}-ik)u(x)=0$

uniformly in all directions

$\hat{x}=\frac{x}{|x|}$

(above, $i$ is the imaginary unit and $|\cdot |$ is the Euclidean norm). Here, it is assumed that the time-harmonic field is $e^{-i\omega t}u.$ If the time-harmonic field is instead $e^{i\omega t}u,$ one should replace $-i$ with $+i$ in the Sommerfeld radiation condition.

The Sommerfeld radiation condition is used to solve uniquely the Helmholtz equation. For example, consider the problem of radiation due to a point source $x_0$ in three dimensions, so the function $f$ in the Helmholtz equation is $f(x)=\delta (x-x_0),$ where $\delta$ is the Dirac delta function. This problem has an infinite number of solutions. All solutions have the form

$u=c{u}_{+}+(1-c)u_{-}$

where $c$ is a constant, and

$u_{\pm }(x)=\frac{e^{\pm ik|x-x_0|}}{4\pi |x-x_0|}.$

Of all these solutions, only $u_{+}$ satisfies the Sommerfeld radiation condition and corresponds to a field radiating from $x_0.$ The other solutions are unphysical. For example, $u_{-}$ can be interpreted as energy coming from infinity and sinking at $x_0.$

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