Radial Dependence of Radiation from a Bounded Source

The stated result is likely considered to be “obvious” by most physicists [1]. Yet, there have been recent claims [2, 3, 4, 5, 6] that electromagnetic waves from a bounded source can be generated whose amplitude falls off as 1/ √ r over portions of solid angle in far zone. Mathematically, there exist cylindrical waves whose amplitude falls off as 1/ √ ρ, where ρ is the radial distance in a cylindrical coordinate system whose axis is the axis of symmetry of the source, which latter has infinite extent along the axis. An example is Čerenkov radiation; however, the amplitude of Čerenkov radiation from a finite path length falls off as 1/r for r large compared to the path length [7]. Also, mathematical plane waves, whose amplitude is independent of distance only their direction of propagation, can be generated by sources of infinite extent in the plane perpendicular to the direction of propagation. An example of plane waves for which misconceptions abound is the case of the so-called Bessel beam [8, 9, 10]. However, when the source of the waves is localized to a bounded three-dimensional region, there are restrictions on the character of the waves. One aspect of waves from a bounded source that is too often overlooked is that such waves cannot be unipolar [12]. Here, we reconfirm that the amplitude of waves from a bounded source fall off as 1/r as distances large compared to the size of the source. We will make the desired demonstration in the context of scalar diffraction theory, which gives a prescription for calculation of the amplitude of a wave of a pure frequency ω based on knowledge of the amplitude of the wave on a surface that encloses the observation point, provided that there are no charges or currents within the enclosed volume [1]. We suppose that the source of the waves is in the vicinity of the origin of a spherical coordinate system (r, θ, φ), and that the source lies entirely within a sphere of radius r0. We take this sphere (plus the “sphere at infinity”) to be the surface that encloses the point of observation at radius r r0. Since the amplitude of the wave on the “sphere at infinity” A different argument that the amplitude falls off as 1/r for a charge in uniform circular motion with speed v > c/n in a medium of index of refraction n is given on p. 4 of [11].