Vladimir F. Tamari

A simple type of waveform is described. Possessing both a forward and radial momentum, the wave resembles the bow wave of a boat. It will be shown that geometrical rays, Huygens' wavelets and the Fresnel-Kirchhoff wave are all special cases of bow waves, depending on the ratio P of the forward and radial velocities.


What is the geometrical shape of a photon, regarded as a wave? It is neither a spherical wave - a Huygens wavelet - emitted by the atom, nor the Fresnel-Kirchhoff wavelet with the inclination factor, since neither wave can be used as an exact solution, for example in describing diffraction phenomena.Towards finding a possible answer to this question, a type of wave is proposed as follows: The atom ejects particle-like energy from the origin along the z-axis, with a forward velocity Z, and pulsating sinusoidally with a frequency f. At the same instant the particles start expanding radially as a spherical wave with a radial velocity R. The photon continues to be formed by such pulses, and after time intervals of 1/f, co-phasal wave shells are created and expand in space, even after the original atomic pulse is finished. Depending on the ratio P=Z/R various types of waves can be formed, as in Fig. 1. To conform to reality, however, it is necessary to assume that Z=S-R=C, the speed of light. The formation of these waves can be demonstrated by dropping pebbles at regular intervals down from a bridge into a moving stream. The expanding circular ripples keep their circular shape, but move forward with the stream's velocity. The bow of a moving boat also creates a pattern similar to P > 1 in Fig. 1. When Z= 0, P= 0 and the Huygens spherical wave is created. The ray of geometrical optics is created when bow waves posses no radial velocity and P= ∞. When 0 < P <1 the bow wave pattern greatly resembles the graph of (1+ cos θ)/2, the Kirchhoff inclination factor. Can the right choice of P give us an exact solution for the photon-wave ? This speculative paper will not attempt to answer this question, but some qualitative implications of bow wave formation will be examined below.

Figure 1


Along the z-axis, bow waves travel at a velocity C, the speed of light, and at 1/f time intervals, the co-phasal shells will be separated by the wavelength λ , as in Fig. 2. However, it was found that measured along a line inclined θ to the z-axis, the bow-wave geometry creates a phase shift when the field is measured along a line such as OQ2 giving :

Can such phase shifts provide an alternative explanation to the 'Wolf effect'.1 Similar phase shifts are also theorized in the author's Streamline Diffraction Theory 2 near the diffracting edge. Another aspect of bow waves is that the Z and R vectors add. up to create a backwards traveling wave, such as at points Q3 and S3 of Fig. 2. Presumably such waves are not observed because they are absorbed and possibly re-emitted by surrounding atoms?

Figure 2


A new type of wave has been described that might provide a useful model of a photon emission. The search for the type of bow wave that fits observed physical phenomena such as the diffracted field depends on finding the correct value of P, and modeling the resultant computed field.


1. Emil Wolf, in a paper that appeared in Physical Re view Letters, Nov. 13, 1989..

2.Vladimir Tamari, "The Cancellation of Diffraction in Wave Fields", Optoelectronics Vol.2, No.1, June 1987. Mita Press, Tokyo.


*This paper has been sent as a late and unofficial contribution to the Conference "Huygens' Principle 1690-1990: Theory and Applications" Scheveningen, The Hague, The Netherlands. November 1990. Congress Bureau: University of Twente, P.O.Box 217, 7500 AE Enschede, Professor Van Jroesen, The Netherlands.

+Note June 2003: the qualitative speculations in this paper lead to the author's 1993 paper United Dipole Field which was published in 2003 as:,in which this present paper was mentioned as a reference.