The following Galilean transformation equations are presented as a contrast to what have commonly become known as the Lorentz transformation equations.  As spoken to earlier the essential component of the Lorentz equations is more properly identified as a transvaluation.  The ability of the Lorentz equations to equate the principle of relative velocity with the law of the transmission of light, i.e., from one frame of reference to another, is solely based upon the manipulation of time and as such is wholly dependant upon the aforementioned transvaluation factor [- (v/c²) x], introduced into the equations in the form of an inequality.

In contrast thereto, the classical Galilean transformation equations presented herewith, satisfy both the principle of relative velocity and the law of the transmission of light while still maintaining an absolute standard of time.  The effect of the Doppler wave deformation is added in the form of an equality, therewith preserving the integrity of the transformation.  The justification for the addition of the Doppler effect equality is predicated upon understanding that the standard photon wave that K measures differs from the Doppler effected photon wave that K' measures.

The following Galilean transformation equations are presented in terms of the photon wave's linear velocity as well as in terms of its angular velocity.  A photon wave velocity schematic is presented with each set of equations and provides a comparison of the physical characteristics of the standard photon wave with those of the Doppler effected photon wave.

Assume K is stationary relative to a photon source.  Assume K' is traveling at linear velocity v away from that same photon source.  It can then be said that the linear velocity of the photon wave relative to K' will decrease by the magnitude of K' 's velocity v.  This decrease in relative velocity is identified as the principle of relative velocity.  However, concurrently with this decrease in relative velocity, the photon wave will deform relative to K' therewith causing an increase in its linear velocity relative to K'.  This deformation of the photon wave and associated increase in linear velocity relative to K' is identified as the Doppler effect.  The magnitude of each of these effects will be equal to the magnitude of K' 's velocity v.  The net result of these two offsetting phenomena produce a photon wave with a constant linear velocity relative to K' regardless of the magnitude of K' 's velocity.

Conversely if K' is traveling toward the source of a photon wave the principle of relative velocity will cause an increase in the linear velocity of the photon wave relative to K' while the Doppler effect will produce an equal and offsetting decrease in the linear velocity of the photon wave relative to K'.

Since K is stationary relative to the photon source, the photon wave as encountered by K will exhibit neither the effect of the principle of relative velocity nor the effect of the Doppler wave deformation.  Hence the characteristics of the wave encountered by K can be construed as representing a standard wave since the characteristics of the wave remain constant with respect to K.  

Linear Velocity Equations (see figure 6)   

When a Galilean transformation is applied to the co-ordinate systems K and K' in accordance with (1) the principle of relative velocity (2) an absolute standard of time (3) the Doppler (d) photon wave deformation effect and (4) the law of the transmission of light; the linear velocity of a Doppler (d) photon wave relative to K' is described as follows.  Where

x' = x - v t              (1) principle of relative velocity

t' = t                      (2) absolute standard of time

x = c t                   (4) law of transmission of light - standard wave encountered by  K

then since

x' = x - v t                  substituting for x where x = c t, yields

x' = c t - v t                dividing by t' = t, yields

x' = c t' - v t'               adding Doppler effect equality delta x' = v t', yields 

x'+delta x' = c t' – v t' + v t'

or

x'+delta x' = c t'  

From the above Galilean transformation equations and the associated linear velocity schematic it is evident that the Doppler (d) photon wave encountered by K' exhibits a frequency and wave length that sustains the wave's constant linear velocity relative to K'.  That is to say,  

x'+delta x' = c t'

or

(f + delta f) (l) t' + (f + delta f) (delta l) t' = (f)_d (l)_d t'.  

Hence it can be theorized that the Doppler (d) photon wave's linear deformation is equal in direction and magnitude to K' 's relative velocity v.  The resulting Doppler (d) frequency and wave length manifest this linear deformation as a constant linear velocity relative to K'.  As is shown in the following angular equations, the resulting constant linear velocity relative to K' is maintained only at the expense of a variation in the Doppler (d) photon wave's angular velocity.

Angular Velocity Equations (see figure 7)   

When a Galilean transformation is applied to the co-ordinate systems K and K' in accordance with (1) the principle of relative velocity (2) an absolute standard of time (3) the Doppler (d) photon wave deformation effect and (4) the law of the transmission of light; the angular velocity of a Doppler (d) photon wave relative to K' is described as follows.  Where

x'_a =  x_a - 2 v t                (1) principle of relative velocity

t' = t                                (2) absolute standard of time

delta x'_a = v t'                 (3) Doppler (d) photon wave deformation effect 

x_a = c_a t                          (4) law of transmission of light - standard wave encountered by K

x'_a+delta x'_a = (c_a- v) t'  (4) law of transmission of light - Doppler (d) wave encountered by K'

then since

x'_a = x_a - 2 v t                 substituting for x_a where x_a = c_a t, yields

x'_a = c_a t - 2 v t               dividing by t' = t, yields       

x'_a = c_a t' - 2 v t'              adding Doppler effect equality delta x'_a = v t', yields

x'_a+delta x'_a = c_a t' - 2 v t' + v t'              

or

x'_a+delta x'_a = (c_a - v) t'

Consistent with above equations and the associated angular velocity schematic it is evident that the average angular velocity of the Doppler (d) photon wave varies from that of the standard photon wave to the extent of    K' 's relative velocity v, that is to say,  

x'_a+delta x'_a = (c_a - v) t'

or 

(f_a+ delta f_a) ( l_a) t' + (f_a+delta f_a) (delta l_a) t' = (f_a)_d (l_a)_d t'. 

Wherefore it can be theorized that the Doppler (d) photon wave's angular deformation is equal in magnitude to K' 's relative velocity v.  The resulting Doppler (d) frequency and arc length manifest this angular deformation as a change in the wave's angular velocity relative to K'.  

Summary Comment

In summation it can be said that as a consequence an observer's velocity relative to a photon wave's source, the linear dimensions of the photon wave as encountered by the observer elongate (compress) at a rate equivalent to the observer's relative velocity.  The result of this Doppler effect linear deformation of the photon wave is to offset or negate the linear effect of the principle of relative velocity.  However, the principle of relative velocity will not be denied and its effect is ultimately substantiated by the deformation of the photon wave's arc and therewith an associated change in the angular velocity of the wave.  This change in the angular velocity of the Doppler effected photon wave is also equal to the observer's relative velocity and confirms the validity of the principle of relative velocity.  Accordingly the nature of these interacting phenomena is to produce a photon wave that exhibits a constant linear velocity with respect to all observers, and as such, is recognized as the law of the constant propagation velocity of electromagnetic phenomena.