In 1842 Christian Doppler proposed that frequency of waves change for an Observer moving relative to Source. In case of light: f = f’(1+v/c)

f= frequency received by Observer

f’= frequency emitted by Source

v = relative velocity of Source and Observer

Doppler focused on frequency and didn’t take into consideration the cause of it, namely the change of the Time Frames of events relating to Observer moving relative to Source.

Real Doppler Effect depends on 4 factors: 1- Speed of waves mediating events in space; 2- Relative speed of Source in space; 3- Relative speed of Observer in space; 4 The length of Time Frame of the observed event (determines the angle of field Alfa for Observer for trigonometric calculations).

The Time frames of events emitted by Source change for the Observer by the next formula:

T = T’+ t – t’ = T’ + (x – x’)/c

- where T= Perceived Time frame of Observer for received event;

T’= Real Time frame of Source for emitted event

t’ = time needed by the first wave of event to reach observer;

t = time needed by the last wave of event to reach observer;

x and x’ = the distances covered by the last and first waves

D = T/T’ - where D = Real Doppler factor;

Trigonometric form for stationary Observer: D = 1 + Sv/c

-where ’S’ Sine factor = (sinGamma-sinBeta)/sinAlfa

v= relative velocity of Source and Observer; c = velocity of light

Beta = inner angle of movement of Source (- Theta used in classic Doppler effect);

Alfa= angle of event seen by Observer; Gamma=180-Alfa-Beta (see Doppler Effect vs. Special Relativity)

lim S = cosBeta; lim D = 1 + cosBeta v/c

Identities: lim (sinGamma-sinBeta)/sinA = cos Beta = cos(-Teta);

D = factor of Classic Doppler effect when Beta = 180 or 0 degree;

Because T’ cannot be zero the result of Classic Doppler effect is false when Source nears Observer in an angle.

D=Lorentz’s factor when Gamma=90 degree and Observer moves away from stationary Source and T=t;

D = 1/Lorentz’s factor when Beta = 90 degree and Source moves toward stationary Observer, and T ’ = t’

Identity Angle

When cosBeta = v’T’/2tc then Beta = Gamma S=0 and D = 1. At this angle T’=T .

In Real Doppler Effect the Identity Angle depends on Time frame, velocity of waves, velocity of Source and distance, while Classic Doppler Effect fix this angle as 90 degree because cos90 = 0.

The equations of Real Doppler Effect:

T =DT’ - where T= Time Frame of event measured by Observer; T’= Emitted Time Frame of Source

f = f’/D - where f= average frequency measured by Observer; f’=frequency emitted by Source

v = v’/D -where v = average speed of Source measured by Observer

l = Dl’ - where l = average wavelength measured by Observer, l’= wavelength measured by Source

L = L’+ v (t’ - t) - where L= length of Source measured by Observer in the direction of movement

L’= length of Source measured by Source;

t’- t = time difference of waves reaching Observer started from the front and from the back part of Source.

Main Differences to Special Relativity (SR)

Real Doppler Effect transforms events mediated by waves to the perception of Observers and to the realities of Sources, or calculates relative velocities by analysing frequency changes, while Special Relativity transforms the realities of Observers and Sources to the speed of light to keep it constant.

According Special Relativity the maximum reachable relative velocity is the constant speed of light.

Doppler effect doesn’t set a limit, even calculates by higher relative velocities when converting time or frequencies e.g. when observer moves toward Source: D= c/(v+c); T=DT’; f= f’/D = f’(c+v)/c.

Sandor Fofai