Annus MirabilisEinstein in 1905
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special relativity

The equation explained

X=X0sin⁡Φ,L=L0sin⁡Φ,Y=Y0sin⁡Φ,M=M0sin⁡Φ,Φ=ω(t−ax+by+czV).Z=Z0sin⁡Φ,N=N0sin⁡Φ,\begin{aligned} X &= X_0 \sin \Phi , &\quad L &= L_0 \sin \Phi , \\ Y &= Y_0 \sin \Phi , &\quad M &= M_0 \sin \Phi , &\quad \Phi &= \omega \left( t - \frac{a x + b y + c z}{V} \right) . \\ Z &= Z_0 \sin \Phi , &\quad N &= N_0 \sin \Phi , \end{aligned}
In wordsIn the stationary system, each component of the electric force and of the magnetic force is its electric or magnetic amplitude times the sine of one phase, which advances with the time at the angular frequency and falls behind along the coordinates in the direction the wave travels, given by its direction cosines, at the velocity of light.OverviewA plane light wave in the stationary system KK: every component of the electric and magnetic force oscillates with the same phase Φ\Phi, which travels along the wave normal at the velocity of light. This is the wave that § 7 will look at from the moving system.Full explanationThe paper places a source of electrodynamic waves very far from the origin of KK, so that near the origin its waves may be represented, to a sufficient approximation, by plane waves. (X0,Y0,Z0)(X_0, Y_0, Z_0) and (L0,M0,N0)(L_0, M_0, N_0) are the amplitudes of the electric and magnetic force, and aa, bb, cc the direction cosines of the wave normal, the cosines of the angles it makes with the three axes. All six components rise and fall together with sin⁡Φ\sin \Phi. The phase Φ=ω(t−ax+by+czV)\Phi = \omega \left( t - \frac{a x + b y + c z}{V} \right) is the same on every plane ax+by+cz=consta x + b y + c z = \mathrm{const}, and such a plane moves along the normal at the velocity VV. The angular frequency ω\omega says how fast the phase advances in time.Every stepΦ=ω(t−ax+by+czV)\Phi = \omega \left( t - \frac{a x + b y + c z}{V} \right)The phase of the wave. At a fixed place it grows steadily with tt, at the rate ω\omega.ax+by+cz=Vt+consta x + b y + c z = V t + \mathrm{const}Where is the phase the same? Wherever t−ax+by+czVt - \frac{a x + b y + c z}{V} is the same: on a plane perpendicular to the direction (a,b,c)(a, b, c), which moves along that direction a distance VtV t in the time tt. So the wave travels at the velocity of light.X=X0sin⁡Φ,L=L0sin⁡ΦX = X_0 \sin \Phi, \quad L = L_0 \sin \PhiEach component of each force is its amplitude times sin⁡Φ\sin \Phi. The other four components follow the same pattern with their own amplitudes.Also

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