Electromagnetic Radiation
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1 Ordinary cycloidal motion
  1 Definitions
  2 Kinematics of ordinary cycloidal motion
  3 Dynamics of ordinary cycloidal motion
  4 Relations among parameters
2 Electromagnetic radiation
  1 The elementary constituent of electromagnetic radiation (photon)
  2 The motion of the photon
  3 Perceptibility of the photon's electric and magnetic fields
  4 Construction of electromagnetic radiation
    1 Many points on a circumference that rolls
    2 A train of circumferences that roll
    3 Alternate trains of circumferences that roll
    4 Transition to three-dimensional space: the plane wave
  5 Emission of electromagnetic radiation
  6 Generalization: the source also emits photons in non-ordinary cycloidal motion
    1 The dynamics of non-ordinary cycloids of the straight line in the plane
    2 Three-dimensional cycloidal trajectories
  7 Overlapping waves of all lengths and velocities
  8 The observer in motion sees the photons with non-ordinary cycloidal motion
  9 Definitive generalization
3 Sorgente o/e osservatore in moto
  1 Motion of the observer relative to the source at rest
    1 Observer approaching the source along the fixed line that joins them
    2 Observer moving away from the source along the fixed line
    3 Observer in motion in any direction
  2 Motion of the source relative to the observer at rest
    1 Source approaching the observer along the fixed line
    2 Source moving away from the observer along the fixed line
    3 Motion of the source in any direction
  3 Integral motion of source and observer along the direction of the emission
    1 Motion in the same sense of propagation as the radiation
    2 Motion in the sense opposite that of the propagation of the radiation
    3 Motion of the frame in any direction
  4 Conclusions
Appendix
I Interferenza
II Linear relation between the photon's energy and frequency
III Reflection of the photon
IV The transmission of the wave through matter ("transparency")
V Polarization of the wave emitted
VI VI - The minimum ("weak") emission: the "photon"

1 - Definitions


Ordinary cycloidal motion is defined as the motion of any point of a circumference that rolls with constant velocity on a straight line of the plane. The geometrical trajectory described by the point in the course of a single rolling is the ordinary cycloid.

It goes without saying that the motion resulting from several rollings is equipped with periodicity, and implies a translation of the point along an overall direction, which is that of the straight line on which the circumference rolls.



A cycle, both in the spatial and the temporal sense, is the single cycloidal "jump," from one extreme to the other.

More precisely, a period is the duration of the cycle, and the length of the "jump" (d) is the distance covered in each cycle between two successive extremes, which we shall call cuspidal points, or cusps.

Frequency ( f ) is the number of cycles performed in the unit of time. The greater the velocity of rotation of the circumference, i.e., of rolling, the greater is the frequency.

The overall direction of propagation is that of the line r on which the cuspidal points lie (or on a line parallel to it).

We also define the instantaneous velocity () of a generic point of the trajectory, in both the modular and the vectorial sense, represented by an arrow, tangent in that point to the trajectory, whose length indicates the modulus, and whose orientation indicates the direction and the sense.

The useful instantaneous velocity () is the projection of onto the overall direction of propagation; i.e., the component of that is useful for producing advancement in that direction.

The mean useful velocity in the direction of propagation () is the mean of the useful instantaneous velocities, calculated on the basis of a whole or semi-whole number of cycles; it is equal to the "length of the jump" divided by the period ().

The tangential acceleration (a) is the variation of the modulus of the instantaneous velocity, i.e., its derivative, represented by an arrow tangent to the trajectory, whose length gives the modulus and whose direction, coinciding point by point with that of the instantaneous velocity, qualifies it also as vector.

Initial acceleration () is the tangential acceleration at the instant in which it begins a cycle, equal, in modulus, to the tangential deceleration with which the previous cycle terminates, registered at the same cuspidal point and at the same instant.