This effect can be observed in the propagation of any wave or repetitive signal and is, in fact, quite familiar. For example, the sound of an airplane has higher pitch when the plane is flying toward the person hearing the noise, and has lower pitch when the plane is flying away. The motion of the source toward the receiver results in an increase of reception frequency and, therefore, a decrease in reception period, whereas the motion of the source away from the receiver results in a decrease of reception frequency and, therefore, an increase in reception period. Intuitively, if the emitter is moving away from the receiver (or vice versa), each wave crest must travel a larger distance than the previous one, resulting in an increase in the reception interval of successive crests. In the case of a sound wave, which propagates in a physical medium, such as the air, the amount of frequency modification will depend on the velocities of the source and the receiver with respect to the propagation medium, as well as on the propagation velocity of the signal in the medium. In the case of light waves propagating in vacuum, the size of the effect is determined only by the relative velocity of the receiver and the source.
The argument made above on successive crests of a wave applies equally well to successive light pulses emitted by a first inertial observer and received by a second inertial observer. The ratio between the interval of emission of light pulses by the first observer and the corresponding interval of reception of the pulses by the second observer can be used to characterize quantitatively the motion of one observer with respect to the other. If the observers are moving away from each other, this ratio will be larger than one, increasing to arbitrarily large values as the relative velocity of the observers approaches the speed of light. Correspondingly, if the observers are moving toward each other, the ratio in question will be smaller than one, decreasing to arbitrarily small values as the relative velocity of approach of the observers gets close to the speed of light. Bondi adopted the symbol k to refer to this adimensional quantity and the development of relativistic kinematics based on its use came to be known under the name of k-calculus.
The Doppler ratio is usually expressed in terms of the speed of light and the velocity of the receiver with respect to the source. In an approach in which this ratio, dubbed k, is adopted as basic quantity, expressing it in terms of the relative velocity v requires expressing v in terms of k (and the speed of light) and inverting this relation.
Here is the link to the text “Concepts of Special Relativity” in PDF format. The relation between the Bondi (or Doppler) factor k and the relative velocity v is worked out in Chapter 4.
A numerical example of the relationship between the Bondi factor and the relative velocity can be found in Appendix A.2.
In order to understand clearly the concepts and the nomenclature used, it is recommended to read the first three chapters before studying Chapter 4.
An animation software written in the Java language is available. The relation between the Doppler factor and the relative velocity is illustrated by the third animation proposed.
If you are interested in a particular topic of Special Relativity, here is a list of the topics covered in the text and in the software. Clicking on an item in this list, will open a page which briefly introduces that topic and indicates the parts of the text and of the software in which it is treated.