we finally get
This is the general form (2) from which we started, but with constants entirely defined by , \, , , it contains what is usually understood by the principle of Doppler, so far it is true.
If it is possible to neglect ϰ² next to ω², then q = 1 and we very simply obtain:
The condition (1') is in this case:
and with the assumed negligence it is only to the extent necessary to be fulfilled, that the term, which is multiplied in , is of the first order.
If, besides the illuminating surface, the observer is also in motion, such as with the constant velocity ϰ' in a direction given by the direction cosines α', β ', γ', then the displacements u, v, w,, which are only related to a coordinate system X', Y', Z' moving with the observer, i.e., we must replace in (12) or (13) by , by , by .
With those findings we give some applications.
1) Let a plane parallel to the YZ-plane be set in vibrations in accordance with the law
then the motion propagated in positive X-axis is given by: