I am not sure whether some of the following is relevant to the sort of questions you are trying to raise. Equally, some more qualified members may wish to correct some of my comments, but for what it's worth….
Are you assuming the wavefront is moving at constant velocity defined by the medium? Would the energy be dissipated over an ever-expanding wavefront defined by an expanding sphere? In many cases, energy sets the frequency [f], but the medium sets the propagating velocity [v] and wavelength = v/f. However, this seems to pre-suppose the existence of some sort of medium. If the wavefront acts more like a blast wave then a taylor-sedov solution is often considered, but have been told it gives a different result for the radius as a function of time in comparison to the Friedmann equation. The Friedmann solution is said not to describe an explosion because it describes a uniformly expanding sphere at any point in time. The uniformly expanding sphere idea seems to be supported by the cosmological principle of homogeneous and isotropic space. Not sure what scope this leaves for what is implied as the `original disturbance`.
With reference to my comments above, observations appear to suggest that the universe is essentially homogeneous and isotopic on the large-scale and expanding. This seems to lead to the qualification that states that the big bang was not an explosion, hence the need for the expanding space idea where two points in space, i.e. distant galaxies, move apart due to space between them expanding. The uniform expansion of each unit volume of space would also seem to explain how superluminal speeds can exist without violating special relativity. On the other hands, much of 20th century science refutes the existence of space as a propagating medium, i.e. no aether, in the conventional sense and therefore light (EM) waves are considered self-propagating in a vacuum. Of course, this may only seems to underline your questions. As far as I know, cosmology cannot directly answer your questions and you may have to turn to quantum or string theory for an answer, at least, in respect to the possibility of the vacuum having some sort of energy structure that might then explain how space expands and where it gets the energy to support this expansion.
However, as indicated at the outset, some more qualified members may be in a position to clear up the confusion I have suggested.