Either way speed of sound or speed of light. It takes time for things to take effect.

So my reasoning is the following. A light beam travels a distance x in one frame, this is an event that occurs in space-time. The equation in one frame is x-ct=0. In the other frame x'-ct'=0. The only way the same event can satisfy these two equation is when (x-ct)=k(x'-ct'). This is because at t=t'=0, and x=x'=0 (when the light beam is emitted) is also an event that has to satisfy the relationship. So the disappearance of the x-ct coincides with the disappearance of x'-ct'. The two systems are coupled.

I recall in Einsteins original 1905 paper on special relativity, he derived things a bit differently. To be honest, his derivations were a bit confusing and they require a lot of thinking to understand. There are much more easier methods that you can use to build the Lorentz Transformations. (e.g. Griffiths Intro to Electrodynamics has a good derivation of the transformations).

I did some research and I found this. Is this what you are refering to?en.wikipedia.org/wiki/…

If you could please refer me to the paper, I may be able to assist you further.