What is the difference between an electromagnetic wave and other waves that amounts to the EM wave following the Special Relativity. I have been reading about the Special Relativity for some time, and the arguments provided for light having a constant speed w.r.t. to all observers in all the inertial frames moving relative to each other seem to work on other waves too in my understanding. What is the point I am missing out on ? What separates the other waves from an EM wave regarding the 2nd postulate of Special Relativity ? If it is the EM nature, please elaborate.
all electromagnetic waves are light and abide by the laws of special relativity. This is because they are governed my maxwell's equations which do not change based on relavistic transforms (Lorentz transforms).
Maxwell's equations also define the wave equation that predicts EM waves and thus the speed of light.
The wave equation that produces sound waves is not lorentz invariant. You can catch up to a sound wave and travel at the same speed of it. This is what jets do when they break the sound barrier. They literally fly next to the sound waves they are generating. Such a thing is not possible with light since, no matter how fast you travel, you will never catch up to a photon.
The same also follows with other waves.
One other thing to consider about electromagnetic waves is that there is no known medium that they travel through except space-time itself. General relativity explains that the effects of both special relativity and gravity is the medium of space-time curving. Light is bound to space-time, traveling in the shortest path around it. again, such is not true for other waves.
That's an excellent question I've thought about myself too. I considered mathematical analogues of Lorentz transformations & Minkowski metric for acoustic waves in an acoustic medium, not as fundamental but as a mathematical model. Turns out others have to. This is usually done for general relativity rather than special relativiy. This is called Analog gravity. It also goes by names like acoustic/sonic metric and phenomena like sonic black holes. e.g. https://en.wikipedia.org/wiki/Acoustic_metric and the links therein or just google those terms.
There are even people doing table top experiments with these acoustic analogs to study phenomena like Hawking radiation which would otherwise be impossible to study directly.
You can flip this on it's head and consider the analog model as more fundamental and consider classical spacetime (& Lorentz invariance) not as fundamental but as emergent (this is done in several quantum gravity theories like String theory, loop quantum gravity, causal dynamical triangulation, etc.) but in this particular approach they follow a condensed matter model analog for spacetime where it is considered to be a flowing superfluid. e.g. see https://en.wikipedia.org/wiki/Superfluid_vacuum_theory
Here is also an interesting paper about special relativity and how it's effects can emerge from classical (non-relativistic) wave mechanics: https://arxiv.org/pdf/1405.3979.pdf. This is in the same spirit as analog gravity/GR models but specifically for SR.
So in summary: the previous answer is correct (as least for our current understanding of spacetime as fundamental) but there is real value in also considering acoustic analogs either as an alternative mathematical description of fluid dynamics/acoustics or even as a possible (speculative) fundamental model of spacetime.