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What we learn about physics can usually be divided into two parts: study of Particle or Wave. We later learn wave-particle duality, etc.

Particle is somewhat intuitive, but what's so special about a wave that makes it fundamental? Why not other things?

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You can see waves as the continuum version of the point-particle harmonic oscillator: imagine a tiny oscillator at each point in space and those oscillators are coupled linearly (see this answer, or this, or even this). Therefore, the wave equation describes the dynamics of small oscillations in an "elastic" medium. This medium is not necessarily a material medium (for example, disturbances in both the electromagnetic and gravitational fields propagate as electromagnetic and gravitational waves, respectively).

The harmonic oscillator is so widespread because it is often a good linear approximation to more complex problems, at least for small amplitudes when you can "linearize" the problem (see e.g. this linearization procedure that gives rise to Hooke's law and, therefore, to harmonic/wave phenomena). The same for waves (i.e. solutions of the wave equation... but be careful that this term is also used in a broader sense, see this answer).

Extra comment: The wave equation is hyperbolic, meaning that they typically allow for finite-velocity propagation of information (a fundamental request in relativity), allowing us to build relativistically causal models. As far as we know, everything at the fundamental level must obey this principle, so this is also why wave-like equations (more generally, hyperbolic equations) pop out so much in field theory.

Closely related posts: Of course, there are also other reasons why the wave equation is so pervasive (this has been already discussed in many posts), for example: Why are oscillations so ubiquitous in nature?, Why is the harmonic oscillator so important?, Why is the wave equation so pervasive?, Question about the Wave equation, Why do waves occur?.

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Although there are exceptions you can think of travelling mechanical waves as a mechanism of transferring energy and momentum from one position to another without the transfer of the medium through which the wave is moving.
From this idea you can build to have waves which do not need a medium (electromagnetic), waves which do not transfer energy and momentum (standing waves), other types of wave (shock waves, etc).

As waves have common properties like velocity, frequency, wavelength, amplitude, etc, it means that analysis of one type of wave lends itself to an easier analysis of other types of wave.

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