Are mechanical waves regarded as radiation? By definition, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium.
However, in wikipedia, only 4 types of radiation are mentioned: EM wave, particle radiation, acoustic radiation and gravitational radiation.
I am wondering why mechanical waves are included in them?
 A: Mechanical waves (including acoustic waves, which are mechanical waves in a certain frequency range) transmit energy through a material medium, so they fit the general definition of radiation that you quote.
The Wikipedia article says that radiation includes the four examples given, but this is not an exclusive definition.
I can perhaps see an argument for excluding one dimensional waves (vibrating strings, organ pipes) from the definition of radiation, since a one dimensional wave propagates but does not radiate out into two or three dimensions. But it is consistent with the definition of radiation that you quote to classify two or three dimensional mechanical waves such as low frequency surface waves in a liquid and seismic waves as forms of (mechanical) radiation.
A: Indeed the idea of 'acoustic radiation' in the Wikipedia article is rather a stretch.
The word 'acoustics' is used in the context of macroscopic phenomena. By 'macroscopic' I mean that while the properties of the medium arise from emergent properties of the constituent particles together, there is no window onto the properties of the constituent particles. For example, for the propagation of sound in air the speed of the propagation follows from the elasticity of air and the density of air, and it isn't necessary to know whether air consists of molecules.
However, in the case of solids there is, at the level of the constituent atoms of the solid, the phenomenon of phonons.
It is only at the level of atoms that the quantisized nature of phonon states is discernable.
Acoustic waves can be thought of as a superposition of a macroscopic amount of phonons, it's just not practicle. The practical approach is to account for the properties of acoustics in terms of elasticity and density; macroscopic properties.

Still: it can be argued meaningfully that 'acoustic radiation' is a thing, a member of an overarching category of forms of radiation.
