So, I was thinking about how light wave something act like particles, and I thought, if light is made of particles, couldn’t a mechanical-like wave travel through it? Is this possible, and in what situation would it happen
I think you are asking if photons can pass energy from one to another the way massive particles do when a sound wave passes, e.g., through a gas.
The answer is "no" in the vacuum and in ordinary media, because photons pass right through each other under those circumstances. However in the case of extremely intense light and in a nonlinear medium, light can interact with light. In that case, probably there could be something analogous to a sound wave that propagates through a "gas" of photons. I haven't seen any theoretical papers addressing the question; it's an interesting idea.
So, I was thinking about how light wave something act like particles,
Light is a superposition of zillions of photons, and photons are elementary particles of zero mass.
and I thought, if light is made of particles, couldn’t a mechanical-like wave travel through it?
There are a number of laser light experiments, where the frequency is fixed, that show interference of light waves. Note that interference is not interaction, just that the amplitude in space of the supposed beams changes , and these changes can be made periodic and be fitted with a different wave pattern.
In this video one can see how the pattern of interference changes, for example. One could use interference to generate wave patterns. In this sense light can be the medium of another waver.
Actually there is something simpler, as we are using radio frequency electromagnetic waves to transfer music, this is also mathematically described by wave solutions.
So the "particles of light" are used to carry other wave solutions.
Is this possible, and in what situation would it happen
Photons do have properties that describe both waves and particles, but in the case of photons, these characteristics usually are for separate situations. Photons do have wave characteristics, usually when they travel. But the particle characteristics of photons are visible when they interact with other particles, usually when they are absorbed by an atom (electron).
Usually the medium that you are talking about is made of particles that have rest mass. Usually these mediums are made up of composite particles, atoms and molecules. We do not know of mediums that are made up of only elementary particles that do not make up composite particles. What you are talking about would be a sea of electrons or a sea of photons. There are no mediums like that, at least not in nature. The only occurrence of these type of elementary particle mediums are:
sea of photons after the big bang
inside a neutron, there is a sea of quarks and gluons. Yes, it is a misconception that neutrons are made of three quarks. Those are what you get if you net out the sea of ever changing quarks and gluons, and the three quarks left are called valence quarks.
inside a neutron star
inside a black hole (where gravity is so strong, that composite particles cannot exist in their normal form because of spaghettification and the crush of gravity)
Now a medium that you are talking about would be made of photons, but those photons do not have rest mass and are traveling at speed c in vacuum when measured locally. How could a wave travel inside a medium that travels at speed c?
The only solution that we know about is GWs. According to QM, gravitons are hypothetical particles, that like photons, do not have rest mass, and travel at speed c.
These GWs are possibly made up of gravitons, so that gravitons are the quanta of GWs. If gravitons are the quanta of GWs, then the GWs themselves are traveling in a medium made up of gravitons.
Please see from wikipedia:
However, if gravitons are the quanta of gravitational waves, then the relation between wavelength and corresponding particle energy is fundamentally different for gravitons than for photons, since the Compton wavelength of the graviton is not equal to the gravitational-wave wavelength. Instead, the lower-bound graviton Compton wavelength is about 9×109 times greater than the gravitational wavelength for the GW170104 event, which was ~ 1,700 km. The report did not elaborate on the source of this ratio. It is possible that gravitons are not the quanta of gravitational waves, or that the two phenomena are related in a different way.