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You're confusing completely different types of noise. Palm is correct in that the only unifying characteristic between them is that they are generally unwanted sounds.
White noise (and its derivative, pink noise) is one special case that has a few creative uses. It has completely random amplitude, giving it equal energy at every frequency, much like white light. It therefore has no fundamental frequency and thus no pitch, so logically the pitch cannot be changed simply by resampling. However, you can give it a pitch by filtering it, same way you can give white light a colour by shining it through tinted glass. You can "change" the pitch by varying the filter characteristics.
White noise is electrical noise that comes from a combination of shot noise and thermal noise in an electric circuit. For a recording engineer, this is always undesirable as it can interfere with recordings from low-amplitude devices like microphones, and is difficult to filter out (it's random!). Most of the time it sounds like hiss because the low and ultra-high frequencies don't transmit as easily.
You ask "is it a sound wave" - it actually isn't. It's an electrical signal and only becomes a sound wave once it gets amplified and pumped out a speaker. The sound does not exist in nature. Yeah yeah, it kind of sounds like rushing water, but it isn't.
There's also ground noise (hum) in AC circuits. If the current draw is consistent then you often can filter this. It's always at the line frequency which is at 50/60 Hz.
Air conditioners, heaters, fans, and so on, are producing mechanical noise, not electrical noise. It's not random, it's periodic, proportional to the speed of the motor. It's also affected by vibrations with the container or surrounding structure, which gives the noise low-frequency oscillations that sound like "beats" or "swells". That's what makes them so headache-inducing and difficult to ignore. I've never heard of anyone using this kind of noise in a creative sense, especially not in music production. Aside from the unifying characteristic of being an unwanted sound, it has no relationship to white noise.
I'm glad for your sake that you kept your mouth shut in class, because if your teacher was talking about air conditioners and you'd piped up with this discussion of white noise in sound synthesis, you'd have sounded pretty pompous and stupid.
Hope that helps.
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