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From what I understand, sound is simply the jostling of the molecules that make up the air in a specific pattern, widely known as waves. I also know that these are longitudinal waves. If we were to look at "heat", it'd simply be the jiggling of molecules that make up the air that are moving in all directions. Mathematically put, the sum of all the vectors in 3D space that would describe each molecule would equal zero.

Now if we are to look at sound waves, we could in principle, describe each molecule with a vector as well. For example: The air that is in the compression portion at a given dt would have a vector that is larger in magnitude than the uncompressed parts. We can also have standing sound waves by reflecting them off the walls of our hypothetical room, let's say is 20 ft^3.

Say we have special speakers on each of the three walls that are orthogonal to each other. What overlap of sound waves from all the speakers would be the best to heat the room? Would I be correct in saying that the arrangement of sound waves would have to make all the molecules in the room move in all directions because that is the concept of what heat is?

Going further with the example, couldn't we overlap sound waves in such a way that a certain part or parts of the room are colder or hotter? Can't we cancel out sound waves in certain places and therefore efficiently control heating? Couldn't this save energy because you wouldn't have to heat the entire room? Preferably, I'm looking for a mathematical answer.

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There are two ways to think about the temperature of air.

One way is to measure the distribution of air molecule velocities, and relate this to a temperature using the Maxwell-Boltzmann distribution. However no (reasonable) amount of messing around with sound waves is going to reproduce a Maxwell-Boltzmann distribution.

The other way is to say air is hot if it heats things it's in contact with. The heating is due to transfer of energy from the air molecules to vibrational modes in the object being heated. If you generate a sound wave then that wave carries energy so it will heat anything that absorbs the sound. If you look at things this way then using sound waves to make the air hot is really easy since any old sound wave does it. All you need is to put enough energy into the sound, i.e. make it loud enough, to transfer the required energy to whatever it is you're heating.

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