How do they know about the pith and that it's a wave?
What you're maybe confused about is that textbooks often oversimplify and depict sound as a pure sine wave. That only rarely happens, and is usually man made.
Day-to-day sounds can be approximated as the sum of such perfect sine waves. Fourier transformations are a well known method for that.
That approximation usually isn't perfect. But it's good enough to be usefull.
We record them. as some others have pointed, we measure all kinds of sources, be it a sound wave which is recorded by a mic, where we have some diaphragm, which physically moves, and depending on how the mic works (for example, a mic with a working principle of capacitence would use this motion of diaphragm as one of the capacitors plate, and measure change in capacitence by measuring voltage across the plates)(also, i am not saying this is how all mics work, just something i came up with for a example, but it works in theory).
At a much better level, for complex sources, for example, you gently resting your hand on a table, the sound currents produced are not strong enough, so one could potentially amplify, but doing that also amplifies noise (which means erroneous data here, and not like bad sounds), so we go to a better measurement technique, for example seismography (very complex stuff, which i am not going to go into).
in a lab setup, we may actuallu pick a material, and excite it. Think of like plucking a guitar string, but instead, we use for example a piezoelectric material, and give a input source voltage (which will vary periodically) and use it to excite a wave, and we can, for example optically measure the motion of material (if for example, we start with a block, we can measure it's compression and expansion). though this requires quite high quality cameras, this can be done.
acoustics as a phenomenon originates as large scale effect of molecules vibrating (i will not go into detail about vibrations here, because that is way out of scope), but think of it like this - you may know that electricity, or to be precise current, is just flow of electrons, so if electrons move slightly on application of a electrical voltage, and large scale effect is current, then sounds or all mechanical motions can be roughly considered as large scale motion of these vibrations. at this level, there are not really any optical cameras that can measure, so what we do spectroscopy (fancy words warning - we shine different wavelengths of light, and material absorbs wavelengths corresponding to particular vibration's resonant frequencies)
tl:dr - we record the motion
Virtually everything scientists know about sound began with microphones which output a voltage wave with the same 'pitch' as the sound wave, and the piezoelectric transducer which enabled extremely sensitive measurements. It's a deep rabbit hole I expect if we go into much detail.
It certainly helped when the cathode-ray tube was able to trace the waves visually in real time.
absolute nonsense
In the 6th century BCE Pythagoras accurately described the overtone series. Aristotle understood that sound was a pressure wave in the air. In the 1st century CE roman architects were designing music halls around their acoustics.
The field started being much more active during the Renaissance and onward. Most of the big names in the field of acoustics predate the invention of microphones.
I get that, but OP wants to know HOW they knew it was a wave, and I only have modern examples that show the 'how'. If you can expand on how Pythagoras 'knew' or 'demonstrated' it was a wave, you should add that to your reply lol.
The point of my reply was to debunk what you said. I've not read that text and I am no expert. I can only speculate.
If you know that sound is caused by something vibrating, as was known, and you also know that when something moves through air it pushes the air away, you can reasonably conclude that sound is a vibration in the air caused by a vibration in an object.
But I don't know how he came to that conclusion, or what, of any he ran experiments he ran to show it.
My point is that anyone saying you need a microphone to learn about acoustics are talking nonsense. You have to already understand acoustics to even build the first microphone.
I gotcha. I have a bedroom which resonates at a certain frequency of my voice, and it could be compared to the resonance of a wave when you get the driving frequency just right. I didn't need a microphone to quickly realise this from my knowledge, but the ancients were extremely well-versed or understanding of it without using current modern terms or techniques.
how did Aristotle know it was pressure wave in the air? Did he say any reason for that
At that time it was a hypothesis. But the motion of snares was one of his inspirations.
In any case, it's impossible to "proof" anything in physics, you can only disprove things. If enough people fail to disprove a hypothesis, for a long enough time, a hypothesis is generally accepted as true.
An early experiment that supports the hypothesis, was in the 17th century, when experiments in vacuum became possible. Robert Boyle put a mechanical bell inside a vacuum chamber and, as the bell rang, no sound was transmitted. Highly suggesting that the presence of a medium to transfer the motion is important. (1)
Changing the type or amount of gas changes the pitch, again suggesting the medium is important.