Before the XIX century acoustics were based on the precepts of Anicius Manlius Severinus Boethius. He was a Roman scholar, Christian philosopher and politician of the V-VI century aC. His works were of reference in Europe throughout the Middle Ages until the XVIII century. One of his books, De Institutione Musica, translated as The Fundamentals of Music, was the book for all matters concerning music. In this book Boethius compares sound propagation to the propagation of water waves. Nowadays Physics indeed encompass both phenomena under mechanical wave propagation.
The propagation of mechanical waves offers a new explanation of how musical instruments work completely different from those offered by the traditional musical acoustics.
How do musical instruments produce sounds? All musical instruments have a part that vibrates. That part can be strings, reeds, membranes and more other things, like the lips of a trumpet player, and in generic it is called "vibrating corpus". The vibration of the vibrating corpus produces mechanical waves in the substance of the instrument. While propagating in the substance of the instrument those mechanical waves will encounter lots of times the surfaces of the instrument, roughly said the end of the instrument and the beginning of the air around and inside it, which is a change of density from a solid material into gas.
One part of those waves will be reflected on those surfaces, remaining in the substance of the instrument, and another part will get refracted through those surfaces into the air, becoming sound.
Thanks to their cavities (resonance boxes and the inside of pipes) musical instruments amplify the sound they produce, making it louder. This amplification occurs as follows: the air within the cavities gets from every refracting point the same waves in the same frequency. So many waves coming from everywhere to the same and small volume of air result in the superposition of all those waves, getting amplified. That means that while keeping their frequency, their peaks become higher and their troughs become deeper, being this translated as a louder sound. And these amplified waves go out of the cavity through the holes of the resonance box of stringed instruments, and through the bottom and lateral holes of wind instruments.
In every instrument the sound production process is exactly the same, depending the timbre of each instrument on the type, material and shape of both vibrating corpus and the whole instrument.
Sound is the translation that our brain does of the information perceived by our eardrums, the sensitive membrane of the aural system. What our eardrums perceive are mechanical waves. Normally we perceive these waves when they do propagate through the air. But these waves can sometimes be perceived while they propagate through solids, for instance when you listen to the tone of a tuning fork and as it vibrates you stick it to your ear, or when a doctor auscultate you with a stethoscope; or through liquids, as whales do in the ocean...
But to make it now easier I will call sound mechanical waves propagating through the air, which is the case if you want to listen to music in a concert hall or in your living room.