The formation of the wave theory of sound

Looking at archeological findings we assume that, since the Babylonians and early ancient Greek philosophers, there have been systematic experiments with the sound of vibrating strings in monochords and excited air columns in flutes. At the same time, many ancient theaters have also been built with outstanding acoustic properties all over the Greek Empire, some of them still in use today. Was all of this just a result of countless iterations using simple trial and error experiments with sound over centuries or were those advanced musical instruments and buildings based on a practically usable theory of sound? Was there already a link between the vibration of strings and oscillations of air, analogies with observations of waves on the water surface or explanations of sound production and perception, when theaters such as the one in Epidaurus were built?