Acoustic phonetics is a subfield of phonetics which deals with acoustic aspects of speech sounds. Acoustic phonetics investigates properties like the mean squared amplitude of a waveform, its duration, its fundamental frequency, or other properties of its frequency spectrum, and the relationship of these properties to other branches of phonetics (e.g. articulatory or auditory phonetics), and to abstract linguistic concepts like phones, phrases, or utterances.
The study of acoustic phonetics was greatly enhanced in the late 19th century by the invention of the Edison phonograph. The phonograph allowed the speech signal to be recorded and then later processed and analyzed. By replaying the same speech signal from the phonograph several times, filtering it each time with a different band-pass filter, a spectrogram of the speech utterance could be built up. A series of papers by Ludimar Hermann published in Pflügers Archiv in the last two decades of the 19th century investigated the spectral properties of vowels and consonants using the Edison phonograph, and it was in these papers that the term formant was first introduced. Hermann also played back vowel recordings made with the Edison phonograph at different speeds to distinguish between Willis' and Wheatstone's theories of vowel production.
Further advances in acoustic phonetics were made possible by the development of the telephone industry. (Incidentally, Alexander Graham Bell's father, Alexander Melville Bell, was a phonetician.) During World War II, work at the Bell Telephone Laboratories (which invented the spectrograph) greatly facilitated the systematic study of the spectral properties of periodic and aperiodic speech sounds, vocal tract resonances and vowel formants, voice quality, prosody, etc.
On a theoretical level, speech acoustics can be modeled in a way analogous to electrical circuits. Lord Rayleigh was among the first to recognize that the new electric theory could be used in acoustics, but it was not until 1941 that the circuit model was effectively used, in a book by Chiba and Kajiyama called "The Vowel: Its Nature and Structure". ( This book by Japanese authors working in Japan was published in English at the height of World War II.) In 1952, Roman Jakobson, Gunnar Fant, and Morris Halle wrote "Preliminaries to Speech Analysis", a seminal work tying acoustic phonetics and phonological theory together. This little book was followed in 1960 by Fant "Acoustic Theory of Speech Production", which has remained the major theoretical foundation for speech acoustic research in both the academy and industry. (Fant was himself very involved in the telephone industry.) Other important framers of the field include Kenneth N. Stevens, Osamu Fujimura, and Peter Ladefoged.