Who discovered the sound wave and when was that?

The nature of sound has been a subject of study since ancient times. As early as 500 B.C. it was well known that sound is produced by vibrating objects. For example the Greek Philosopher and mathematician conducted experiments around 500 B.C. conducted experiments on sound produced by vibrating strings. Around 350 B.C. Aristotle stated that the sound is carried to our ears by movement of air. But the realization that such movement was in the form of waves was not there.

The understanding that sound is the result of waves created in air was first developed by Leonardo da Vinci arounf 1500 A.D. This concept was then developed by other people. Around 1600 Galileo demonstrated that pitch of sound depends on its frequency. However even then the nature of sound as form of waves was not established beyond doubt. This realization came slowly with contribution of many different scientists. For example, around 1640 Marin Mersenne, a French mathematician, measured speed of sound in air. Around 20 years later Robert Boyle established that sound cannot travel without a medium, which is an essential characteristics of any kind of wave. Later Sir Issac Newton established the correct relationship between speed of sound in a medium and the density and compressibility of the medium. Thus we can say that many people contributed to developing the understanding the nature of sound as waves.

Leonardo DaVinci in (1500) discovered that sound travels in waves. Galileo, an Italian astronomer and physicist, was the greatest contributor to our understanding of sound. He demonstrated that the frequency of sound waves determined the pitch.

Marin Mersenne was the first to measure the speed of sound in air (1640). Robert Boyle discovered that sound waves must travel in a medium (1660) and this lead to the concept that sound is a pressure change. Newton was the first to formulate a relationship between the speed of sound in gases by relating the density and compressibility in a medium.

Pythagoras, the Greek philosopher and mathematician could be acredited to be the earliest researcher  particularlarly on the vibrating aspects of the sound around 500 B.C. Further Archytas of Grece made studies on sound related with its pitch. Even the famous pholosopher, Aristotle made suggestions that sound makes movements  and air carries it.But it was Galeleo , the father of accoustics,who reveals to us that the characteristic that the frequency of the sound wave is the determinig factor of the pitch of the sound.Thus analogy of the wave and oscillatory aspects are a concurrent developments along with musical science along the history.

Pythagoras, ancient greek  philosopher and mathematician, studied the strings' vibrations, somewhere around 500 B.C.

Another greek philosopher, Aristotle, made an assumption about sounds which are moving along with air moving, reaching our ears.

In 1600's, Galileo has demonstrated that the pitch is determined by the sound waves frequency.

Again, an important discovery linked to sound waves is the phonograph, invented by Thomas Edison, in 1877. 

Ironically, the sound wave was discovered back as far as Leonardo da Vinci’s  (1452–1519) era.  He discovered that sound travels in waves.  He was a brilliant scientist, artist, and thinker and contributed many discoveries in the field of science.  He was a man before his time who thought up inventions and ideas that came to be built far later by other engineers and scientists.

Christian Doppler later discovered that sound waves could be compressed or expanded, thereby, further expanding on da Vinci’s theory.

Sound waves have been rethought and the theory re-analyzed by many scientists throughout the years. 

What an interesting question! In my opinion, it is difficult to say that someone "discovered" something such as sound waves, which are a natural thing that occur in nature, but I digress. 

To answer your question, it is believed that Galileo was perhaps the first one to have dealings with sound waves; he was sort of the one who started the excitement of sound waves. 

Later Leonardo Da Vinci also did quite a bit of work with sound waves, as he discovered that sound travels in waves. 

Hope this helps!

Sound waves were later farther develloped with the invention of the radio.