I am searching for an explanation of how drums produce sound waves. What type are these waves and how do they spread in the air?
Like all sound waves in air, the waves emitted by a vibrating drum head are called compression or longitudinal waves.
As the drum head vibrates upwards, it pushes on the air next to it, which pushes on the air next to it, and so on. then when the drum head vibrates downwards, it pulls on the air next to it, which then pulls on the air next to it, and so on.
The net result is a train of waves moving away from the drum head in a series of zones of slightly compressed air each being followed by a zone of slightly rarified air. these sound waves move away from the drum head in all directions, becoming weaker in strength as they travel away from their source.
This is a simplified picture of the process. The detailed physics of drum head vibration is more complex than this, and much has been written about it.