Is there a specific branch of physics that studies waves? Is there a branch of physics that studies waves and how they propagate through air, wires etc.?
 A: Acoustic physic deals with mechanical waves. But as CuriousOne says, many areas of physics uses waves in some way, so it's hard to pinpoint a "wave-only" physics.
A: Waves are in nearly every part of physics: sound waves, light waves, ocean waves, all of hydrodynamics really, all of magnetohydrodynamics too, stellar pulsations, gravitational waves, alternating current, coupled oscillators of any sort$\ldots$ The list goes on.
One way of seeing that the ubiquity is not surprising is the following: Quantities smoothly changing in space and time are well described by partial differential equations. Most simple physical setups lead to second order linear equations (sometimes there are deep reasons for this). Such equations can be elliptic (like Poisson's equation, leading to boundary-value problems) or hyperbolic. The latter describe time evolution of a system (as opposed to figuring out something's configuration based on its surface properties), and the most common hyperbolic equations are the wave equation and the diffusion equation. So right away if you are trying to see how something evolves in time, there is a good chance something like the wave equation will enter at some level.
For this reason, some physics curricula (including the one I had) have a dedicated course on abstract waves early on, since the techniques and insights learned can be applied to a great diversity of problems.
