When a tuning fork is struck, how does the struck tine induce vibrations in the secondary tine? Background
I'm currently writing a book on music theory and I'd like to include some background information on the physics of sound waves. My example of choice is to explain compression and rarefaction using a tuning fork example (seeing as this is a simple device that musicians will be familiar with). I understand the vibrational motion of the tines once the tuning fork has been struck however I'm confused with how it starts.
Question
When the first tine is struck against an object (table) it would move towards the second tine. My assumption is that this would increase the pressure in-between the tines and move the second tine outwards but this would mean the tines are moving in phase. So how do the tines end up moving together and apart (out of phase)?

My second thought is that perhaps the vibrations are induced by the wave moving up and down the stem   and not the air particles... however my ability in physics runs out after this thought!
I would really appreciate any diagrams to help my understanding too if possible
 A: The exact mechanics of how the tuning fork vibrates is complicated*-however once set vibrating , the equilibrium motion is easy to understand-the to and fro movement of tuning fork prongs.
Your emphasis is on explaining compression and rarefaction-for this how the tuning fork reaches its equilibrium vibrational motion isn't important. Infact you don't even need the second prong(buzz of fly wings for example). You are right in thinking that the second prong, the one not struck, is not set in motion by the interveining air. In fact the prong would vibrate in pretty much the same way in vacuum too.
Once the prong starts vibrating at a fixed frequency, it moves rapidly towards and away from its nearby air molecules. Air is a fluid and compressible. So the rapid movement compresses and "stretches" the nearby air volume. These generates local pressure variations which are what we call rarefactions and compressions. Its the pressure variation which travels away from the fork towards the listener. 
Here are some generic images illustrate the point:  
fig 1.Two vibration modes

fig 2. Compressions and rarefactions


*A fork is a elastic body. An absolutely rigid body comprises of constituent atoms/molecules that absolutely don't budge from their initial positions. A very nearly rigid body or elastic is the one which can be slightly deformed under applied stress(~force). This is characterized by the moduli of elasticity of these bodies. Things which are nearly but not perfectly rigid are capable of vibrating. When struck, the atoms/molecules are set abuzz gyrating slightly about their equilibrium positions. 
Macroscopically, we see this as propagation of sound through the material. In fact the whole body is set vibrating. The exact vibration state is determined by how much of each normal mode of the body the initial impulse "awakens". For each body, there exist some natural ways in which it can vibrate. These are called normal modes. They depend on object geometry. Each normal mode has a characteristic frequency and motion. When struck, objects vibrate in a superposition of these normal modes so the motion can be quite complicated and the sound produced multi-tonal.
Fortunately for high Q systems like tuning forks, when struck, the fork settles into one particular frequency-the normal mode its designed for.
Further Edits
$1$
On how the second prong is set into motion after the first prong is struck
The short answer is that its connected to the first.
This question is in general about how motion spreads to other parts of a body when impulse is applied to only a particular part.
Elastic solids like a tuning fork, comprise of atoms (or molecules) in a certain structure. Most solid materials we encounter in daily life have some sort of equilibrium arrangement of atoms. This arrangement is responsible for stability of 
matter. For a metallic tuning fork, its a metal lattice. 
The constituents are held in equilibrium positions by inter atomic forces. At low temps**, the atoms can be approximated as staying in place.
When an impulse is delivered to any point on such an object, the atoms at the place of impact get displaced from their equilibrium positions. The displaced atoms being linked to other atoms via inter atomic forces, push on them in turn. This sets of a chain reaction throughout anything in contact with the point of impact: 
the solid itself, the air around it too.So the vibrations from one tong of the struck tuning fork travel to the other part via its bulk and set it in motion.
This displacement wave which is set off in the material is the sound that travels through it. This sound reflects off of the object's boundary and traverses across it again and again and so on. In this transient phase, the exact motion is, at least in theory, just the evolution of certain superpositioned normal modes.  By design, in case of a tuning fork, eventually the body is left vibrating in just one dominant mode while the energy in most other modes has decayed away faster.
This treatment of analysing transient motion in terms of normal modes(eigenmodes) of interfering sound waves has the benefit that one doesn't have to track  down the individual motion of typically $10^{23}$ atoms to make predictions about sound frequency. Only geometry is needed.

**compared to melting point
A: The tuning fork has a number of eigenmodes, elementary ways it can stably oscillate. The lowest frequency and energy eigenmode (the fundamental mode) is the one where both tines are moving in opposite directions, with the biggest displacement at their ends and nearly no displacement at the forking point. When struck, the vibration will first be just across one tine. This is not itself an eigenmode and it will tend to transfer energy to other eigenmodes - in particular, the low frequency eigenmode with both tines moving. The transfer presumably happens because the vibrating tine will cause pushing and pulling on the forking point that stimulates the oscillation in the other tine.
See also the images at this blog post (dealing with the other odd issue of why the frequency doubles when you press down the fork on a hard surface).
A: I think you may envision this, in a simplified way, as a system of two masses $m$ (the tins) connected to a bigger mass $M$ (the fork handle) with springs. Imparting an initial force to either of the $m$ will utimately set the whole system in motion according to one of its eigenmodes. During the transient, movement is gradually transferred to the second mass $m$ via the movement of the bigger mass $M$ and the generated forces in the springs. Of course in the real case, mass are ditributed and springs are "flexural " springs, and what is transferred are bending moments.
