How do particles in a wave make the next particles move? I only know few things about waves, but this question just popped up in my mind.
I already know that particles transfer energy to each other,but I do not know HOW they do it!
The truth is that I really want an explanation using energy transfer and an explanation using the forces that one particle applies to the one next to it.
I am talking about an analysis for each type of wave. Longitudinal,transverse and surface waves.  Thank you in advance :)
 A: A rope is the best example. 
Think of a rope as a chain of particles attached to each other.


*

*You now grab the particle in the end and lift it up. Your hand applies the force that causes the acceleration of the particle.

*As that particle is starting to move up, it will pull the next particle beside it by exerting the same force as the hand applies.

*This second particle replies by starting to accelerate, and as it starts moving, it will pull the third particle along.

*And so on 


Whatever you exert of force on the end particle will therefore be sent all the way through to each particle in turn. 
When your hand reaches the top of its movement (the amplitude of the wave), it will stop moving and thus it will stop the motion of the first particle also. It now exerts a force downwards, so the first particle will stop and starts a downwards acceleration. 
As before these same forces are sent all the way through all particles in turn. 

This is for a transverse wave. For a longitudinal wave, the same reaction chain of forces will happen, but it would rather be a compression force.
During an earthquake, the inner particles close to the point of expansion are being compressed. Think of a bath of elastic balls.


*

*If you push so fast on the first ball, that it will not have time to move or anything, it will get squeezed a bit.

*this first ball will now try to "unsqueeze" and return to original size. Therefor is will push in the next ball.

*this second ball now gets a bit squeezed. In the same way, this ball will try to return to the relaxed state and release it's stored elastic energy. To do that it must push on its neighbour, the third ball in the row. 

*and so on. 


I hope this makes it clearer. 
A: Electrostatic forces, a.k.a attractive forces,
between particles.
(for a transverse wave) On a micro scale, when the wave is initiated, a particle is displaced perpendicular to its neighbouring particle. Because the displaced particle and the neighbouring particle are electrostatically attracted to each other, the neighbouring particle is pulled perpendicular to the next particle along.
The original particle is moved the opposite direction, and so the neighbouring particle is pulled in this direction.
You were correct that the medium in which the wave propagates is irrelevant/unnecessary to give an answer.
