Why are waves formed in a string? Perhaps this is a very naive question. If that is the case, I apologise.(I would still like to hear your answer ).
Assume that I took the end of a very long string and moved it up and down. A number of waves are generated.
My question is, why are they formed? I know that this is a way of propagating energy. But, why does it do that in this manner?
I mean, why is it not like a standing wave? ie. The disturbance causes the particles at the end to oscillate up and down and further disturbances solely affect these oscillations. (The energy I give by moving the end of the string is responsible for these oscillations ).  Why do these disturbances travel in a particular direction?
Another question is,  when I hit a drum using a stick, that sound is heard by a person standing far behind. Why? I mean, if the diaphragm was situated in the vertical direction perpendicular the ground , shouldn't only the region of air situated infront of the air undergo compression and rarefaction and consequently, shouldn't only a person standing infront of the drum hear that sound?
 A: I'll start with your main question about how waves work.
Let's imagine that the rope is not continuous, but instead consists of little lumps of mass that are all connected together in a line with little springs. You have one end of this lumpy "rope" in your hand and you suddenly pull it upwards; what then happens?
As your hand moves upward, the spring closest to your hand stretches because the lump of mass next door to it resists movement, but as that mass lump is gradually set into motion upwards, it then begins stretching the next spring down the line, which then starts to pull on the next mass lump, which then starts stretching the next spring, etc., etc. and the disturbance of stretched springs and moving masses propagates away from your hand. This is a WAVE and its speed depends on how stiff those springs are and how heavy the mass lumps are.
Now we slice the springs into tinier pieces and cut the masses into smaller bits and hook them all together and continue this process until we get a springy, continuous rope with its mass smoothly distributed along its length and BINGO! you now have a real rope which supports wave propagation!
Regarding the vibrating drum head: Each tiny portion of the moving drum head (which has springiness and mass!) is connected to a little slab of air right next to it, and each little slab of air is then connected to another little slab of air right next to it- just like the rope example, except now we are in THREE dimensions. Each of those little slabs has springiness (air is compressible) and it has mass too, and so once again you get wave motion being coupled from the 2-dimensions of the drum head to 3-dimensional space. In this case, the waves are compressive and they are free to spread sideways because there are no walls or edges to get in the way.
So wave motion is all about connecting little springs and masses into a continuous network in one, two, or three dimensions, and then shaking one end of it.
