Why wave pattern travels forward? Suppose I oscillate one end of a taut string sinosuidally the particle in contact with my hand performs shm but  consequently I see a pattern emerging out from my end and travelling forward with a velocity called wave velocity. Why the wave pattern moves forward? What imparted that wave velocity and over what it is imparted?
 A: Uh? The energy spreads to nearby parts of the string because they are connected in such a way that facilitates energy sharing; when this part of the string moves then nearby parts also have to move.
This does not have a preferred directionality, as you can see if you fix both sides of the string and wiggle the middle; the waves do not only go to one side. 
If you are asking how this emerges in the physics, briefly: we assume that the string is not transporting its energy longitudinally through its length but only transversely. This means that if we graph the string in 2d as a time-parametrized family of curves $y(x,t)$ then $\partial y/\partial x$ is constant the contact force is necessarily longitudinal and not what we are looking for; so our desired force is proportional to $\partial^2y/\partial x^2$ and then Newton's laws take on the form $${\partial^2y\over\partial t^2}=c^2{\partial^2y\over\partial x^2},$$for some constant $c^2$ that can be seen to be positive in some limiting cases.
The above can then be seen to be satisfied by any combination $$y(x,t)=f_+(t-x/c) + f_-(t+x/c)$$ for any two single variable functions $f_{{+},{-}}$ describing a forward and backward traveling wave profile. So your oscillation is free to describe any combination; you effectively just cannot see it because the $f_+$ that you would be inducing to exist is being carried away to some $x > L$ where you cannot see it; you only see the $f_-$ that is traveling towards this fixed point at $x=0$.
But, that does not mean that you never see anything moving forward. Fixing a specific end like $y(0,t)=0$ then forces a coordination between these two functions as $f_-(t) = -f_+(t),$ so that, from that fixed point’s perspective, incoming signals are inverted as they are returned back. So whatever you send at such a fixed point should be returned back to you inverted after a time $2L/c.$ And if you don’t absorb it, if you hold your hand to be a fixed point $y=0$ again, then it will reflect off your stationary hand and travel back to the origin twice-inverted (so not inverted at all) and then return to you at time $4L/c$ after you initially caused the disturbance, inverted again. And this could go on and on and on.
One can then use a variant of what we call a Green's function analysis (recognize that the operator which takes $g$ to $f_-$ should be a linear operator in $g$, consider a $g$ which is a sudden, very short time impulse, see what response that has in $f_-$, then use a superposition of such simple $g$-functions to describe an arbitrary $g$-function) to just guess that if you are forcing $y(L, t) = g(t)$ for some forcing function $g$, then probably the answer is $$\begin{align}f_-(t) &= g\left(t-{L\over c}\right) +g\left(t-{3L\over c}\right) +g\left(t-{5L\over c}\right) +\dots\\f_+(t) &= -f_-(t)\end{align}$$for which we find that $$y(L, t) = 
g\left((t + L/c)-{L\over c}\right) - g\left((t - L/c)-{L\over c}\right) + g\left((t + L/c)-{3L\over c}\right) - \dots$$
in which we can see that indeed, there is a telescoping series pattern where the first term from $f_+$ cancels the second term from $f_-$ and the second term from $f_+$ will cancel the third term from $f_-$ and so on, forcing $y(L, t) = g(t) + 0 + 0 + 0 + \dots$ as desired.
I think this is technically not the only possible solution but due to the power of Green’s function analysis, it is the right one. (So for an example of another solution that looks like it probably works, it looks like you might be able to make a similar cancellation with $$u(t) = g(t + L/c) + g(t + 3L/c) + g(t + 5L/c) + \dots,$$ with the string somehow seeing into the future all of the things which you might later do and proactively cancelling them out. In Green’s function analysis we would say that our assumption about the boundary conditions with the string being at rest before we interfered with our $g(t)$, this forced us to choose a “delayed” (or in the majority of literature, “retarded”) Green’s function that looks into the past, rather than an “advanced” one that looks into the future or a hybrid of the two that looks into both.)
