Abstract concept of wave propagating on a string I'm a real beginner in physics with a really basic doubt about waves.      
Suppose i have a string ( perfect elastic material ) whose left-end i can manipulate ( i can change its heigth ) and whose right-end can either be fixed, loose or actually be non-existant ( the string goes indefinitely towards the right direction ) .     
I'm interested in the effect of manipulating the heigth of the left-end of the string.
Let's consider two cases.
i) The heigth starts at $c_0$ and i indefinitely ( non-stop ) make it grow .
ii) The heigth starts at $c_0$ and i lift it to $c_1 > c_0 $ ,  keeping it constant at $c_1 $afterwards.    
What would be the effect of i) and ii) in the remainder of the string ?
I was thinking that we could consider a wave would be generated, that is,  the action i) or ii)  would be propagated sequentially to all points of the string,  starting at the consecutive point to the right of the left-end, forming a kind of wave.   
My motivation for thinking of it as a wave, is that we can perceive that effect as simply energy traveling through a medium ( the string ) withouth  causing a permanent change in its constituents ( the points of the string ) ... and that is the common feature that all waves we know ( electromagnetic, surface waves, etc ) share.
P.S : Here we can consider no damping ratio at all ( the energy is transfered withouth loss ) .      
But is it really a wave ?  Can we talk about frequency,period, wave-length and velocity of wave in both cases i)  and ii) ?         
In the i) action, i was think of considering , only theoretically,  the period of the generated-wave as infinite and the frequency as 0.  What about the wave-length and the velocity of propagation of  the wave ?   On what parameters would it depend ? Would it depend solely on the properties of the string ( density, tension,etc ) , or would it maybe also depend on the velocity that the i) action is realized ?   
In the ii) action, i was thinkg of considering the period of the generated-wave simply as the ammount of time that it takes for the heigth $c_0$ grow to $c_1$,  and  considering the frequency as the inverse of that .   But again,   What about the wave-length and the velocity of propagation of  the wave ?        
Thanks a lot , and sorry for elaborating the question in a simplistic way ( i don't have enough physic's knowledge to elaborate it better ) .     
P.S : What motivated this question was exploring with the following web app :
http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html 
 A: From Wikipedia: 

In physics, a wave is disturbance or oscillation (of a physical
  quantity), that travels through matter or space, accompanied by a
  transfer of energy. Wave motion transfers energy from one point to
  another, often with no permanent displacement of the particles of the
  medium—that is, with little or no associated mass transport. They
  consist, instead, of oscillations or vibrations around almost fixed
  locations. Waves are described by a wave equation which sets out how
  the disturbance proceeds over time.

If we use that definition, in both cases you will have a wave, because there is a perturbation that moves and transfers energy without mass transport. But is not a sinusoidal wave, that is one that has a sine shape with endless peaks and valleys. However, Fourier's theorem shows that any shape can be decomposed into a collection of sinusoidal waves of different frequencies. What this means is that in your example the wave has multiple frequencies (an infinite number to be more precise) or wavelengths. 
