@Willie Wong, so that animation (I though could help) finally confuse it all.. To understand what I want to ask, perhaps we shouldn't look at the red dot, I want to know the up-down axis speed of the wave, watching at the left side of the animation, you can see "first" left most pixel going up and down, this is the vertical speed I want to really calculate, and this speed is not zero even for a simple monochromatic traveling wave, thanks Willie

you said k, don't mind, but this is the result I get, A (k-1) w cos(t w-k x) so, if k=1, then rise speed = 0 ? mmmm something is wrong

This is the way in wolfram alpha to do that wD(ASin(kx - wt),x)+D(ASin(kx - w*t),t)