# How to calculate the Rise speed of a travelling wave?

Perhaps it's a newbie question.., as you can see in this wikipedia animation

Phase velocity (red) and group velocity (green) are described

( * sorry, new users aren't allowed to use image tags)

both show "speed" in the Propagation direction

but

How I calculate the rise speed of a cicle red point in transversal direction?

For example in a very simple wave like this

y(x,t) = A* sin(kx - w t), should I calculate derivative of y having x=constant?

What about in general? How to calculate the Rise speed?

EDIT:

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

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Once you know the phase velocity $\omega$, you know that along the $x$ axis the red dot is described by $x(t) = \omega t$. Then you just apply the chain rule. The value of $y(x,t)$ at the red dot is a function of time $y(x(t),t)$. So $$\frac{d}{dt}y(x(t),t) = \frac{\partial}{\partial x}y(x(t),t) \cdot \frac{d}{dt}x(t) + \frac{\partial}{\partial t} y(x(t),t)$$ which we more succinctly write as $$\omega \partial_x y + \partial_t y$$
Then you are just looking at the vertical position change fixing the $x$-coordinate, which means that it is exactly given by just the partial derivative in the time-direction, $\partial_t$. –  Willie Wong Jul 26 '10 at 22:07