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

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

http://upload.wikimedia.org/wikipedia/commons/b/bd/Wave_group.gif

( * 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?

Thanks for any advice

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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1 Answer 1

up vote 3 down vote accepted

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 $$ \omega \partial_x y + \partial_t y $$

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To clarify, the position/velocity of a point on the wave can be modelled as SHM (simple harmonic motion), which is what Willie Wong is effectively doing. –  Noldorin Jul 22 '10 at 13:14
    
How can I compute the solution to a simple case, for example in wolfram alpha ? I think I would need to specify k=constant, w=constant –  Hernán Eche Jul 22 '10 at 13:20
    
k does not matter. –  Noldorin Jul 22 '10 at 13:54
    
This is the way in wolfram alpha to do that w*D(A*Sin(k*x - w*t),x)+D(A*Sin(k*x - w*t),t) –  Hernán Eche Jul 26 '10 at 15:41
1  
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

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