Velocity in D'Alembert solution is the same as virtical velocity?

[yungman]

One dimensional wave equation:

$$\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}$$

Where c is the vertical velocity of the vibrating string.



This will give D'Alembert solution of $$u(x,t) = \frac{1}{2}[f(x+ct) + G(x+ct)] + \frac{1}{2}[f(x+ct) + G(x+ct)]$$

Where $$u(x,t) = \frac{1}{2}[f(x+ct) + G(x+ct)]$$ is the wave moving left with velocity c and $$\frac{1}{2}[f(x+ct) + G(x+ct)]$$ is wave moving right with velocity c.


From the above, this mean the vertical velocity ( let's call u(x,t) axis ) of the vibrating string is the same as the propagating ( along x-axis ) velocity of the wave.

Question:

1) If the string is vibrating in the fundamental frequency ( single freq.). The velocity at different point is different because every point is vibrating at the same frequency and the point in the middle travel a farther distance than the points close to the end.

2) Is it really true the propagation velocity same as the vibrating velocity?

 

[yungman]

Anyone have some insight?