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

  1. Apr 21, 2010 #1
    One dimensional wave equation:

    [tex] \frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}[/tex]

    Where c is the vertical velocity of the vibrating string.



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

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


    From the above, this mean the vertical velocity ( lets 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?
     
    Last edited: Apr 22, 2010
  2. jcsd
  3. Apr 22, 2010 #2
    Anyone have some insight?
     
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