- #1
yungman
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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 ( 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?
[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 ( 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?
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