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

In summary, the one dimensional wave equation describes the vertical velocity of a vibrating string, with c being the velocity of the propagating wave. The D'Alembert solution shows the wave moving in both left and right directions with velocity c. This means that the vertical velocity of the string is equal to the propagation velocity of the wave. However, in the case of a string vibrating at the fundamental frequency, the velocity at different points may vary due to the different distances traveled by each point. It is still a question whether the propagation velocity is truly the same as the vibrating velocity, and further insight is needed.
  • #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?
 
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  • #2
Anyone have some insight?
 

1. What is the D'Alembert solution?

The D'Alembert solution is a mathematical technique used to solve wave equations in physics, specifically in the study of fluid dynamics. It is named after French mathematician Jean le Rond d'Alembert, who first proposed the method in the 18th century.

2. How does the D'Alembert solution relate to velocity?

The D'Alembert solution involves representing a wave as a sum of two waves, one propagating in the positive direction and one in the negative direction. The velocity in the D'Alembert solution refers to the speed at which these two waves travel.

3. What is vertical velocity?

Vertical velocity refers to the speed at which an object moves up or down in a vertical direction. It is often used in the study of fluid dynamics to describe the movement of fluids in pipes or channels.

4. How is velocity in the D'Alembert solution related to vertical velocity?

In the D'Alembert solution, the velocity refers to the speed of the wave components in the positive and negative directions. This is not the same as the vertical velocity, which is specifically the speed of an object moving up or down in a vertical direction. However, in some cases, the vertical velocity can be calculated using the velocity in the D'Alembert solution.

5. Why is velocity in the D'Alembert solution important?

The D'Alembert solution is an important mathematical tool in the study of fluid dynamics, which has numerous applications in physics and engineering. Understanding the velocity in the D'Alembert solution can help scientists and engineers analyze and predict the behavior of fluids in different scenarios, such as in pipes, channels, and waves.

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