Boundary condition of wave impact on mass

[Trevorman]

Homework Statement

Two elastic bars are joined. A step wave is coming in from left. Derive the shape and magnitude of the reflected wave if the right bar is approximated by a rigid body (point- mass) that is free to move in the axial direction.

The Attempt at a Solution

I have problem with the boundary conditions, I know that a free will reflect the wave as a tensile wave and that a fixed end (eg wall) will reflect the wave as a stress wave. But what if the "wall" could move? Should i use the conservation of momentum and investigate or is there a equation that can describe this?

 

[mark726]

Hello,

Thank you for your question. In this case, we can use the conservation of momentum to determine the shape and magnitude of the reflected wave. Since the right bar is approximated by a rigid body (point-mass), we can assume that the mass of the bar is concentrated at a single point. Therefore, the momentum of the bar can be represented by the product of its mass and velocity.

When the step wave reaches the junction of the two bars, it will exert a force on the right bar, causing it to move in the axial direction. This force will also create a reaction force on the left bar, causing it to move in the opposite direction. The magnitude of these forces can be determined using Newton's second law, F=ma, where F is the force, m is the mass, and a is the acceleration.

Since we know that the right bar is free to move in the axial direction, we can assume that its velocity is equal to the velocity of the step wave. Therefore, the momentum of the right bar can be represented as p=mv, where p is the momentum, m is the mass, and v is the velocity of the step wave.

Using the conservation of momentum, we can equate the momentum of the right bar before and after the collision. This means that the momentum of the reflected wave will be equal to the momentum of the step wave. From this, we can determine the velocity of the reflected wave, which will also be the velocity of the right bar after the collision.

Once we have the velocity of the reflected wave, we can use this to determine the shape and magnitude of the reflected wave. Since the right bar is approximated as a rigid body, the reflected wave will have the same shape and magnitude as the right bar. Therefore, we can use the velocity of the right bar to determine the velocity and displacement of the reflected wave.

I hope this helps. Let me know if you have any further questions.