Propagation of transverse pulse on a string

[Terry Bing]

Homework Statement

A horizontal string at tension T is tapped at the midpoint to create a small transverse pulse. What happens to the pulse as time passes? If the pulse is instead created at a point other than the midpoint, what happens to it? Neglect damping.

Homework Equations

Speed of transverse waves on a string stretched to a tension T is $v=\sqrt{\frac{T}{\mu}}$, where $\mu$ is the mass per unit length of the string.

The Attempt at a Solution

In the first case (pulse at midpoint), by symmetry, the pulse splits into two smaller pulses moving in opposite directions at the speed $v$ mentioned above.
In the second case (pulse not at midpoint) the same thing should happen, because even though the pulse is closer to one of the boundaries, it is not aware of where the boundaries are till it reaches it. Any transverse mechanical disturbance cannot propagate faster than the speed $v$.So there is no way the pulse is initially influenced by where the boundaries are.
Is this line of reasoning correct?

 

[ehild]

Homework Statement

A horizontal string at tension T is tapped at the midpoint to create a small transverse pulse. What happens to the pulse as time passes? If the pulse is instead created at a point other than the midpoint, what happens to it? Neglect damping.

Homework Equations

Speed of transverse waves on a string stretched to a tension T is $v=\sqrt{\frac{T}{\mu}}$, where $\mu$ is the mass per unit length of the string.

The Attempt at a Solution

In the first case (pulse at midpoint), by symmetry, the pulse splits into two smaller pulses moving in opposite directions at the speed $v$ mentioned above.
In the second case (pulse not at midpoint) the same thing should happen, because even though the pulse is closer to one of the boundaries, it is not aware of where the boundaries are till it reaches it. Any transverse mechanical disturbance cannot propagate faster than the speed $v$.So there is no way the pulse is initially influenced by where the boundaries are.
Is this line of reasoning correct?

Yes, but what happens when the pulses reach the ends?

 

[Terry Bing]

Yes, but what happens when the pulses reach the ends?

Assuming there are no losses (Reflection coefficient is 1), they are inverted and reflected. Shape and size of the pulses remains the same, propagation speed remains the same (since it is a property of the medium). So the pulses just bounce about, inverting at each reflection. And whenever and whenever they meet, they superpose to give the initial larger pulse (only for an instant).
Is this correct?

 

[ehild]

Assuming there are no losses (Reflection coefficient is 1), they are inverted and reflected. Shape and size of the pulses remains the same, propagation speed remains the same (since it is a property of the medium). So the pulses just bounce about, inverting at each reflection. And whenever and whenever they meet, they superpose to give the initial larger pulse (only for an instant).
Is this correct?

Where do they meet?

 

[Terry Bing]

Where d they meet?

In the first case, they would meet at the midpoint again.
In the 2nd case, if say the pulse was initially at a distance d from one boundary, then after an even no. of reflections, they will meet at the same initial position of the pulse. After an odd no. of reflections, they will meet at a distance of d from the other boundary to give an inverted pulse there.

 

[ehild]

In the first case, they would meet at the midpoint again.
In the 2nd case, if say the pulse was initially at a distance d from one boundary, then after an even no. of reflections, they will meet at the same initial position of the pulse. After an odd no. of reflections, they will meet at a distance of d from the other boundary to give an inverted pulse there.

Correct!

 

[Terry Bing]

Correct!

Thanks!