Why Does a Wave Reflect at a Free End with Unchanged Polarity?

[kelvin490]

A wave pulse on a string moving from left to right towards a free end will reflect and propagates from right to left with the same speed and amplitude as the incident wave, and with the same polarity.

My question is, why the slope and the vertical force must be zero at the free end? If the restoring force is zero why the free end can move? Some textbooks explain by imaging a wave with same polarity traveling in opposite direction but I think it's just an analogy and the phenomenon is better be explained by force interaction.

Some ref:
p. 489 University Physics
https://books.google.com.hk/books?id=7S1yAgAAQBAJ&pg=PA489&lpg=PA489&dq=string+is+now+stretched,+giving+increased+tension,+so+the+free+end+of+the+string+is&source=bl&ots=oSDs7tqt4M&sig=5zzzUMnsC8dG_qdObx9nJxQRPkg&hl=zh-TW&sa=X&ei=MT2VVYbREdP-8QWssoDgAw&ved=0CCQQ6AEwAQ#v=onepage&q&f=false

http://www.acs.psu.edu/drussell/Demos/reflect/reflect.html

 

[Orodruin]

Consider what is going on at the free end. Think of the situation where the end is attached to a mass m. The acceleration of the mass would be proportional to the transversal force from the string. As the mass goes to zero (which is what you would have for an actual free string), the force needs to go to zero for the mass not to have infinite acceleration. For the transversal force to go to zero, the angle the string makes with the horizontal must go to zero.

 

[kelvin490]

Consider what is going on at the free end. Think of the situation where the end is attached to a mass m. The acceleration of the mass would be proportional to the transversal force from the string. As the mass goes to zero (which is what you would have for an actual free string), the force needs to go to zero for the mass not to have infinite acceleration. For the transversal force to go to zero, the angle the string makes with the horizontal must go to zero.

Thanks. It is a common way to explain why the slope and force is zero at the free end. But I still have some more questions (may be stupid questions): Why the end particle of the string moves if there is no force ever acted on it? Why the end of the string can move to a height which is two times the amplitude of incident wave? What makes it move back and create a reflected wave without polarity change?

 

[Orodruin]

Why the end particle of the string moves if there is no force ever acted on it?

You have to go back to the assumption that the end is massless here. The force equation would tell you that 0 = 0a, which is always satisfied, so just from this you do not get a constraint.

What makes it move back and create a reflected wave without polarity change?

You are here talking about a single wave pulse. When it propagates along the string, the front is being pulled down (and therefore decelerated). When you reach the free end, this no longer happens and you reach a higher amplitude.