Why Does a Wave Reflect with Double Amplitude at a Free End?

[Rishavutkarsh]

Hi everyone, the fact that a wave reflects from a string when it's end is free and the end of the string rises to a height of 2A seems to confuse me a bit. Even though I find it somewhat intuitive I would appreciate a detailed explanation of the same. Thanks in advance.

 

[gsal]

Hhhmmm...traveling waves? I seem to recall a lesson about this in one of my power systems class several decades ago...maybe I can "translate" what I remember...hope this rambling helps in some way.

When you have a source of energy that somehow has produced a wave in a string, I presume several scenarios are possible.

First, the string is not massless or friction less and so, with a long enough string or rope the wave will eventually dissipate and not even reach the end of the rope.

With a "short" rope and a free end ("zero impedance"), I thought the wave would disappear (this is what I seem to recall for electrical traveling waves, anyway), but maybe with the whipping at the (non-massless) rope end reflects some of the wave back?...don't know.

With a "short" rope and the other end firmly tied to a post ("infinite impedance"), the wave is supposed to reflect back at 180 degrees out of phase...meaning, if the wave hump is going say along the top of the horizontal and it hits the post, it comes back along the bottom of the horizontal.

With "impedance" values in between zero and infinite, the wave is supposed to reflect some and refract some...for example, say you have a thin string tied to a thick string...a wave coming along the thin string reaching the tie will "see" a change in "impedance"...part of the wave will propagate (refract) to the thick string, excite it and produce a wave there, though smaller...and part of the wave will reflect back through the thin string itself (kind of similar as with the post but much smaller scale).

hope this helps in some way

 

[Rishavutkarsh]

The fact that the wave reverses with a phase change of 180 degrees is something I am quite comfortable with. But however the second part with a rope at free end the wave reflects without any phase change and the end point rises at double the amplitude is something that seems to confuse me. Thanks for the reply though.

 

[gsal]

I see. Think about it though...how did you produce the wave in the first place? By raising your end of the rope high up and lowering it; once the wave approaches the other free end, when there isn't enough rope (mass and downward force) to contain the energy of the wave, the wave raises and launches the rope high up in the air placing some potential energy and internal tension into it...now, the rope end is high up in the air similar to the position where your end was when creating the wave and, so, when the end comes back down and the internal tension springs back...a similar wave is created on the same side of the horizontal. ...just a thought.

 

[Rishavutkarsh]

I've already thought it this way and to be honest it does seem intuitive to me. I actually was looking for a concrete mathematical proof. Nevertheless thank you I appreciate all the help.

 

[olivermsun]

If you are trying to understand the "mathematical" reason for "why" the rope reflects the way it does at an endpoint, I suggest starting by writing out the boundary conditions at the endpoint — for both the "free" rope end and the "anchored" end.

 

[Philip Wood]

It may help to think about a line of trolleys joined by springs. The trolleys are orientated transverse to the line joining them; they can move only at right angles to this line. Now imagine the propagation of a transverse wave when a trolley at one end of the line is displaced. When the transverse disturbance (wave) reaches the other end of the line (assumed 'free' with no springs or trolleys beyond it), what will happen to this final trolley? Not being restrained by further springs, it will be displaced further. Hand-waving, but can be dealt with mathematically.