Please help, conceptual motion of a transverse wave on a string question

[agonydrum]

I am losing my mind over this, it seems the longer i think about it the further i get from a definitive answer. It started with me trying to understand the variables the would result in a standing wave, ie: what needs to occur, why, and how it occurs.
At first i was confused because it seemed that the normal force at the fixed end should just halt all motion since it is an application of dampening but i was told that something happens which distorts the reflected wave in some way? I am not sure what could be happening to distort the wave.

Second, I don't understand how exactly the reflected wave is created in the first place, like i said above it seems to me that the energy should just be dissipated at the fixed end by the normal force not reflected upside down.

The last concept I am just unsure about, I did some quick math with a simplified wave force diagram and it seems that the nodes could only be located were the two opposing waves first intersect, but that would mean you could generate nodes even if the reflected wave didnt math the energy of the incident wave?

Any help would be greatly appreciated I've spent days obsessing over this and i get too fixated to just move past it and finish the chapters.

 

[Andrew Mason]

In order for a string to vibrate, it has to have tension so it is either a string fixed at both ends (like a guitar string) or something equivalent (eg. a string with a weight hanging from it). Both ends are effectively fixed. If they are fixed, they can't move so the ends are nodes. The only wave that can exist on such a string is one that has nodes at both ends.

There is no need to think of a reflected wave. Just think of it as a vibration of the whole string. It vibrates at a particular frequency because of the tension, length and mass per unit length of the string. See: http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

AM

 

[Nugatory]

Second, I don't understand how exactly the reflected wave is created in the first place, like i said above it seems to me that the energy should just be dissipated at the fixed end by the normal force not reflected upside down.

Consider the mechanism that leads to waves in a string in the first place: We model the string as a spring; when we displace a segment the string stretches increasing tension; this tension accelerates the displaced segment towards the centerline; but by the time it gets there the segment has picked up some velocity so overshoots and is displaced to the other side; and the cycle repeats.

This is basically the interaction that you're describing when you write down with the differential equation for the wave (and it's also very similar to the dynamics of a harmonic oscillator). Viewed this way, there's no energy dissipation at the fixed end; the end is tugged way and then the other but it doesn't move so $d$ is zero in $W=Fd$ and no work is done.

That's for an ideal spring/string described by the classical wave equation and with no damping. In any real physical device, some energy will be dissipated in friction at the fixed point (very similar to the dynamics of a damped harmonic oscillator) so the reflected wave will have slightly less energy and amplitude than the incident wave. Thus, to set up the standing wave and keep it going we have to drive the system, adding energy to replace that lost by damping. Otherwise...

I did some quick math with a simplified wave force diagram and it seems that the nodes could only be located were the two opposing waves first intersect, but that would mean you could generate nodes even if the reflected wave didn't match the energy of the incident wave?

Saying that the energies don't match is equivalent to saying that the amplitudes are not the same. Thus, the two waves won't be able to exactly cancel at the points where you expect the nodes to appear, and a standing wave won't form.

 

[olivermsun]

I don't understand how exactly the reflected wave is created in the first place, like i said above it seems to me that the energy should just be dissipated at the fixed end by the normal force not reflected upside down.

It isn't so surprising. Think of what happens when an elastic ball strikes a wall. It bounces; the energy doesn't just dissipate. Similarly, think of what happens when waves in a swimming pool hit the end (the wall): they also reflect, they don't just dissipate on the wall. Ocean waves encountering a cliff or sea wall reflect as well (alhough waves that impinge on beaches do tend to dissipate—for more complicated reasons).

 

[Philip Wood]

If the string is fixed to a fixed anchorage, as the wave arrives at the end, it can't move the anchorage and can't do work on it, so can't lose energy to the anchorage. But the anchorage will exert a force on the string to the force equal and opposite to the force which the string exerts on the anchorage, and it is that force on the string which sends a wave back down the string, reversed in phase at the anchorage.

As Andrew Mason remarked, there is no necessity to regard a stationary or standing wave as a combination of progressive waves traveling in opposite directions. It is just one way of thinking about what is going on.

 

[agonydrum]

I really appreciate all of you posting on here to try and help me understand this, some of your explanations have helped me see areas where i was modeling the system incorrectly but I am still having trouble understanding the reflected wave. I understand generally that the reflection is generated by reflecting the force in the X and Y at the fixed end, what i don't understand is how the forces in the Y don't cancel immediately. It seems that if you look at what's happening in a second by second shot than the normal Y force would have to be double the incident force since the new amplitude is a change of 200%. Once to cancel out the incident wave and a second to generate the reflected wave.

 

[olivermsun]

One thing that may be helpful to realize is that, in the idealized string problem, the anchor can exert as much force as is necessary to enforce the boundary conditions of zero displacement and zero velocity (and hence zero energy transmission) at the end of the string.

 

[Nugatory]

It seems that if you look at what's happening in a second by second shot than the normal Y force would have to be double the incident force since the new amplitude is a change of 200%. Once to cancel out the incident wave and a second to generate the reflected wave.

It's a bit like bouncing a ball off a wall - there's no problem getting enough force to change the velocity of the ball by 200% with an idealized perfectly rigid wall.

 

[Philip Wood]

what i don't understand is how the forces in the Y don't cancel immediately.

The forces I was talking about in #5 were on different bodies (the wall and the string) and therefore don't cancel in a physical sense.

One of the things that makes it difficult to get one's head round what's happening is that it's easy to flip inadvertently between two valid approaches: (1) considering the string's motion in terms of net forces on the parts of the string, (2) considering the motion to be the resultant of superposed progressive waves traveling in opposite directions.

In approach (2) the string actually moves up and down at the end attached to the wall – for each of the progressive waves. But the resultant displacement here is zero. Another thing that you have to watch with approach (2) is that the angle at which the string meets the wall for the individual waves, and therefore the Y component of the force between string and wall due to individual waves, isn't the same as for the resultant wave.