Htykuut

So I was using this wave simulation (https://phet.colorado.edu/sims/html/wave-on-a-string/latest/wave-on-a-string_en.html) when this happened; using the settings: Oscillator, High tension, no damping. Any frequency or amplitude will do as long as it's divisible by 5 I think it's because of calculations.

After sometime, the wave form would collapse on itself leaving a plain rope and restarting. (It has to be with the settings since the beginning)

Is there a name for this?;

Could it happen in the real world or is it purely hypothetical since there's no system like that;

Why with low tension doesn't happen?;

Was it purely coincidental?

That's a pretty neat effect! It's not a bug in the simulation, it is correctly solving the equation.

To get a feel for what's going on, recall how a standing wave is formed. If you hold one end of a string and fix the other end, and give your end a wiggle, that wiggle will propagate down the string, bounce off the fixed end, return and bounce off the end you're holding, and so on. In practice, this would damp out quickly, but for an ideal string it would go on forever.

Now, if you continuously wiggle the string, the same process happens for each wiggle. After you've wiggled the string up and down $10$ times, there are $10$ wiggles continually bouncing back and forth.

If you're driven the string at one of the standing wave frequencies, then by the time the first wiggle comes back, you will be producing another with the same phase. They reinforce each other, producing a standing wave whose amplitude grows and grows. If you don't drive at a standing wave frequency, there will be a phase difference. For example, by the time your first wiggle has come back, you might be creating another one $90^circ$ out of phase. The next will be $180^circ$ out of phase, and the next $270^circ$ out of phase, and all four of these will superpose to exactly zero, giving a stationary rope.

In general, this will happen whenever you drive at a rational multiple of the fundamental frequency. The reason the effect doesn't work when you change the tension setting is because that changes the wave speed and hence the frequencies, so you're no longer driving at a rational multiple of the fundamental. It doesn't violate conservation of energy, because for the last two you will be doing negative work on the rope. The effect may even be observable for simple multiples of the fundamental on a real string.

If you want to give this effect a name, it's simply the usual destructive interference, but with the neat twist that a wave you're putting in now is destructively interfering with a wave you put in earlier.

At first glance I was with @John Rennie: it has to be a bug, but after playing around I think this just passive cancelation (which isn't a named phenomenon, but active cancelation is).

If you add a slight amount of damping, the effect goes away, but the wave still gets pretty small. That's not what I would expect from a bug, with the caveat that predicting the behavior of bugs is tricky.

Back to no damping: if you mix in an frequency component without hitting "restart", the phenomenon disappears, but you can see it might be happening to each component at different times. This is what linear superposition should show for a real phenomenon.

Also: it's hard to land on a resonance with this simulator. With no damping, $Q=infty$, and the resonance is infinitely narrow so you will never hit one.

You might need to work out the details, but he is where I am. I put the thing in "pulse mode" and verified that you can "catch" a reflected pulse, so I believe that after enough reflections, the system "catches" itself.

You start with a pure sine wave:

$$ y_0(x, t) = sin(kx-omega t) $$

then you add a delayed reflection:

$$ y_1(x, t) = sin(-k(x-L)-omega t + phi_1) $$

where you chose $phi_1$ so that it cancels the 1st wave.

Keep doing this an at some point, does everything cancel for one "cycle", but does:

$$sum_1^{n}y_n(x, t) = 0 $$

$forall x in (0, L]$ an $ t_n < t < t_{n+1}$

where $t_i$ is the time of the $i$-th reflection?

I think the answer is "yes".

i see that there is a secondary wave "riding" on your wave. Its amplitudes goes higher and in a certain point two waves extinguish themselves. Since you dont have any damping, no energy loss... the simulation doesnt seem to have a bug.

This is a very interesting observation, and it is actually a particular case of a much more general theorem, which applies to all Halmitonian Systems or Energy Conserving Systems: The Poincaré Recurrence Theorem (PRT). This important theorem of classical mechanics states that Halmitonian Systems, will, after a sufficiently long but finite time, return to a state arbitrarily close to the initial state. You may find a prove in the book Mathematical Methods of Classical Mechanics, by Vladimir Arnold

In this particular simulation, when you switch off damping, and the tension is sufficiently large, PRT applies. But how can that be if the system motion is forced and energy does not conserve? You may imagine the forcing system as a mass-spring system with an arbitrarily large mass. In this case, the string motion will be forced by another mass in harmonic motion, but this harmonic motion will not be significantly perturbed by the string if the mass is sufficiently larger than the string mass. The system as a whole is now conservative, and PRT applies.

It is important that the frequency of the harmonic oscillation is much lower than the resonance frequency of the string. Otherwise, the string motion amplitude will rise too much and it will eventually perturb the mass-spring motion. That is why the tension of the string must be large.

Many lab real and non-trivial systems reveal this beautiful phenomenon with good approximation. I give here a link to a system of pendulums which return approximately to the initial state (link https://youtu.be/yVkdfJ9PkRQ) at video time 1:27';