The Newtonian model of string vibration traces to Euler who asserted the string can have the shape of any curve that can be drawn free hand.
The Hamiltonian model traces to Brook Taylor who asserted in 1714, based on his description of harmonic motion by a pendulum as isochronous, that every point on the string passes though the center of motion at the same point in time.
These predictions are subject to experimental confirmation.
Now a days, everyone is on Euler's side, and Taylor's equations for the string are not generally known. Euler and Bernoulli attacked Taylor's version. The issue has never been settled as to who was correct. I think it is obvious Taylor was right. The wave equation has only one solution because if S is the string then dS=0. That is why it is called a stationary wave.
We can predict that an image of the string taken at two different positions on the string will show the same orbit (at dynamic equilibrium) for the Hamiltonian string but the action at two points will never synchronize in the Newtonian.
So my question is how can I form the image of a point on the string during vibration. Given the frequency of the string is maybe 440 Hz, here are my questions:
1. Do I form the image inductively or photographically?
2. How many frames a second will I need?
3. Can I make an image of the string by plugging an electric guitar into an oscilloscope?
In the Hamiltonian model points on the string only move in the z-y plane but in the Newton points move in xyz directions. Could a laser measure the length of the string during vibration?
I think the experiment I am proposing is technically possible. There must be someone that know how to do it.
There is no experimental evidence of simultaneous multi-modal string vibration, in spite of what they say on Physics Stack Exchange. Showing waves on hanging ropes or strings that are driven by an oscillator do not count.
I sorry I don't accept either Youtube demonstration.
The first example simulates d'Alembert's equation but violates the string boundary condition dx/dt=0 because the string is allow to stretch. It assumes the string acts like a whip but has two fixed ends. The stationary string cannot elongate distance at rest, and it also cannot elongate as a stationary wave. There is no experiment evidence to support this nonsense that two waves add to one.
The boundary condition dx/dt =0 describes an orbit where if something comes around once, it will cycle.
Gauss said that when a curve develops on a surface the distance of the arclength does not change regardless of how the surface twists or bends. This principle must also apply to the string under energy conservation.
According to the principle of harmonic motion every point on the string must pass through the center of motion at the same time. That isn't happening in the cartoon.
We can be sure the string string video is phony because no string has amplitude that great. It is just a cartoon.
Think about it this way. If a string arrives at the center and it is not in a straight line then the string has extra potential enegry that will just transfer to kinetic energy in the next cycle.
If the string arrives at the limit of displacement at different times the conservation of the angular momentum and position will simply make any delta x go to zero under the principle of least action.
Doesn't make you wonder when the equation of motion for the string is 300 years old?
But back to the electric guitar. Does the impedance measure y-z motion, or just a scalar value. Obviously the pickups see the string phase differently but that does not prove there is more than one mode. Least action proves that.
Also if frequency is 440 Hz then 440 frames a second will only be one frame per cycles which will always find the string in the same phase.
Also the two waves on a string look good for waves in a plane.
Now try and wrap your mind around how two waves work in 3-dimensions!
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