My exploration of the string Euler characteristic made me wonder if the Euler characteristic is related to energy conservation, then why not consider string thermodynamics. Not heat but free energy.
The Gibbs phase rule F = c - p + 2 for the string reads degree of freedom 1 = 1 component - 2 phases +2.
The rule is a tautology on one component because one degree of freedom implies two phases, and two phases implies one degree of freedom.
The string energy can only be defined under one degree of freedom.
So experimental evidence unequivocally shows two distinct energy phases: amplitude expansion and amplitude contraction.
Clearly the two phases are determined by the same closed system.
Note the 1 degree of freedom Lagrangian is E = T + U, not E = T - U
Phase I: When the deformed string is released, the baseline potential energy U is increases to U + U'(t). Energy conservation is the same as volume preservation, so the shape of the manifold minimizes surface area. This forces the excess potential energy U'(t) into kinetic action T(t) so that U + U'(t) > U + T(t).
Phase II When all excess potential energy is transferred to kinesis, the normal curvature of the smooth manifold is restored but with a surface that is moving. Then the kinetic energy T(t) runs down to zero. The base line potential energy cannot run down.
So the time-invariant standing wave has a covariant derivative which gives the string velocity, and therefore the invariant frequency, too.
This proves that the frequency and amplitude are both determined by the Gibbs free energy change which drives amplitude decay.
It is therefore proven that frequency and amplitude are dependent on the same closed potential system.
I have attached sketches of the string energy cycle at rest, deformed, expansion, and contraction.
If anyone would like to help write these equations better, I would appreciate it. My calculus has limits. I think someone could really do some interesting things here. The field is wide open for discovery and original research (in spite of what they tell me on Stack Exchange).
If you do write the string energy equations, go over and lay them on physics stack exchange for me.
If the system is at equilibrium, there are numerous relationships among these
variables. We want to know how many independent relationships there are among them. Each such relationship decreases by one the number of independent intensive variables that are needed to specify the state of the system when all of the phases are at equilibrium. Let us count these relationships.
• The pressure must be the same in each phase. That is,
,
, …,
,
, …,
, etc. Since
and
implies that
, etc., there are only
independent equations that relate these pressures to one another.
• The temperature must be the same in each phase. As for the pressure, there are
independent relationships among the temperature values.
• The concentration of species
in phase 1 must be in equilibrium with the concentration of species
in phase 2, and so forth. We can write an equation for phase equilibrium involving the concentration of
in any two phases; for example,
(In
On the string we have no temperature or pressure. We are only concerned with free energy created with the string is deformed.
We have one component and one degree of freedom, so two phases.
This can also be done using n points for components and n lines of motion for phases plus a position vector as one more component.
So one degree of freedom equals n - (n + 1) + 2.
The two observed phases, amp expansion and contraction, have an equilibrium on zero mean curvature when amplitude peaks.
I don’t know how to understand the Gibbs free energy equation but I think it has an analogous form in a potential mechanical system.
When you vibrate a string at a frequency different from its natural frequencies (the frequencies that form standing waves), several things can happen:
1. Non-Standing Wave Formation: The string will not form a stable standing wave pattern. Instead, the vibration will create a transient wave that may not maintain a consistent amplitude and shape.
2. Damping and Energy Dissipation: The energy of the vibration might dissipate quickly, leading to a damped response. The string may not sustain the vibration effectively, especially if the frequency is far from a natural frequency.
3. Harmonic Overtones: If the f
…
I just realized the figure above showing the sound envelop where amplitude expands and contracts is a pseudosphere! It has constant negative Gaussian curvature with a singularity on the peak at equilibrium.
This shows the evolution and involution of the surface motion on the wave. The wave is the potential energy field create by the internal stress of tension. The wave is always there but the surface can move.
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