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What's the minimum (physics first) to get an oscillator?
AndyBeez:
--- Quote from: westfw on May 23, 2023, 06:25:34 pm ---
--- Quote ---The prototype harmonic oscillator in physics discussions is a spring
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How about a pendulum ? Or does the externality of gravity complicate things?
--- End quote ---
Or a planet orbiting a star? Velocity relative to the star comes out as a sine wave... of sorts. Also electrons orbiting the nucleus too. For a pure audio sine wave, how about the physics of a tuning fork? Really, everything is in oscillation. Or resonating. Except for that RF oscillator you designed and spent hours building :(
T3sl4co1l:
--- Quote from: TimFox on May 27, 2023, 05:15:15 pm ---An interesting difference between resonators such as quartz crystals and simple oscillators (mass and spring) is that resonators can have multiple resonant frequencies, while simple oscillators have only the one.
One-dimensional resonators (organ pipes, violin strings, resonant transmission lines, etc.) have harmonic overtone frequencies (integer multiples of the lowest or fundamental resonant frequency), but three-dimensional resonators (quartz crystals, resonant cavities, etc.) have overtones that are not harmonic frequencies.
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Sort of. Several of these are dispersive so have nonharmonic overtones. Acoustic resonators are a good example, or better to say: acoustic waveguides. Most waveguides I suppose, share this property?
Bulk material vibrations are especially rich, having three directions each longitudinal, transverse, and shear, all with different wave velocities, all which couple to each other to varying degrees depending on geometry (and in nonlinear combinations, because displacement determines the moment on which longitudinal force acts, etc.).
EM structures with interaction between nearby elements (not just directly adjacent) also works. Consider the helical waveguide for example.
Or for more quantum examples, consider the 1/n^2 spectrum of atomic levels; the resonator is highly irregularly shaped (a spherical 1/r potential well) so different states have very different waves. At least until you get to the Rydberg sort of levels where states (or modest superpositions thereof) start to look like classical orbits.
Tim
TimFox:
Also, in the classical simple harmonic oscillator, obviously the minimum possible energy stored in the system is zero, where the displacement and velocity are both zero.
In quantum mechanics, such a state violates Heisenberg, since both the position and momentum are zero, with no uncertainty.
In the quantum harmonic oscillator (a prototype for many physical systems), the minimum state is given by
E0 = (h f)/2 , where f is the natural frequency of the oscillator and h is Planck's constant.
and the discrete set of energies is given by
En = (n + 1/2) h f
for non-negative integers n
The spacing between these states is h f
(This is usually written h-bar omega, but I'm too lazy for proper typesetting.)
TimFox:
Separate issue about overtones and harmonics:
Sometimes people confuse the harmonic frequencies of a simple (non-dispersive) one-dimensional system (such as a violin string) with the harmonic frequencies that appear in the Fourier series for the decomposition of any periodic waveform.
They have the same form (integer times fundamental frequency), but originate differently.
AndyBeez:
...and ring modulation ???
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