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Will Hooke's law $F = -kx$ applied to a large mass-spring grid array such as:

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provide the full and complete mechanistic explanation for classical wave behavior, including the 2nd order wave behavioral equation?

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Simulation (pulsing wave source at center, immovable edge masses):

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Is it fair to say that common wave phenomena such as diffraction, interference, radiating speed, etc... all follows naturally from Hooke's law applied to large systems?

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In short, it does not. The "high frequency" behavior of the spring/mass system is different from the prediction of the wave equation. It is a simple example of a system with a cutoff scale (there can be no wavelengths shorter than (twice) the distance of the masses) vs. one without. The wave equation is independent of the length scale. For "long" wavelengths (many times the grid constant) the two sets of equations produce many of the same effects, but even that is deceptive because the wave equation is isotropic, but the spring-mass system has a 90 degree rotation symmetry. In three dimensional lattice systems (crystals) the mechanical properties are anisotropic and they depend on the crystallographic point group of the lattice.

One can turn your question upside down, of course and ask "Do real-world materials obey a classical wave equation?" and then the answer becomes negative, as well. No, real materials are often not isotropic and even their classical vibration spectrum does not follow the linear dispersion relation of a classical wave equation at high frequencies. The proper quantum mechanical treatment modifies even that and the "phonon" spectrum that we end up with is the one that gives us the proper low energy (low temperature) vibration properties of materials (like specific heat capacity).

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  • $\begingroup$ thank you. I attached a simulation result of applying Hooke's law to individual masses in a grid array. is it fair to say that within non-extreme bounds, common wave phenomena such as diffraction, interference, radiating speed, etc... all follows naturally from Hooke's law applied to large systems? $\endgroup$
    – James
    Oct 18, 2022 at 8:52
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    $\begingroup$ @James Yes. Along with another common wave phenomenon, dispersion, which is not captured by your "classical" wave equation, but is exhibited by your model as you approach the cutoff frequency. $\endgroup$
    – John Doty
    Oct 18, 2022 at 12:50
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    $\begingroup$ @James It is not necessary to simulate these systems numerically. One can formulate the equations of motion for these systems and diagonalize the resulting matrix equations. The classical and quantum theory of these systems can usually be found in books about solid state physics if you are interested in the details. $\endgroup$
    – FlatterMann
    Oct 18, 2022 at 14:19

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