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An Abstraction of the Wave Equation

Wave Energy (work in progress)

We consider the wave equation under a vaccum and ideal circumstances and draw conclusions from ti regarding a mapping between phonetic audio and phonemes where ΔE E = 0.
E=μA2ω22LE = A^2 ^22L

The properties of the differential (gradient) are the bearers of information for our change-of-basis matrix. Therefore, we take the partial derivatives of EE with respect to each variable:

∂E∂A=μAω2L∂E∂ω=μA2ωL∂E∂L=−μA2ω2L2∂E∂μ=A2ω22L E A = A^2L\\ E = A^2 L\\ E L = - A^2 ^2L^2\\ E = A^2 ^22L

Amplitude–Frequency Relationship

We notice that two partial derivatives are very similar to each other with respect to AA and ω. We want to analyze the relationship of these variables between each other given an arbitrary wave where the energy is held constant.

Thus, we take the product of the partial derivative with respect to ω multiplied by the inverse of the partial derivative with respect to AA:

∂E∂ω⋅∂A∂E=∂A∂ω=μA2ωL⋅Lμω2A=Aω E A E = A = \\ A^2 L L^2 A = A

Amplitude and frequency appear to be directly proportional to each other with respect to this wave, meaning that for energy to be conserved the amplitude must increase at the same rate on which the frequency is decreasing; this establishes a relative relationship between amplitude and frequency.

Wavelength as Intermediary

We need to consider the other non-constant variable, wavelength. We once again take the product of the partial differential with respect to LL and the inverse of the partial with respect to ω:

∂E∂L⋅∂ω∂E=∂ω∂L=−μA2ω2L⋅LμωA2=−1ω E L E = L = \\- A^2 ^2L L A^2 = -1

This immediately suggests that wavelength is directly inversely proportional to the frequency of a wave. By the definition of the transitive, we can also imply that wavelength is inversely proportional to the amplitude of a wave.

This immediately proves that wavelength is an intermediary that holds no relative mention in relation to the energy of the wave. Therefore, wavelength is not necessary to determine the behavior of a wave given a known frequency, wavelength, and linear density given that dE=0dE = 0.

A Conserved Relationship

We want to derive a mathematical relationship between amplitude and frequency that will hold constant under these conditions. We expand the definition of the partial:

∂ω∂E=∂ω∂A⋅∂A∂E+∂ω∂μ⋅∂μ∂E+∂ω∂L⋅∂L∂E∂E∂A⋅∂ω∂E=∂A∂ω+∂E∂A ⁣(∂E∂μ⋅∂μ∂ω+∂E∂L⋅∂L∂ω)∂E∂A⋅∂ω∂E−∂A∂ω=∂E∂A ⁣(∂E∂μ⋅∂μ∂ω+∂E∂L⋅∂L∂ω)∂E∂A⋅∂ω∂E+∂ω∂A=∂E∂A ⁣(∂E∂μ⋅∂μ∂ω+∂E∂L⋅∂L∂ω)∂E⋅∂ω∂E+∂ω=∂E ⁣(∂E∂μ⋅∂μ∂ω+∂E∂L⋅∂L∂ω)∂ω+∂ω=∂E ⁣(∂E∂μ⋅∂μ∂ω+∂E∂L⋅∂L∂ω)2∂ω∂E=∂E∂ω+∂E∂ω2∂ω∂E=2∂E∂ω∂ω∂E=∂E∂ω  ⟹  energy is frequency with respect to the gradient with respect to this system E = A A E + E + L L E \\ E A E = A + E A\!( E + E L L )\\ E A E - A = E A\!( E + E L L )\\ E A E + A = E A\!( E + E L L )\\ E E + = E\!( E + E L L )\\ + = E\!( E + E L L )\\ 2 E = E + E \\ 2 E = 2 E \\ E = E \\ energy is frequency with respect to the gradient with respect to this system

I have observed a very reliable mapping between phonetic audio and its respective phonemes.

WIP - Full POC Soon!