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the Generalized Helmholtz Equation
The Helmholtz equation arises in many contexts in the attempt to give a mathematical explanation of the physical world. These range from Alan Turing's explanation of animal coat patterns to Schrödinger's time-independent equation in quantum theory.
The Helmholtz equation per se is
where k is a constant. The Generalized Helmholtz equation is that equation with k a function of the independent variable.
In one dimension the Helmholtz equation just has the sinusoidal solution of φ(x) = A·sin(kx)+B·cos(kx). In one dimension the Generalized Helmholtz equation has a sinusoidal-like solution of varying amplitude and wavelength.
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