San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Studies Concerning the Solution of
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

∇²φ = −k²φ

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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