When Laplacian of a function is equal to some constant times the function itself, does anything interesting happen? Or does it only indicate that the function is somehow oscilating (in terms of its derivative)?
motoole2
This partial differential equation (Laplacian of f = f * K) is a bit special; when the constant K is negative here, this equation is referred to as the Helmholtz differential equation, which is used throughout physics (and graphics) to model the propagation of light. The space of solutions to this equation involves linear combinations of functions of the form a * exp^{sqrt{-1} * dot(x,k)}. The vector k is known as the wave vector, and is related to our constant as follows: |k|^2 = K.
But the functions are not always oscillating. What happens if the constant K is positive?
When Laplacian of a function is equal to some constant times the function itself, does anything interesting happen? Or does it only indicate that the function is somehow oscilating (in terms of its derivative)?
This partial differential equation (Laplacian of
f=f * K) is a bit special; when the constantKis negative here, this equation is referred to as the Helmholtz differential equation, which is used throughout physics (and graphics) to model the propagation of light. The space of solutions to this equation involves linear combinations of functions of the forma * exp^{sqrt{-1} * dot(x,k)}. The vectorkis known as the wave vector, and is related to our constant as follows:|k|^2 = K.But the functions are not always oscillating. What happens if the constant
Kis positive?