I have the Helmholtz equation
$$\nabla^2f = -k^2f $$
I am trying to solve it as a second order differential equation using a numerical method.
However, I am unable to find an application of it, other than its special case as the Laplace equation. Could someone tell me another area where I can use the above equation.
Best Answer
Consider the general wave equation, $$\frac{\partial^2u}{\partial t^2}=\alpha\nabla^2u$$ If you assume that the solution is separable, $u\left(x,\,t\right)=U(x)\cdot T(t)$, then you end up with, $$ \frac{1}{T(t)}\,\frac{\partial^2T}{\partial t^2}=\frac{1}{U(x)}\alpha \nabla^2 U $$ Since these two equations are equal to each other, then we can say, $$ \frac{1}{T}\frac{\partial^2T}{\partial t^2}=-k^2=\frac{\alpha}{U}\nabla^2U $$ The right two terms are then the Helmholtz equation: $$-k^2U=\alpha\nabla^2U\Rightarrow\left(\nabla^2+k^2\right)U=0$$ (where we absorbed the $\alpha$ into $k$ in the right side).
Thus, basically any wave equation with homogeneous initial conditions should end up as a Helmholtz equation due to separability (and probably also due to the superposition principle).