What is an intuitive explanation of Helmholtz equation in acoustics?. What are the prerequisites to understand Helmholtz equation?

Question

I understand it as the wave equation for a specific frequency component. I have a big picture of partial differential equations but still don't get the Helmholtz equation. Is Helmholtz equation and theorem are same ?

Answer 1

The Helmholtz equation (a.t. Wikipedia:) $$\nabla^2A + k^2A = 0$$ Where $A$ is an amplitude, that is: a scalar. The $\nabla$ ( nabla ) is the vector operator that does partial differentiation wrt all available spatial dimensions. In this context $\nabla^2$ must mean outer product followed by inner product or in other words: the Laplacian ($\Delta$) resulting in a scalar value, because if it did not, then the addition to the scalar $+k^2A$ would not make sense

---

The Helmholtz theorem (or decomposition): Any vector field $\bf F$ being sufficiently smooth (twice partially differentiable) can be written as the sum of the gradient of a scalar field $\Phi$ and the vector gradient of a vector field $\bf A$ : $${\bf F}=-\nabla \Phi + \nabla \times {\bf A}$$

Mathematicians may use + instead, but it is no biggy as simple sign change of $\Phi$ alters that.

---

So the theorem (decomposition) is true mathematically for any field that fulfulls the smoothness requirements but the equation (as far as I know) is a mostly physics related differential equation. Like a law of physics.