Helmholtz equation

The Helmholtz equation is a second-order elliptic partial differential equation

∇2ψ=−k2ψ\nabla ^{2}\psi=-k^{2}\psi

for some constant kk. It is a special case of Poisson's equation where f=−k2ψf=-k^{2}\psi. It is also the eigenvalue equation of the Laplacian operator, of eigenfunction ψ\psi and eigenvalue −k2-k^{2}.

As a wave equation

This equation appears quite commonly in the study of waves, as it is a special case of the wave equation when the wavefunction is separable between time and space. In fact, consider the wave equation

(∇2−1v2∂2∂t2)u(r,t)=0\left( \nabla ^{2}- \frac{1}{v^{2}}\frac{ \partial ^{2} }{ \partial t^{2} } \right)u(\mathbf{r},t)=0

for a wavefunction u(r,t)u(\mathbf{r},t). Assume that uu is separable as

u(r,t)=A(r)B(t)u(\mathbf{r},t)=A(\mathbf{r})B(t)

Substitute this above to get

(∇2−1v2∂2∂t2)A(r)B(t)=0⇒∇2A(r)A(r)=1B(t)v2∂2B(t)∂t2\left( \nabla ^{2}- \frac{1}{v^{2}}\frac{ \partial ^{2} }{ \partial t^{2} } \right)A(\mathbf{r})B(t)=0\quad\Rightarrow \quad \frac{\nabla^{2}A(\mathbf{r})}{A(\mathbf{r})}=\frac{1}{B(t)v^{2}}\frac{ \partial ^{2}B(t) }{ \partial t^{2} }

We now employ the common technique of separation of variables: both sides of the equality are of course equal to each other, but they depend on different variables. It makes no sense that varying, say, r\mathbf{r} would also vary the right-hand side, which is only dependent on tt. The converse also applies. Thus, the two sides must be constant. Let's call this constant −k2-k^{2} for reasons that will be apparent later1. In the meantime, you can already start to read kk as the wavenumber.

We are now left with two equations in one variable each:

∇2A(r)A(r)=−k2,1B(t)v2∂2B(t)∂t2=−k2\frac{\nabla^{2}A(\mathbf{r})}{A(\mathbf{r})}=-k^{2},\qquad \frac{1}{B(t)v^{2}}\frac{ \partial ^{2}B(t) }{ \partial t^{2} } =-k^{2}

The equation in AA rearranges to

∇2A(r)=−k2A(r)\boxed{\nabla^{2}A(\mathbf{r})=-k^{2}A(\mathbf{r})}

This is the Helmholtz equation. Evidently, it must be the equation for the spatial part of a wavefunction u(r,t)u(\mathbf{r},t). But don't stop here, see the other way around too: the eigenfunctions of the Laplacian are always the spatial part of some wavefunction. This very interesting characterization helps to understand why the Laplacian appears so incredibly often when it comes to wave phenomena; it basically "contains" wavefunctions, so of course it would appear all the time.

The second equation in BB to

∂2B(t)∂t2=−k2v2B(t)\frac{ \partial ^{2}B(t) }{ \partial t^{2} } =-k^{2}v^{2}B(t)

If we define ω≡kv\omega\equiv kv and use dot notation for time derivatives we meet an old friend:

B¨(t)=−ω2B(t)\boxed{\ddot{B}(t)=-\omega ^{2}B(t)}

The temporal part of a spherical wave must therefore be a harmonic oscillator! This is why we made the weird choice of picking −k2-k^{2} as a constant in the first place. In fact, the Helmholtz equation itself is quite reminiscent of a harmonic oscillator, just instead of the time derivative we have the Laplacian.

Thus, in order to fully determine the nature of a spherical wave, one must solve a Helmholtz equation for space and a harmonic oscillator for time. By the way, if the wave is monochromatic, ω\omega is its angular frequency.

Footnotes

  1. The shape of the constant is completely arbitrary. We choose −k2-k^{2} only because it makes the solutions more meaningful, but we could've just as well called it kk or k3+83kk^{3}+83k. ↩