# Wave equation in inhomogenous medium - Missing term?

When it comes to the electromagnetic wave equation in linear inhomogenous media, I find many sources who will just use the usual formula: $$(\Delta - \frac{\epsilon \mu }{c^2} \frac{d^2}{dt^2})\vec{E} = 0$$ and render $\epsilon$ and $\mu$ being position dependend: $$(\Delta - \frac{\epsilon (\vec{x}) \mu (\vec{x}) }{c^2} \frac{d^2}{dt^2})\vec{E}(\vec{x}) = 0$$ Why are we allowed to do that? When I derive the wave equation from Maxwell's equations, I require $\nabla \vec{E} = 0$, but in inhomogenous Media the equation is $$\nabla (\epsilon(\vec{x}) \vec{E}(\vec{x}))=0$$ which leads to $$\nabla \vec{E} = \nabla( \log{\epsilon} )$$ This would be an additional term in the wave equation, and it's similar with $\mu$. Can somebody tell me why this additional terms are neglected, what approximation is made and when this is valid?

• Yes, you are correct. There is a term $-\nabla (E \cdot \nabla \ln \epsilon)$ on the RHS. Nov 30, 2016 at 23:17
• I have also faced this annoying problem. I think people implicitly assume $\epsilon$ is slowly varying or something and say $\nabla \epsilon \sim 0$. Check out these notes: opt.zju.edu.cn/zjuopt2/upload/resources/…. -- They were very helpful for me, hopefully they will be for you too. Equation (5.2--14) in particular is what you are looking for. Nov 30, 2016 at 23:19
• I have a difficulty to think of this at the edge of 2 media, where $n$ is not continuous, and $\nabla n$ would be some kind of $\delta$ function. Nov 30, 2016 at 23:31
• indeed, ∇n would be infinite at an infinitesimal thin interface across two media. I guess one could remedy the situation applying some tapering to obtain a more gradual boundary. Jun 12, 2022 at 15:52

You are correct that there is a term on the RHS of $-\nabla(\vec{E}\cdot \nabla \ln \epsilon)$.
In order to neglect this you compare it's magnitude with $\nabla^2 \vec{E} = -k^2 \vec{E}$.
So, just looking at magnitudes, we can say that the RHS of your wave equation is $\sim 0$ if $$|\nabla(\vec{E}\cdot \nabla \ln \epsilon)| \simeq kE \nabla \ln \epsilon \ll k^2 E$$ $$\nabla \ln \epsilon \ll k$$ $$\nabla \epsilon \ll k\epsilon =2\pi \frac{\epsilon}{\lambda}$$
So as long as a typical length scale for a change in $\epsilon$ is much longer than the wavelength of light, then this approximation is good. You certainly could not use this approximation (re your comment) where there are abrupt changes in refractive index, where you would have to employ the usual continuity conditions for the electromagnetic fields.
• @QuantumWisp I can derive the laws of geometric optics by treating the boundary condition as sharp, with an abrupt change in $\epsilon$ at the boundary and a uniform (but different) $\epsilon$ either side. The problem you have posed would not necessarily enter into it. Dec 1, 2016 at 16:58