The Robin Boundary Condition

 This post is more for my own understanding than making maths accessible, as the topic requires a bit of Partial Differential Equation (PDE) knowledge. My 4th year project concerns, succinctly, the equation common to wave propagation and heat flow. I plan to write a post on my project at some point and another motivating the wave and heat equations but I'm going to start at the end here and provide physical intuition for something called the Robin boundary condition (RBC), what my project focuses on specifically. I have read a large number of sources that refer to the RBC as describing a "partially insulated boundary" or an "elastically supported boundary" but it took me a very long time to find a source providing any intuition as to why this equation describes those physical phenomena so thank you to the textbook of Strauss.

The RBC is a linear combination of the Neumann and Dirichlet boundary conditions (I promise to talk about these in the post on the wave and heat equations). More specifically, it states

on the boundary, where the first term denotes the normal derivative to the boundary (could be replaced with grad u dot n) and alpha is some real parameter. If alpha = 0 then this is the Neumann condition and if alpha = ∞ then this is the Dirichlet condition and this makes sense to be a combination of the 2. But I wouldn't be able to go the other way around and derive this from the description "partial insulation" or "elastically supported". So let's derive it.
For ease we'll derive it in one direction and specifically for the right end point of a thin, insulated rod submerged in a reservoir of heat. In the Neumann condition the right end point is insulated so no heat flows out of the rod and in the Dirichlet condition the end point is held in a reservoir of fixed temperature so large that it remains at a fixed temperature because it has no effect on the reservoir. But if the reservoir is of finite size then there is heat transfer between the two, which obeys Newton's law of cooling. Newton's law of cooling tells us that the heat flux to the environment is proportional to the difference in temperatures between the end point of the rod (u) and reservoir (Q). Fourier's law tells us that the heat flux is proportional to the x-derivative of temperature. Combining these, we find at the right endpoint:

Intuitively, if Q < u then heat flows out of the rod to the right. Alpha describes the conductivity. Then we can simply rearrange this and absorb alpha*Q into the units to recover the RBC!

What about in the wave equation? In this interpretation the end point of a vibrating string is attached to a spring with natural length 0. Then the spring stores elastic potential energy proportional to its extension (good old Hooke's law) and its extension is u. Then conservation of energy at that point gives us the results, although the derivation uses the energy momentum tensor and some things I don't fully remember from Mathematical Physics II so I'll omit it.