Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 4 / b / ii / Solution

In the two constant regions defineThe normalized positive-frequency modes areup to the common spatial Fourier normalization. They satisfy the unit Wronskian conditionwhich is the mode form of the Klein-Gordon inner product. Because the mode equation contains no delta function at $\eta =0$, both $\Phi$ and ${\Phi }^{\prime }$ are continuous there.