Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 79 / 2 / i

Work in a meridional $R,z$ plane and denote the interface height by $Z(R)$, to distinguish it from an observation coordinate. PutThe interface is $Z=(\alpha s+{\alpha }_{f}D)/d^2$, so $Z^{\prime }={\alpha }_{f}R/D$. Its unit normal into the fluid is proportional to $(-Z^{\prime },1)$. Consequently the angle between this normal and the vertical satisfiesThe outgoing source P wave has ray direction proportional to $(R,Z)$, tilting outwards by $\arctan (R/Z)$ from the vertical, while the normal tilts inwards. Thus ${\theta }_P=\arctan (R/Z)+{\theta }_z$. The tangent addition formula givesDirect substitution simplifies its denominator to $s(\alpha D+{\alpha }_{f}s)/(d^{2}D)$ and its numerator to $\alpha R(\alpha D+{\alpha }_{f}s)/(d^{2}D)$. HenceThe denominator is positive, so these are the physical incidence angles in $[0,\pi /2)$; the transmitted P wave reaches the fluid rather than propagating back into the solid.