The given figure shows the geometry of a seismic ray path inside the Earth, with the P-wave velocities of 10 km/s, 11 km/s and 12 km/s corresponding to the layers L1, L2 and L3, respectively. If the angle of incidence (\(\theta\)) at the L1-L2 boundary is \(40^\circ\), then what is the angle of refraction at the L2-L3 boundary? (answer in nearest integer) (In the figure, \(r_1 = 3100\) km is the radius, measured from the Earth's centre O, of the L1-L2 boundary, and \(r_2 = 3000\) km is the radius of the L2-L3 boundary.)
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Apply ordinary Snell's law at each boundary (same radius on both sides), and separately use \(r\sin\theta=\)constant for the straight ray segment travelling between the two different radii inside L2.
This is a spherically layered Earth, so the problem has to be handled in two distinct steps: an ordinary (flat) Snell's law refraction at each velocity boundary, plus a geometrical correction because the ray travels as a straight line between two boundaries that sit at different radii from the Earth's centre.
Step 1 - Refraction at the L1-L2 boundary (radius \(r_1\)). Both the incident and refracted rays touch the boundary at the same radius \(r_1\), so ordinary Snell's law applies exactly as in flat-layered media: \[\frac{\sin\theta_2}{v_2}=\frac{\sin\theta_1}{v_1}\ \Rightarrow\ \sin\theta_2=\frac{v_2}{v_1}\sin\theta_1=\frac{11}{10}\sin40^\circ=1.1\times0.6428=0.7071\] \[\theta_2=\sin^{-1}(0.7071)\approx45^\circ\]
Step 2 - Straight-line travel through L2, from radius \(r_1\) to radius \(r_2\). Inside layer L2 the velocity is constant, so the ray is a straight chord. For any straight line, the perpendicular distance from the centre O to the line - equal to \(r\sin\theta(r)\) at any point on the line - never changes. Applying this at the two ends of the chord (radius \(r_1\) where the angle is \(\theta_2\), and radius \(r_2\) where the angle is \(\theta_2'\)): \[r_1\sin\theta_2=r_2\sin\theta_2'\ \Rightarrow\ \sin\theta_2'=\frac{r_1}{r_2}\sin\theta_2=\frac{3100}{3000}\times0.7071=0.7308\] \[\theta_2'=\sin^{-1}(0.7308)\approx46.9^\circ\]
Step 3 - Refraction at the L2-L3 boundary (radius \(r_2\)). Again both rays share the same radius, so ordinary Snell's law applies: \[\frac{\sin\theta_3}{v_3}=\frac{\sin\theta_2'}{v_2}\ \Rightarrow\ \sin\theta_3=\frac{v_3}{v_2}\sin\theta_2'=\frac{12}{11}\times0.7308=0.7972\] \[\theta_3=\sin^{-1}(0.7972)\approx52.85^\circ\]
Rounded to the nearest integer, the angle of refraction at the L2-L3 boundary is \(\boxed{53^\circ}\), option (C).
