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.
There is a faster, single-formula way to reach the same result by recognising that this problem is really just a spherical version of Snell's law, governed by one conserved quantity along the whole ray: the ray parameter (impact parameter) \[p=\frac{r\sin\theta(r)}{v}\] This quantity is preserved for two separate reasons that both hold here - it is preserved across a velocity discontinuity at fixed radius (that is just Snell's law), and it is also preserved along a straight chord travelling through a layer of constant velocity (that is pure geometry, since \(r\sin\theta\) is the fixed perpendicular distance of a straight line from the centre). Because L2 is both a discrete velocity layer AND traversed as a straight line, \(p\) carries through it unchanged, so it must be equal at the very start of the path (in L1, at radius \(r_1\), angle \(40^\circ\)) and at the very end (in L3, at radius \(r_2\), angle \(\theta_3\)): \[\frac{r_1\sin40^\circ}{v_1}=\frac{r_2\sin\theta_3}{v_3}\] Solving directly for \(\theta_3\), skipping the intermediate L2 angle entirely: \[\sin\theta_3=\frac{r_1}{r_2}\cdot\frac{v_3}{v_1}\cdot\sin40^\circ=\frac{3100}{3000}\times\frac{12}{10}\times0.6428=1.0333\times1.2\times0.6428=0.7971\] \[\theta_3=\sin^{-1}(0.7971)\approx52.85^\circ\approx\boxed{53^\circ}\]
This single-step calculation lands on exactly the same value as the step-by-step Snell's-law-plus-geometry method, confirming option (C).
