Question

A P-ray arrives at the mantle-core boundary at an angle 25º with respect to the normal. At what angle to the normal does it enter the core? (P-wave velocity in the lower mantle is 13.7 km/s and outer core is 8.1 km/s)(Round off to two decimal places)

Answer 1

Correct Answer: 14.35

To determine the angle at which a P-wave enters the core, we apply Snell's Law, which relates the angles of incidence and refraction to the velocities of the wave in the two media. 

The formula is: \( \frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2} \) where \(\theta_1 = 25^\circ\) is the angle with respect to the normal in the mantle, \(v_1 = 13.7 \text{ km/s}\) is the P-wave velocity in the mantle, \(v_2 = 8.1 \text{ km/s}\) is the P-wave velocity in the core, and \(\theta_2\) is the angle with respect to the normal in the core, which we need to find. 

Rearranging for \(\theta_2\), we get: \( \sin \theta_2 = \frac{v_2}{v_1} \cdot \sin \theta_1 \). 

Substituting the values: \( \sin \theta_2 = \frac{8.1}{13.7} \cdot \sin 25^\circ \approx 0.59124 \cdot 0.42262 = 0.24983 \). 

Calculating \(\theta_2\): \( \theta_2 = \sin^{-1}(0.24983) \approx 14.35^\circ \). 

Thus, the P-wave enters the core at an angle of 14.35° with respect to the normal.