A seismologist analyzes wave velocities through layers of Earth, finding that P-wave speed increases linearly with depth. At 10 km depth, speed is 6 km/s; at 30 km, it’s 7.5 km/s. What is the speed at 22 km?

["Understanding P-Wave Velocity Through Earth’s Layers: A Linear Model and Its Application at 22 km Depth", "Seismologists play a crucial role in decoding Earth’s internal structure by analyzing how seismic wave velocities change with depth. One key insight is that P-wave (primary wave) speeds generally increase linearly with increasing depth—especially through the rigid layers of the mantle. This foundational observation helps scientists infer composition, temperature, and pressure conditions beneath the surface.", "Our analysis focuses on a simplified but powerful model: P-wave velocity increases linearly with depth. Using two known data points—at 10 km depth, the P-wave speed is 6 km/s; at 30 km depth, it rises to 7.5 km/s—we can derive the equation of this linear relationship and predict wave speeds at intermediate depths, such as 22 km.", "### Deriving the Linear Velocity Model", "We assume a linear relationship of the form:", "[\nv(z) = m \cdot z + b\n]", "where:\n- ( v(z) ) = P-wave velocity at depth ( z ) (in km)\n- ( m ) = slope of the velocity-depth relationship\n- ( b ) = intercept (velocity at surface, ( z = 0 ), assumed 0 km/s for idealized linearity)", "Using the given points:\nAt ( z = 10 ) km: ( v = 6 ) km/s\nAt ( z = 30 ) km: ( v = 7.5 ) km/s", "First, calculate the slope ( m ):", "[\nm = \frac{7.5 - 6}{30 - 10} = \frac{1.5}{20} = 0.075 \ ext{ km/s per km}\n]", "Now, confirm the model:\n- At ( z = 10 ): ( v = 0.075 \ imes 10 + b = 0.75 + b = 6 \Rightarrow b = 5.25 )\nThus, the linear model simplifies to:", "[\nv(z) = 0.075z + 5.25\n]", "However, this suggests a velocity of 5.25 km/s at the surface, which is below typical values. Instead, since the observed increase is only 1.5 km/s over 20 km, a better assumption is that velocity increases from 6 km/s at 10 km to 7.5 km/s at 30 km — but since linearity is approximated, recalculating slope with corrected interpretation:", "Check the change: 7.5 – 6 = 1.5 km/s over 20 km → 0.075 km/s per km, but since velocities rise from surface downward, and 10 km is already at 6 km/s (consistent with real crustal values), the model should reflect velocity increasing monotonically with depth.", "Thus, using linear interpolation between the two points is robust for rapid estimation:", "[\nv(z) = 6 + 0.075(z - 10)\n]", "At ( z = 22 ) km:", "[\nv(22) = 6 + 0.075 \ imes (22 - 10) = 6 + 0.075 \ imes 12 = 6 + 0.9 = 6.9 \ ext{ km/s}\n]", "### Conclusion: P-Wave Speed at 22 km Depth", "The linear velocity model, validated by observed P-wave increases through Earth’s upper layers, predicts that P-wave speed at 22 km depth is approximately 6.9 km/s. This value supports seismological understanding of wave propagation and magma-induced velocity variations in the mantle transition zone.", "For real applications, more complex models incorporate nonlinear effects and phase changes, but linear approximations remain essential for quick interpretation of seismic data, monitoring crustal dynamics, and studying tectonic processes.", "---", "Key Takeaways:\n- P-waves accelerate linearly with depth in homogeneous layers.\n- At 10 km depth, speed = 6 km/s; at 30 km, 7.5 km/s.\n- Interpolation gives P-wave velocity of 6.9 km/s at 22 km.\n- This approach underpins seismic tomography and Earth modeling efforts.", "Stay tuned for next updates on refining velocity models using real global seismic networks!", "---", "Keywords: seismologist, P-wave velocity, Earth layers, earthquake waves, seismic modeling, wave speed, depth relationship, geophysics, seismology education, crustal structure, mantle transition zone."]