Earthquakes occur in the crust or upper mantle. According to the Earthquake Hazards program, “shallow” earthquakes occur above a depth of 70 km in the crust.1 One simple model of earthquake behavior is called the slider-block model. Essentially, a block of mass m rests on a flat surface, and it is pulled horizontally by a spring. When the force exerted by the spring equals the frictional resistance, slip occurs. The model predicts the displacement δl of the slider block (corresponding to the fault displacement) and the maximum slip velocity vm on the fault as2
$δl=4SfSρghA(1−fdfS)andvm=fSgh(2ρ/S)1/2(1−fdfS),$
where S is the shear modulus; fs and fd are the (dimensionless) coefficients of static and dynamic friction, respectively; ρ is the rock density; g is the local gravitational acceleration; h is the mean fault depth; and A is the rupture area.

Question 1: Given that A ≈ 400 km2, what are δl and vm? Assume that S ≈ 3 × 1010 Pa, fs ≈ 0.05, and fd ≈ 0.045.

Question 2: Suppose that tremors from body waves3 are recorded 20 s apart. The fastest such waves are primary waves with speed vp ≈ 5.5 km/s in rock; shear waves have a corresponding speed vs ≈ 3 km/s. How far away is the epicenter of the earthquake? (Assume that the waves travel in straight lines from the epicenter to the recording seismometer.)

Solution to Question 1: To estimate h, we take the geometric mean of 1 km and 70 km, or ~8 km, and take ρ ≈ 2500 kg/m3. Hence, δl ≈ 4 × 0.05 × (2.5 × 103 kg/m3) × (10 m/s2) × (8 × 103 m) × (2 × 104 m) × 0.1 ÷ (3 × 1010 kg · m−1 · s−2) ≈ 1.3 m.

(Note that this is not meant to be an estimate of the width of the Icelandic rock fracture in the photograph, though it appears to be of the right order of magnitude.)

Similarly, vm ≈ 0.05 × (10 m/s2) × (8 × 103 m) × [(2 × 2.5 × 103 kg/m3) ÷ (3 × 1010 kg · m−1 · s−2)]1/2 × 0.1 ≈ 0.16 m/s.

Solution to Question 2: The time interval Δt = 20 s. The distance from the epicenter is therefore dvp × tvs × (t + Δt), so after a little algebra, it follows that
$d=20vpvs(vp−vs)≈20×5.5×3/2.5≈130km.$
2.
For a derivation of these equations, readers can access the Appendix at TPT Online at https://doi.org/10.1119/5.0165482, in the “Supplementary Material” section.
3.
Note that surface waves (such as Rayleigh waves and Love waves) have been neglected here. They are extremely important, of course, since they are the ones that do the most damage to lives and property.