Direct Waves vs Head Waves

Harvard EPS55

Comparing travel times and cross-over distances for direct crustal waves (Pg, Sg) and critically refracted head waves (Pn, Sn)

Moho Discontinuity (35 km) Surface Source 100 km 200 km 300 km 400 km Pg Sg Pn (along Moho) Sn (along Moho) P-crossover S-crossover
Pg (Direct P-wave)
Pn (Head P-wave)
Sg (Direct S-wave)
Sn (Head S-wave)

Wave Velocities

Wave Type Crust Upper Mantle
P-waves 6.0 km/s 8.0 km/s
S-waves 3.5 km/s 4.5 km/s

The velocity contrast at the Moho creates the conditions for head wave propagation.

Direct Waves (Pg, Sg)

Travel directly through the crust from source to receiver.

T = D / Vcrust
  • Simple straight-line path
  • First arrivals at short distances
  • Travel time increases linearly with distance
  • Sg arrives after Pg (slower velocity)

Head Waves (Pn, Sn)

Critically refracted along the Moho boundary.

T = 2h·cos(ic)/Vc + (D-2h·tan(ic))/Vm
  • Travel down to Moho, along it, then up
  • First arrivals at long distances
  • Benefit from higher mantle velocities
  • Critical angle: sin(ic) = Vc/Vm

Cross-over Distances

Where head waves overtake direct waves as first arrivals:

  • P-waves: ~140 km (Pn overtakes Pg)
  • S-waves: ~180 km (Sn overtakes Sg)
  • S-wave crossover occurs at greater distance due to smaller velocity contrast
  • Creates distinct phase arrival patterns on seismograms