Elastic Waves: P and S Modes in a Solid

An isotropic elastic solid carries two independent body waves: a compressional P wave at $v_P = \sqrt{\frac{\lambda + 2\mu}{\rho}}$, in which the material is alternately squeezed and stretched along the travel direction, and a slower shear S wave at $v_S = \sqrt{\frac{\mu}{\rho}}$, a transverse distortion. A point source launches both; the scene colours the compression (divergence of the displacement) and strains a reference grid, with the analytic P and S wavefront rings overlaid, while a station at 45 degrees records a seismogram where the P arrival leads the S by $d\left(\frac{1}{v_S} - \frac{1}{v_P}\right)$, the basis of earthquake distance ranging. Lowering the shear modulus toward zero (a fluid) makes the S wave vanish, which is why a liquid outer core casts an S-wave shadow.