Major earthquakes produce high-frequency body waves that impinge over extended periods of time. These waves refract into nearly vertical paths in the low-velocity shallow subsurface. Scaling relationships of laterally homogeneous models for exactly vertical waves illustrate features that arise in fully three-dimensional numerical models and the real Earth. Flow-law rheologies yield testable hypotheses, especially when different types of seismic waves interact. S-waves produce horizontal shear tractions on horizontal surfaces. The anelastic strain rate depends nonlinearly on the horizontal shear traction. The horizontal shear traction is the product of the shear modulus times the difference between total strain and anelastic strain. Damage where anelastic strain decreases the shear modulus and the shear modulus heals after shaking is finished can also be included. Anelastic strain continues when the material is driven at constant stress and stresses relax at constant strain. In contrast, the widely used Masing rules make the counter-intuitive prediction that no further strain occurs in when the material is maintained at constant stress. Conveniently, natural experiments allow appraisal. The Coulomb ratio of dynamic to lithostatic stress is approximately the dynamic (resolved horizontal) acceleration in g’s. The anelastic strain rate for frictional materials increases rapidly with shear traction. The resolved peak horizontal acceleration (peak ground acceleration, PGA) of S waves thus clips in g’s at the effective coefficient of friction. Strong tensional P waves then suppress S waves. Anelastic strain commences at low stresses for muddy soils, but increases slowly with stress. Nonlinear attenuation increases slowly at high shear tractions. Accelerations over 1 g can occur, especially when waves reverberate with the shallow layer. P waves have little effect. That is, the rheology is nonlinear viscous not Coulomb. Records from the Kumamoto 2016 strong earthquakes from KIK-net station KMMH16 display the expected effects for drained muddy soil. Reverberating S waves should interact with shallow distributed slip above the fault trace.