Equations are derived which govern the mode of propagation and change in strengths of wave fronts whose behaviours are controlled by quite general systems of quasi-linear hyperbolic equations. The general theory is applied to discuss conditions at the head of a sound wave of arbitrary initial form and strength and also at the head of gravity waves on sloping beaches. Conditions under which shocks and bores occur are derived. At expansive waves the non-linearity produces “forgetfulness” at “large” times.