  We identify a new type of shock wave by constructing a stationary expansion
shock solution of a class of regularised shallow water equations that include
the Benjamin-Bona-Mahoney (BBM) and Boussinesq equations. An expansion shock
exhibits divergent characteristics, thereby contravening the classical Lax
entropy condition. The persistence of the expansion shock in initial value
problems is analysed and justified using matched asymptotic expansions and
numerical simulations. The expansion shock's existence is traced to the
presence of a non-local dispersive term in the governing equation. We establish
the algebraic decay of the shock as it is gradually eroded by a simple wave on
either side. More generally, we observe a robustness of the expansion shock in
the presence of weak dissipation and in simulations of asymmetric initial
conditions where a train of solitary waves is shed from one side of the shock.
