Accelerations exceeding 20g in surface waves have been observed both in experiments and in numerically computed flows with a free surface. The present paper describes a family of analytic solutions which display such behaviour. They are expressible in parametric form as z = F sinh ω + iG cosh ω + γω + iH, where F, G and H are functions of the time t only, and γ is linear in t. ω is a complex parameter which is real at the free surface. The functions F(t) and G(t) satisfy two nonlinear, coupled ODEs, which can be solved numerically. Typically the solutions pass through an ‘inertial shock’, or singularity in the time, where the displacements vary as t2/3, the velocities as t1/3 and the accelerations as t-4/3. In this class of solution the free surface develops a cusp as t → ∞. In a special case, F and G vary as t4/7 and the cusp is reached in finite time. Gravity is neglected, but plays a part in setting up the initial conditions for the highly accelerated flow.

In future papers it will be shown that more general solutions exist in which the acceleration is momentarily large but bounded.