In this paper we propose a novel method for communication over underwater acoustic channels that exhibit simultaneously large delay spread and Doppler spread, such as those found in the surf zone. In particular, we propose a coded pulse-shaped multicarrier scheme that converts the doubly dispersive channel into an inter-carrier interference (ICI) channel with small ICI spread. The resulting ICI is mitigated using a soft noncoherent equalizer that leverages sparsity in the delay-power profile to generate near-optimal bit estimates with low complexity. The noncoherent equalizer uses a delay-power-profile estimate (rather than a channel estimate) which is obtained from pilots. Numerical simulations with surf-zone-like channels demonstrate performance close to genie-aided bounds.