In fast temporal variations shallow water acoustic (SWA) channels, flexible channel coding rate should be desired in the design of practical error control SWA communication system. Herein, as one of the dynamic coding schemes, rate compatible LDPC (RC-LDPC) codes combing OFDM technique are proposed to improve the system reliability for SWA communications. Based on the automatic repeat request (ARQ) mechanism, the proposed SWA OFDM system consists of three important preprocessing: channel state information (CSI) estimator, signal-to-noise ratio (SNR) estimator and RC-LDPC pattern. The design of RC-LDPC codes and SWA channel profile are described. A look-up table (LUT) of RC-LDPC performances for ARQ-SWA system is built via statistical simulation test. For the proposed scheme, we define and derive the sensitivity to channel time variations and the effects of different performance target BER threshold, which are further confirmed via numerical results. It is shown that RC-LDPC codes have good performances with wide range of rates in SWA channels. Finally, coding rate distributions of RC-LDPC codes in different SNR at BER below 10 ^-4 for Xiamen harbor SWA channel are investigated.