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Time-varying significant multipath interference is the major obstacle to reliable data communication in shallow water acoustic channels. In this paper, the performance of a low density parity check (LDPC) coded orthogonal frequency division multiplexing (OFDM) communication system is investigated for these channels. The initial message for LDPC, decoded by using the belief propagation (BP) algorithm, is deduced for OFDM underwater acoustic channels; based on this deduction, the noise thresholds of regular LDPC codes with different code rates are obtained by using the density evolution algorithm. Furthermore, a communication system model, developed with LDPC code, OFDM and channel interleaver for shallow water acoustic channels, is introduced. The effect of modulation and coding schemes on the LDPC codes performance is investigated by simulation. The results show that the system can achieve remarkable performance in shallow water acoustic channels, and the performance improves with increasing code length and decreasing code rate. The bit error rate (BER) of the system, under conditions with QPSK modulation, 1280-code length and 1/2-code rate, is less than 10-5 when the signal to noise ratio (SNR) is greater than 6.8dB. These values are obtained for a five-path shallow water acoustic channel of Xiamen harbor. [ABSTRACT FROM AUTHOR]