Underwater wireless networks have recently gained a great deal of attention as a topic of research. Although there have been several recent studies on the performance of medium access control (MAC) protocols for such networks, they are mainly based on simulations, which can be insufficient for understanding the fundamental behavior of systems. In this work we show a way to analyze mathematically the performance of an underwater MAC protocol. We particularly analyze a propagation delay tolerant ALOHA (PDT-ALOHA) protocol proposed recently [1]. In this scheme, guard bands are introduced at each time slot to reduce collisions between senders with different distances to the receiver, which have a great impact on acoustic transmissions. We prove several interesting properties concerning the performance of this protocol. We also show that the maximum throughput decreases as the maximum propagation delay increases, identifying which protocol parameter values realize the maximum throughput approximately. Although it turns out that exact expression for the maximum throughput of PDT-ALOHA can be quite complicated, we propose a useful simple expression which is shown numerically to be a very good approximation. Our result can be interpreted to mean that the throughput of PDT-ALOHA protocol can be 17-100% higher than the conventional slotted ALOHA, with proper protocol parameter values