Asynchronous half-duplex Direct-Sequence Code- Division Multiple-Access (DS/CDMA) is a suitable candidate for the MAC protocol design of Underwater Acoustic (UWA) multi-hop sensor networks due to its many attractive features. However, multi-hop communications is very sensitive to the time variability of fading and the snapping shrimp dominated ambient noise in the harsh underwater environment because a broken link in the multi-hop path is enough to disrupt the communications and initiate new route searches. We analyze the theoretical MAC throughput of the network using the Ad hoc On-Demand Distance Vector (AODV) routing protocol with Poisson distributed, constant length packet arrivals at the application layer. We consider horizontal communications of sensors operating in shallow water near coastal regions. Previous empirical studies show that we can model the UWA channel as a slow-varying Rayleigh fading and frequency non-selective channel. The optimal throughput performance of the network under various operating conditions is then examined.