Capacity scaling laws are analyzed in an underwater acoustic network with n regularly located nodes. A narrowband model is assumed where the carrier frequency is allowed to scale as a function of n. In the network, we characterize an attenuation parameter that depends on the frequency scaling as well as the transmission distance. A cut-set upper bound on the throughput scaling is then derived in extended networks. Our result indicates that the upper bound is inversely proportional to the attenuation parameter, thus resulting in a highly power-limited network. Furthermore, we describe an achievable scheme based on the simple nearest-neighbor multi-hop (MH) transmission. It is shown under extended networks that the MH scheme is order-optimal as the attenuation parameter scales exponentially with √n (or faster). Finally, these scaling results are extended to a random network realization.