  Capacity scaling laws are analyzed in an underwater acoustic network with $n$
regularly located nodes on a square, in which both bandwidth and received
signal power can be limited significantly. A narrow-band 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. Cut-set upper bounds on the throughput
scaling are then derived in both extended and dense networks having unit node
density and unit area, respectively. It is first analyzed that under extended
networks, the upper bound is inversely proportional to the attenuation
parameter, thus resulting in a highly power-limited network. Interestingly, it
is seen that the upper bound for extended networks is intrinsically related to
the attenuation parameter but not the spreading factor. On the other hand, in
dense networks, we show that there exists either a bandwidth or power
limitation, or both, according to the path-loss attenuation regimes, thus
yielding the upper bound that has three fundamentally different operating
regimes. Furthermore, we describe an achievable scheme based on the simple
nearest-neighbor multi-hop (MH) transmission. We show that under extended
networks, the MH scheme is order-optimal for all the operating regimes. An
achievability result is also presented in dense networks, where the operating
regimes that guarantee the order optimality are identified. It thus turns out
that frequency scaling is instrumental towards achieving the order optimality
in the regimes. Finally, these scaling results are extended to a random network
realization. As a result, vital information for fundamental limits of a variety
of underwater network scenarios is provided by showing capacity scaling laws.
