Information-theoretic throughput scaling laws are analyzed in an underwater acoustic network with n regularly located nodes on a unit square, in which both bandwidth and received signal power can be severely limited. A narrow-band model is assumed where the carrier frequency is allowed to scale as a function of n. We first characterize an attenuation parameter that depends on the frequency scaling as well as the transmission distance. In the dense network having unit area, a cut-set upper bound on the capacity scaling is then derived. We show that there exists either a bandwidth or a power limitation, or both, according to the path-loss attenuation regimes, thus yielding the upper bound that has three fundamentally different operating regimes. In the dense network, we also describe an achievable scheme based on the simple nearest-neighbor multi-hop transmission. The operating regimes that guarantee the order optimality are identified, where frequency scaling is instrumental towards achieving the order optimality in the regimes.