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Humpback whales summer at high latitudes where they feed and winter at low latitudes where they aggregate for breeding. While on the breeding grounds in Hawaii, male humpbacks space themselves and "sing" the long "songs" that have fascinated scientists and poets. Female humpbacks are outnumbered by males and generally ignore or avoid singing males; however, singers often abruptly stop singing and then swim quickly toward distant nonvocalizing whales. Humpback song has usually been explained as sexual advertisement. The suggestion that song is a form of sonar has been dismissed by cetologists on the grounds that the humpback whale song does not resemble the click trains used by toothed whales for echolocation, and that the signal-to-noise ratios of echoes from humpback song are too low for detection. In this paper, we examine humpback whale song and associated behaviors in the light of modern advances in underwater sound and signal processing, buttressing our earlier conclusion that humpback whale song is a long-range sonar used by male humpbacks to locate other whales on the breeding ground. Simulations with the parabolic equation show that the acoustic environment of the breeding ground (100-500 m water depth, weak surface duct, and sandy bottom) is often excellent for sonar in the 4-5 octave band of humpback song, enabling singers to locate other pods at ranges up to 6 km. All humpback whale behaviors are consistent with the sonar model. In particular, the sonar model explains how singing males find nonvocalizing females, despite the fact that females generally ignore or avoid singers. It also explains why males hardly ever sing while in the company of females or while competing with other males for the position of primary escort. A secondary purpose of the paper is to allay a confusion in the cetology literature between specialized structures for high frequency localization and echolocation itself, which takes place in the brain. Modern sonar algorithms, which d- not require click-type sources and which appear to be mimicked in more robust forms in the brains of bats, suggest that, when dealing with any cetacean, it is always more conservative to presume that echoes are being used than that they are not being used. Many cetacean vocalizations currently thought to be exclusively for communication undoubtedly also have a sonar function. An experiment is proposed to test the humpback sonar model.