Sailors with the U.S. Sixth Fleet call it chicken of the sea. It is a seaborne version of the highway hot-rodders' "chicken" that is frequently played in the crowded Mediterranean by Soviet and American warships. Typically, a Russian vessel will dart and weave among U.S. ships, trying to frighten their skippers into turning sharply to avoid collision. These episodes usually end harmlesslybut not always.

One day last August a Soviet Echo Il-class submarine cruising almost submerged trailed a slowly steaming American frigate, the U.S.S. Voge, for nearly an hour. Suddenly, the sub turned straight toward the Voge and sped up sharply. The...