But neither does the White House want to cede the South China Sea to China, which is what administration officials fear will happen if Beijing continues on its current course. James R. Clapper, Mr. Obama’s director of national intelligence, told the Senate Armed Services Committee last month that by early next year China would “have significant capacity to quickly project substantial military power to the region.”

That could mean that other countries might eventually need Beijing’s permission to traverse the heavily trafficked sea.

And so for the moment, the Obama administration is sending Navy patrols through the Spratlys and other disputed island chains in the region, to drive home the message that the sea is free to all. Some 700 American patrols have gone through in the past year, Navy officials say. Three weeks ago the aircraft carrier John C. Stennis and four other American warships sailed into the South China Sea for routine exercises, meant to convey the message, Pentagon officials said, that the United States is the dominant military power in the region.

Aboard the Chancellorsville last week, the minutes — and the tension — stretched out as the Chinese helicopter pilot refused to answer. The helicopter kept circling and eventually flew back to the Chinese frigate, which then continued toward the American warship. At the helm, Capt. Curt A. Renshaw, who had skipped his morning shower to race up to the bridge when the Chinese helicopter approached, huddled with his officers.

The day before, Captain Renshaw had warned the entire ship over the intercom that the Chancellorsville would be transiting through the Spratlys, and told the crew members to be on their toes and alert to trouble. He had been expecting the Chinese to show up — Beijing, in recent months, has taken to shadowing American warships that have dared to enter the South China Sea.