By Helene Cooper

ABOARD THE U.S.S. CHANCELLORSVILLE, in the South China Sea >> The Navy cruiser was in disputed waters off the Spratly Islands when the threat warning sounded over the ship’s intercom: “Away the Snoopie team. … Away the Snoopie team.”

As the sailors of the “Snoopie team” went on alert and took up positions throughout the ship, a Chinese naval frigate appeared on the horizon, bearing down on the cruiser Chancellorsville last week from the direction of Mischief Reef. More alarming, a Chinese helicopter that had taken off from the frigate was heading straight for the U.S. cruiser.

“This is U.S. Navy warship on guard,” Ensign Anthony Giancana said into his radio from the ship’s bridge, trying to contact the helicopter. “Come up on Frequency 121.5 or 243.”

Ominously, there was no response.

Here in the hot azure waters off the Spratly and Paracel Islands — which encompass reefs, banks and cays — the United States and China are jockeying for dominance in the Pacific. From Mischief Reef, where China is building a military base in defiance of claims by Vietnam and the Philippines, to Scarborough Shoal, where the Chinese are building and equipping outposts on disputed territory far from the mainland, the two naval forces are on an almost continuous state of alert.

Although the South China Sea stretches some 500 miles from mainland China, Beijing has claimed most of it. Tensions have risen sharply, and the topic is expected to dominate President Barack Obama’s meeting in Washington this week with the Chinese president, Xi Jinping.

America’s goal is to keep the South China Sea, one of the world’s busiest shipping lanes, open to all maritime traffic. But administration officials are increasingly worried that tensions will only worsen if an arbitration panel in The Hague rules as expected in the coming months on a 2013 case brought by the Philippines, which has accused China of making an “excessive claim” to most of the sea.

At the Pentagon two weeks ago, the day before a meeting of Obama’s national security team to discuss Chinese expansion in the Pacific, Gen. Joseph F. Dunford Jr., the chairman of the Joint Chiefs of Staff, was talking with Adm. Harry B. Harris Jr., the commander of the U.S. Pacific Command, in the reception area of Defense Secretary Ash Carter.

“Would you go to war over Scarborough Shoals?” Dunford asked Harris, in a conversation overheard by a reporter. If Harris responded, it could not be heard.

The White House and the Pentagon have made clear that they do not want a war with China over a group of uninhabited islands. 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, 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 could 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 U.S. patrols have gone through in the past year, Navy officials say. Three weeks ago, the aircraft carrier John C. Stennis and four other U.S. 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 U.S. 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, Renshaw had warned the entire ship over the intercom that the Chancellorsville would be transiting through the Spratlys, and he 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 U.S. warships that have dared to enter the South China Sea.

On a stand near the captain’s chair, a copy of “Jane’s Fighting Ships” was open to Page 144: “China Frigates.”

“You’ve ever been shadowed before?” Renshaw asked Ensign Kristine Mun, a navigations officer. He turned to Ensign Niles Li, one of several officers who speak Chinese, and wondered aloud at the Chinese helicopter’s refusal to answer the radio message.

Finally, when the Chinese frigate was 6 miles away and clearly visible to the naked eye on the horizon, the ship-to-ship radio crackled with the sounds of accented English. “U.S. Navy Warship 62. … This is Chinese Warship 575.”

And so began an elaborate diplomatic dance.

“This is U.S. Warship 62. Good morning, sir. It is a pleasant day at sea, over.”

No response.

“This is U.S. Warship 62. Good morning, sir. It is a pleasant day to be at sea, over.”

Still no response.

Renshaw turned to Li. “You’re up,” he said. “They can’t pretend they don’t speak Chinese.”

“Chinese Warship 575, this is U.S. Warship 62,” Li said in Chinese. “Today is a sunny day for a sea voyage, over.”

More minutes ticked by. Ensign Anthony Giancana, the junior officer on deck for the morning, was getting antsy. “This is like Opening Day,” he said to no one in particular. “We’ve done spring training.”

Suddenly, the radio crackled again as the frigate responded in Chinese: “U.S. Warship 62, this is Chinese Warship 575. Today’s weather is great. It is a pleasure to meet you at sea.”

Li responded, also in Chinese: “This is U.S. Warship 62. The weather is indeed great. It is a pleasure to meet you, too, over.”

Preliminaries dispensed with, the Chinese ship got down to business, switching to English. “How long have you been since departing from your home port? Over.”

Renshaw was immediately shaking his head. “No, we’re not answering that. I would never ask him that.”

Giancana picked up the radio again. “Chinese Warship 575, this is U.S. Navy Warship 62. We do not talk about our schedules. But we are enjoying our time at sea, over.”

And on it went as the two warships, each loaded with missiles, torpedoes and heavy artillery, confronted each other with an exchange of weather pleasantries at sea. Testing whether the Chinese were openly following, the Chancellorsville made a turn, and its officers stood back and waited.

A shout came from another one of Renshaw’s junior officers: “He just turned, sir!” The Chancellorsville now had a tail. But for how long?

Apparently the Chinese ship wanted an answer to that question as well.

“U.S. Navy Warship 62, this is Chinese Navy Warship 575,” came a new message. “Do you continue to have long-term voyage at sea? Over.”

Another no-no. Telling the Chinese the intended duration of the trip could be an inherent acknowledgment that they had the right to know, Renshaw said. And that is not considered freedom of navigation.

“This is U.S. Navy Warship 62,” Renshaw responded. “Roger, all of our voyages are short because we enjoy our time at sea no matter how long we are away from home. Over.”

As it happened, the Chinese ship had a ready answer for that.

“U.S. Navy Warship 62, this is Chinese Navy Warship 575,” came the reply. “Copy that I will be staying along with you for the following days. Over.”

That was Tuesday. On Wednesday, the Chinese frigate was replaced by a destroyer, which followed the Navy warship until midnight Thursday, when the U.S. vessel exited the South China Sea.

© 2016 The New York Times Company