The 1,200-tonne Cheonan sank off the west coast of the divided peninsula, in an area which has seen deadly exchanges of fire between the navies of the North and South in the past.
But there were no North Korean vessels in the area, and Joint Chiefs of Staff spokesman Park Sung-Woo said there was no sign of any "abnormal movement" in the North.
A group of 150 angry relatives protested outside the Pyeongtaek naval base south of Seoul at the lack of information they have been given, the AP news agency reports.
They pushed their way past security guards, shouting "Liars".
The ship's rescued captain has been recounting what happened.
"There was the sound of an explosion and the ship keeled to the right. We lost power and telecommunications," South Korea's Yonhap news agency quoted Choi Won-il as saying.
"I was trapped in the cabin for five minutes before my colleagues broke the window in and let me out. When I got out, the stern had disappeared."
The divers were due to investigate the cause of the explosion and retrieve bodies, a South Korean military spokesman said.
Navy and coast guard vessels, as well as air force planes, were scouring the area near South Korea's Baeknyeong Island.
The AP news agency quotes an unnamed coast guard official as saying that humans could only survive in the Yellow Sea at this time of year for about two hours. He said the water would be between 3 and 5C.
The Cheonan began sinking at about 2130 local time (1230 GMT) on Friday. Only a small part of its overturned hull remains visible.
A number of the crew jumped into the water, Yonhap said.
"Yells and screams filled the air," witness Kim Jin-ho, a seaman who was on a local passenger ship bound for Baeknyeong, told cable news channel YTN.
"Marines on deck were desperately shouting: 'Save me!"'
Some family members are starting to fear the worst
South Korean President Lee Myung-bak convened an emergency meeting of security officials and said all possible causes for the sinking would be investigated.
He ordered the military to focus on rescuing the sailors.
There were initial reports that another South Korean ship had fired shots toward an unidentified vessel, but officials later speculated the target had been a flock of birds.
The incident comes at a time of tension between the two Koreas. International talks aimed at ending the communist North's nuclear ambitions have been stalled for months.
North and South Korea are still in a official state of war because the 1950-53 Korean War ended only in a truce.
Since then, they have fought three bloody skirmishes in the Yellow Sea.
In January, North Korea fired about 30 artillery shells not far from Baeknyeong. South Korea fired 100 warning shots in response, but no injuries were reported.
South Korea recognises the Northern Limit Line, drawn unilaterally by the US-led United Nations Command to demarcate the sea border at the end of the Korean War. The line has never been accepted by North Korea.
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