A passenger on a United Airlines flight from Chicago to Detroit waited for the stewardess to finish handing out the coveted "half-fare" coupons before he approached several Japanese businessmen seated nearby.

"You won't be needing these pieces of paper," the man reportedly told the foreigners who willingly handed over their coupons. But a second passenger angrily interrupted the first and ordered him to return the valuable coupons.

Since May 28, when United and American Airlines began handing out the discount fare cards to passengers on its domestic flights, the friendly skies and bustling airports around the country have been the scene of coupon mania.

"They're doing all sorts of crazy things," one travel agent said yesterday in reference to bargain-hunting passengers. "You wouldn't believe it."

United Airlines, in a promotional effort to win back passengers after its 58-day mechanics strike, already has handed out 600,000 coupons that entitle passengers to fly half-fare on any domestic flight between July 1 and Dec 15. By June 17. the cutoff date, United expects to distribute 4 million coupons.

What makes the coupons so valuable is that they can halve the fare of any United or American flight regardless of the price of the passenger paid for his initial ticket. That means fliers can purchase a ticket for a little as $21 - United's cheapest flight - and later trade their coupon for half the value of a round-trip, first-class transcontinental flight, the most expensive flight, and save $225.

"They're going like hotcakes," United spokesman Jim Linse said in Washington.

American Airlines, in an effort to keep the passengers it gained during the United strike, matched its competitor's offer with coupons of its own.

The coupons, legally transferable, currently are being bought, sold, scalped and hoarded by passengers who have created a black market for the tiny numbered slips of paper.

Cashing in on the coupon craze are travel agents, individuals who have begun placing coupon "want ads" in newspapers college students and unemployed travelers who spend the day flying from city to city collecting coupons.

"The whole thing is hysterical," said Linda Divall, who flew home to Washington yesterday from Detroit. "They're solid gold.People are hoarding them. And if the stewardess doesn't hand them out immediately after takeoff, the passenger start to get nervous. I saw a lot of little red lights go on. People say, 'Where's the coupon?'"

American Airlines at first handed out the coupons in the terminals. "We stopped that," a spokesman said yesterday, "after we discovered several travel agents who were buying 50 tickets at a time. The flights were always the shortest hops from the airports. The travel agents would throw the ticket away, keep the coupon and never board the flight."

Examples of coupon mania are widespread:

Bill King, a self-described stock market observer who placed an ad in the Los Angeles Times this past weekend that offered "top dollar for airline coupons." King said yesterday the response had been good and he had purchased "less than 100" coupons for $5 each in four days. King said he was planning to hoard the coupons until they increase in value, probably by next month.

Passengers pick the airlines' shortest hops to collect coupons. United's shortest flight is in Virginia, between Newport News and Norfolk separated only by the Hampton Roads harbor. A spokesman said this week that a dozen people had shown up one day for the 20-minute hop, which costs $21 and is a continuation of a longer flight.

Two Secret Service men, waiting at the United baggage claim area at National Airport yesterday, said sadly they couldn't keep their coupons and they could not sell them either. According to a spokesman for the general Services Administration, all federal employes traveling on official business have been ordered to turn in their half-fare coupons to their individual agencies. The government will then distribute the coupons to other employes traveling on official business later in the year.

An American Airlines spokesman said many private companies are doing the same thing, asking the executives to turn in the coupons with their expense accounts so the corporation can use the airline discount.

"It's enough to give travel agents gray hair," said Stephanie Butler, an agent with Thomas Cook Travel, in Washington. Butler said yesterday that the half-fare tickets will have to be processed by hand, and that the 50 percent discount to consumers will reduce the travel agents' commission.

The coupons, she said, "will take more time for much less money."

But Linse, the United spokesman, said yesterday the half-fare coupons were working. "We had three consecutive days last week of record booking across the nation," he said.

Although the airline doesn't encourage the buying and selling of the coupons, it does want to fill empty seats on its flights, "We want them (the coupons) to be used," Linse said yesterday. "A lot of people, though, aren't giving them up."

Like Jim Marshall, who flew into National Airport this week from Detroit displaying a wallet full of half-fare coupons. "I fly 150,000 miles a year for my job," he said. "I plan to use all of these."

Jeff Crawford, 22, of Houston flew into National with a straw cowboy hat and backpack. "Some guy on the plane offered me $10 for my coupon, but I wouldn't sell it," he said, adding that he plans to spend the summer traveling.

Bernice Haight, who flew from Detroit to visit her daughter in Washington this week, said she wouldn't sell her coupon for $100, $200, even $300. "It's mine," she said defensively. "And I'm going to keep it."

All of which Linse finds amusing. "She could have made made more than she was going to save," he said.

Randall Malin, vice president for passenger sales at American Airlines, said yesterday that he had opposed the half-fare coupons. "But we felt we had to match United," he said.

Trans World Airlines considered a similar plan and was embarrased when The New York Times this week mistakenly published a TWA advertisement offering coupons. A TWA spokesman said yesterday that the company was still trying to determine how the ad, promising half-fare coupons and $25 bonus discount, got into the newspaper. The spokesman, repeating the company's position, said the half-fare coupons were "crazy, uneconomical and discriminating." CAPTION: Picture, United Airlines 50 percent discount coupon, given to buyer of regular ticket.