One year after his accession to the throne as the fifth ruler of the modern-day House of Saud, King Fahd has left his countrymen and outside observers alike puzzled by his unexpectedly lackadaisical style of rule and seeming indecisiveness.

Following a smooth transition of power on the death of King Khalid last June 13, the kingdom still awaits a clear sense of direction and purpose from its new monarch whose past dynamism as effective day-to-day ruler had generated great expectations.

"Saudi Arabia is like a horse waiting for his rider to take up the reins," remarked one western diplomat. "And this horse is used to tight reins."

Nothing seems settled or certain in the kingdom today, neither the level of its vastly reduced oil wealth, the effects of a mandatory sharp cutback in spending, the fate of promised political reforms nor the division of responsibilities among leading members of the royal family.

By all evidence, Fahd has assumed power at a time of extreme testing for the kindgom and royal family. The era of oil, which gave rise to the fabulous wealth of Saudi Arabia today, appears to be at least temporarily in decline. The country's annual income of over $100 billion has been cut by perhaps 60 percent in two years.

Saudi influence in the Arab world and in Washington seems to be sliding along with oil prices, while a new wave of Arab radicalism, born of the failed American and Saudi Middle East peace initiatives, now looms as a threat to the kingdom's security.

At home, the king's first year of rule has been marred by a spate of wild rumors of arrests, intrafamily conflicts and coup attempts. Most have been false or exaggerated, but multiple sources confirm there is more than a grain of truth to reports that the king and the new crown prince, Abdullah, have had their troubles working together and even quarreled over their respective roles. Accounts of the rivalry between Fahd and Abdullah, members of different clans within the royal family, have circulated for years, well before each acceded to his present position.

Just how popular the new king is with his subjects is difficult to tell. He was reportedly warmly received last November in Riyadh on his belated arrival there after becoming king.

But the total absence of any crowds turning out to greet him during his long-heralded visit in mid-May to the eastern province seems to indicate little enthusiasm for him, at least in the strategic region where the kingdom's oil resources are centered along with a large minority Shiite Moslem population.

Faced with brewing storms at home and abroad, Fahd has not projected the anticipated image of a particularly strong helmsman, and seasoned royal family watchers here are perplexed.

"I don't think its clear what kind of king he is going to be," remarked another diplomat. "It's been a disappointing year. . . It's not a question of serious blunders, just missed opportunities, really."

Perhaps the biggest disappointment to the reform-minded, western-educated middle class of growing size and strength has been the monarch's catering to the ultraconservative ulema, or spiritual leaders, of the powerful religious establishment. They have unleashed a new crackdown on breaches in the adherence to the kingdom's strict brand of Islam known as Wahabism, apparently with his approval.

Long regarded as the prime mover of the country's headlong rush into the modern era, Fahd had been widely expected to support a liberalization of Wahabi social mores, which still proscribe women from driving a car or working with men.

"People expected him to be more liberal in his ways," remarked a foreign resident living in the eastern province. "This hasn't happened. Nobody knows why exactly. There are many rumors and theories."

Chief among them is that the king feels too vulnerable because of his own reputation as a lover of the western-style good life to oppose now the ulema's campaign for renewed rectitude.

Whatever the reasons, the king has gone out of his way to prove his own Islamic credentials, issuing decrees to enforce prayers during work hours and a stricter segregation of women in public places.

Among the many quirks of the new king has been his apparent strong aversion to the kingdom's capital, Riyadh, the heartland of the Saud dynasty and also of the Wahabi ulema.

Since coming to power a year ago, the king has only resided here a few weeks and did not come to receive his royal welcome until five months after his advent to power. Instead, he has favored the more easy-going seaside city of Jeddah, the mountain-top summer resort capital at Taif or his private desert camp 160 miles north of Riyadh, where he spent six weeks in March and April.

His peripatetic style has also included several long stays abroad, first last summer and then again for five weeks starting in late November. Part of the latter trip was devoted to a successful mediation effort between Morocco and Algeria, but part, too, to vacationing in Spain where he has built one of his many palaces.

Saudi watchers thought at first the king was making use of the solitude of his desert camp and stays abroad to map out a broad strategy of political and other reforms he repeatedly has promised to make. But other than five Cabinet changes and the start of a crackdown on corruption and waste inside the government, no blueprint for a "new Saudi era" has emerged.

The most prominent of the Cabinet switches so far was the "resignation" April 13 of the Saudi Arabian monetary agency chief, Abdel-Aziz Quraishi, after nine years of service, for reasons that remain unclear. His assistant, Hamid Sayari, is now acting governor.

On April 24, Information Minister Abdo Yamani was abruptly dismissed, reportedly for giving too detailed a denial the day before of a French press report of yet another aborted coup. Yamani, known jokingly for years as the kingdom's "minister of denials," was replaced by Ali Shaer, the former Saudi ambassador to Lebanon who is widely regarded as a vast improvement.

Above all other issues, the king has not said a word about the establishment of a shura, or consultative, council and the publication of the basic statutes of government giving the kingdom a kind of constitution and a larger degree of democracy. A draft of these reforms has been on the king's desk now for at least 18 months, and Fahd himself said they would be promulgated by last May or June.

The death of King Khalid was the apparent reason for the initial postponement of any action on the reforms, but there are growing doubts now among Saudis and foreigners alike that Fahd intends to take any decision soon. Nor does there seem to be much visible public pressure any longer for these reforms, while the ulema are believed to constitute a strong lobby against them.

Sorting out reality from fiction is not easy in this highly secretive kingdom, where rumor runs rampant and official denials are seldom convincing. But diplomats of all nationalities seem to concur that widespread press reports this past winter of several coup attempts against the king were "nonsense."

They do confirm other reports of considerable tension between the king and the crown prince over the role the latter should play in the new family lineup and power structure. These reports became so persistent that Prince Abdullah felt obliged in late March to deny them publicly in a remarkably frank interview with the Kuwaiti newspaper Al Siyassah. But he did so in such a way that most Saudi watchers concluded there was more truth than falsehood to them.

While denying there were any "ambitions for power to create conflicts among the kingdom's ruling personalities," Abdullah also glaringly omitted any praise for his half-brother, Fahd, as is the usual royal Saudi practice.

Experienced Saudi observers say the conflict between the two half-brothers, whose relationship has always been somewhat prickly, involves two interrelated issues. First is whether Abdullah is to take on the role Fahd played as virtual day-to-day ruler while he was crown prince for seven years under the ailing Khalid. The second is whether Abdullah should give up his position as commander of the 20,000-man paramilitary National Guard.

Abdullah has reportedly been pushing to have the same authority Fahd wielded as crown prince while refusing to yield control of the guard. The king is said to believe Abdullah should give up his command of the guard just as he was obliged to forgo the post of minister of interior upon becoming crown prince in 1975.

According to one report, Abdullah agreed to yield his command of the guard, but only if he could appoint the successor, and if Defense Minister Prince Sultan, a potential rival for the throne though officially second in line, also stepped down from his defense post.

The betting of most Saudi analysts is that both princes will retain control of their respective military commands, thus preserving the existing balance between the two and peace within the royal family.

But there is also a general feeling that Abdullah, whose abilities were being questioned even in some quarters of the royal family, has proven himself over the past year and will not be opposed as the next king upon Fahd's demise.

"Abdullah has consolidated his position and is now a credible candidate for the throne ," remarked one western analyst assessing his performance. "He has been on the job running Cabinet meetings in Fahd's absence and receiving guests to the kingdom. He has gotten broader exposure as a national leader."

Part of the problem seems to be that Fahd himself has not yet decided what kind of king he wants to be. He clearly has little love or patience for time-consuming ceremonial duties as was obvious during his visit to the eastern province witnessed by this reporter.

The greeting of citizens and partaking in banquets to meet the people, ceremonies that go with such royal occasions, seem to hold little interest for him.

At the same time, Fahd showed a clear preference for dealing with weightier matters of state, chatting throughout a "people's banquet" with a special envoy from Libyan leader Muammar Qaddafi and entertaining other important international guests at his new $1 billion palace complex outside Dhahran.

Criticized at first for keeping all decision-making to himself, the king recently has begun to give the crown prince and others somewhat more authority while still remaining very much both prime minister and king.

Despite tensions among the top members of the royal family, outside observers do not seem to regard them as either destabilizing or serious enough to provoke a dangerous rift within the Saud family. They note that the family has always had a keen sense of survival and never in the past allowed internal squabbles to get out of hand.

Recently, in fact, the king and crown prince seem to have gone out of their way to refute the reports of discord by making sure they are seen together in public and acting in cooperation. In addition, the king has included Abdullah more regularly in his meetings with important visitors such as Syrian President Hafez Assad and U.S. Secretary of State George P. Shultz.

While the royal family sorts out the division of authority among its top members, Fahd has finally begun to make some long-expected government and Cabinet changes, but in piecemeal fashion rather than one big shake-up.

Before the changes at the monetary agency and the Interior Ministry this spring, a popular appointment last October was that of Ghazi Gosaibi, the minister of industry, as acting health minister as well. He has launched a cleanup campaign of corrupt hospital and health officials that has earned him wide praise and the reputation of a determined reformer.

There is intense speculation that other changes will be announced shortly, possibly involving the kingdom's internationally known and highly respected Oil Minister Sheik Ahmed Zaki Yamani and several high ranking royal family members.

The changes already made generally have been well received and raised hope anew that the king may still be serious about giving the kingdom a better image. But how far he really intends to push reform will not be clear until more appointments have been made and concrete actions taken.