But if toppling Colonel Qaddafi is now the more explicit goal, Mr. Obama’s European trip this week has highlighted significant tensions over how much time the NATO allies have to finish a job that is now in its third month.

Mr. Obama has urged strategic patience, expressing confidence that over time the combination of bombing, sanctions and import cutoffs will force Colonel Qaddafi from power. “Time is working against Qaddafi,” Mr. Obama said on Wednesday at a news conference in London with Prime Minister David Cameron of Britain.

But in Europe and in Libya, patience is calculated differently. Many countries are struggling with the rapid pace of operations. Some, like Norway, have already said they will sharply reduce their forces beginning next month. According to NATO officials, Colonel Qaddafi has a calculation of his own: facing a possible indictment by the International Criminal Court, he may soon have few places to go and little to lose by waiting out NATO and betting that European public opinion will tire of the bombing campaign and its costs.