The week brings some very dour news for the Hezbollah killers and their Iranian masters. No, it is not that the Lebanese government has ordered them to take their rocket launchers off the school grounds and out of the hospital parking lots. No, they remain free to use Lebanese as human shields while “world opinion” dizzily blames Israel for the civilian casualties.

The bad news comes from Washington. There our debonair president just had his annual physical, and his blood pressure was a mere 108/68. For a man his age a normal blood pressure is 130/90. Yes, after a weekend of counseling with his advisers on the troubles in the Middle East our Commander in Chief remained notably nonchalant, clinically nonchalant. And just before taking that blood pressure reading he announced that the world pressure on Israel to suspend hostilities in Lebanon “won’t address the root causes of the problem.” President Bush still favored the Israeli offensive to extirpate Hezbollah from Lebanon.

This war against Hezbollah, launched by Israel in response to Hezbollah’s assault on Israeli soldiers and following years of Hezbollah’s refusal to comply with U.N. Resolution 1559 requiring militia’s to disarm has not gone as well as we would like. In the Wall Street Journal Bret Stephens, the extremely well-informed former editor of the Jerusalem Post, announced “Israel is losing this war … if it keeps going as it is, Israel is headed for the greatest military humiliation in its history.” After expressing confidence that Israel will eventually triumph, Mr. Stephens diagnosed Israel’s problems as stemming from hesitancy, early intelligence failures, and over-reliance on what others have called “surgical bombing” as opposed to the early use of infantry.

Now the ground troops have moved in. There is no more talk of ending this in short order but rather of ending this incursion when Hezbollah has been swept from southern Lebanon. The destruction of the Hezbollah infrastructure is being handled by infantry and the soldiers on the ground — with the help of high-tech intelligence gatherers such as drones — are intent on arresting or killing the Hezbollah leadership. Doubtless informants and spies are also at work too.

The fear remaining is that Hezbollah will now be able to portray itself as having withstood an Israeli assault tolerably well. Commentators suggest that without a coup by the Israelis a battered and uprooted Hezbollah will become a legitimate force in Lebanese politics. Well, perhaps, but once the dust has settled on the rubble that Hezbollah invited, perhaps the Lebanese who were wary of Hezbollah to begin with will draw the proper conclusion. Terrorists who refuse to disarm and who fight as guerrillas from civilian neighborhoods are dangerous neighbors. In fact, the way they allowed Lebanese neighborhoods to be used they are worse than dangerous neighbors.They are the enemies of democratic Lebanon.

From the war zone, The New York Sun’s Eli Lake reports that Israeli psychological warfare has been clever and effective. On Hezbollah’s Al-Manar television station Israeli intelligence wizards hack in with threatening messages of imminent death to the Hezbollah leadership and pictures of Israeli air strikes with the accompaniment of a voice saying, “There is no doubt that this is the strongest air force in the area. And we can’t stand up against it.”

Now the Israelis’ goal is to continue to push Hezbollah back and to destroy the terrorists’ infrastructure. A second goal is to instill the fear in prospective enemies that they cannot win. This will take a week or more, the military believes. A question that will remain unanswered for months is what will Hezbollah’s standing be in the Middle East in the long run. One thing is clear. Israel has a very cool ally in the White House.

Mr. Tyrrell Jr. is founder and editor in chief of The American Spectator, a contributing editor of The New York Sun, and an adjunct fellow at the Hudson Institute.