By Holly Yan and Gul Tuysuz, CNN
The U.S.-led airstrikes in Syria on Tuesday were just the beginning. There will be more to come, President Barack Obama vowed.
But on Wednesday, he'll likely have to make a case for them.
Obama will speak at the United Nations General Assembly to defend his decision to bomb terror groups inside Syria without approval from the U.N. Security Council, Congress or an invitation from Syria.
He will address the overall unease in the world given the new dangers posed by groups like ISIS and Khorasan, a senior administration official said.
Much of the president's speech will focus on continuing to build the international coalition he and his administration have already begun to assemble, the official said.
Why not bomb the regime?
While some Syrian residents celebrated the U.S. airstrikes, others expressed frustration that the Syrian regime -- which world leaders blame for thousands of civilian deaths -- goes unscathed.
"I am just wondering why the U.S. didn't bomb the regime's brigades," Foaad Hallak said in Aleppo.
"If the international community is willing to show their good intentions to Syrians, they have to bomb the regime and its militias and also ISIS, and also they have to supply FSA (the rebel Free Syrian Army) with anti aircraft missiles."
Hassan al-Halabi, an activist from Aleppo, supported the attacks against ISIS but also said more needs to be done.
"The people in Aleppo do believe that the international community needs to be responsible against the daily killing machine that is operated by Syrian regime against civilians and kids in all Syrian cities," he said.
Military actions 'necessary and appropriate'
On Tuesday, Obama said the U.S. is doing what it must to "take the fight to terrorists."
"We will not tolerate safe havens for terrorists who threaten our people," he said.
In a letter to the U.N. secretary-general, the U.S. defended its actions, invoking Article 51 of the U.N. charter -- acting when a country is unwilling or unable to handle a threat itself.
"The Syrian regime has shown that it cannot and will not confront these safe havens effectively itself," Samantha Power, the U.S. ambassador to the U.N., wrote in a letter obtained by CNN. "Accordingly, the United States has initiated necessary and proportionate military actions in Syria."
But Iran has already lashed out against the campaign. Russia has criticized the U.S. for its foray into the airspace of a sovereign nation.
And some in Congress -- even from Obama's party -- are grousing as well.
"If Congress allows the president to begin this campaign against ISIL and as he said go on offense against ISIL, without Congress authorizing it, we will have created a horrible precedent," said Sen. TIm Kaine, a Democrat from Virginia.
The airstrikes Tuesday came in three waves, with coalition partners participating in the latter two, Army Lt. Gen. William Mayville Jr. said Tuesday.
The first wave mostly targeted the Khorasan Group, whom Obama described as "seasoned al Qaeda operatives in Syria."
U.S. officials said the group was plotting attacks against the United States and other Western targets.
The second wave, 30 minutes later, involved planes striking northern Syria, with targets including ISIS headquarters, training camps and combat vehicles.
The third wave involved planes targeting ISIS training camps and combat vehicles in eastern Syria, Mayville said.
Bahrain, Saudi Arabia, the United Arab Emirates and Jordan took part in airstrikes on the ISIS targets, the U.S. military said. Qatar played a supporting role, the U.S. military said.
It's too early to say what effect the U.S. strikes had against the Khorasan Group, Mayville said.
The ISIS attacks destroyed targets including fighters, training compounds, command-and-control facilities, a finance center and supply trucks, the U.S. Central Command said.
The airstrike appear to have taken a toll on another terror group, killing the leader of the al Qaeda-linked al-Nusra Front, according to a statement released by the group. It identified the leader as Abu Yousef al-Turki, also known as "The Turk."
The United States has not identified al-Nusra as a group targeted in the strikes.
But al-Nusra statement posted a statement on Twitter, accompanied by a so-called proof-of-death -- a photograph -- of the former fighter.
Now, concern over a possible backlash by the terror groups has prompted the U.S. Department of Homeland Security to warn law enforcement agencies to be on high alert for lone-wolf terror attacks in the United States in wake of the airstrikes, a U.S. law enforcement official with knowledge of the warning told CNN.
The bulletin calls for vigilance as well as scrutinizing social media for anyone encouraging violence in response to the strikes.
It points to the use of social media as a tactic by ISIS to spread its message and call for violence.
It also advises agencies to look for changes in appearance or behavior in those they're tracking, the official said.
Fears on the ground
Al-Halabi, one of the activists from Aleppo, said residents have two fears about upcoming strikes in Syria.
"The first is that they are afraid of having civilian casualties because ISIS' members and fighters are among civilians," al-Halabi said.
"And the second concern is that what will happen after that? Who will replace ISIS, especially that the regime is ready to take control of ISIS' areas?"
CNN's Jim Acosta, Pamela Brown, Jake Tapper, Chelsea J. Carter, Raja Razek, Hamdi Alkhshali, Barbara Starr and Deb Feyerick contributed to this report.