SORRY TO KIDNAP you like this,” Vanderwaal said, as Phillips was escorted into his office. “But I’m sure you’re aware of the situation.”

     “No need to apologize,” Phillips said. “This is a fascinating development! From what I saw, your terrorist was able to selectively apply eight-cycle sound on a case-by-case or class-by-class basis.”

     “I’m not sure I follow,” Vanderwaal said.

     “Well,” Phillips said, “at first, a single Secret Service agent was affected. Then another single agent. Then the rest of them at the same time.

     “Then dignitaries, one at a time.

     “Then whole rows of VIPs. 

     "This is different from Denver or San Francisco or the depot.”

     “How’d you know about the depot?” Vanderwaal asked.

     “They briefed me on the flight from MIT,” Phillips said.

     “In the armored car robberies, they used unidirectional sound -- that is, sound that radiates from a central point in all directions. In both cases, the epicenter, so to speak, was the conical speakers.

     “In Hawaii, individual drivers who listened to their radios were affected -- but since the volume was relatively low, it only affected them. Drivers who didn’t listen to their radios were affected too, along with the people outside their cars for a radius of three blocks. In all of these cases, the sound was unidirectional.

     “When the president couldn’t speak, that was using different technology. No eight-cycle sound was involved.

     "The army had directional sound. But it killed everything in a straight line for five miles.

     “This is different -- it’s not unidirectional, and it’s not directional either. It’s selective -- almost as if a narrow beam of eight-cycle sound is coming straight down from somewhere.”

     “So how would that work?” Vanderwaal asked.

     “I’m not sure. Some of the spectators said they heard low sounds over the speakers. Some said different speakers made different sounds. If that’s true, it might be a clue. In the armored car robberies, there was only one speaker making the sound.”

     “Well,” Vanderwaal asked, “if they were different sounds from different speakers--”

     “It would all depend on what sounds. Remember the frequencies we used in our hypothetical eight-cycle sound generator?” Phillips asked.

     “104, 112--”

     “That’s enough,” Phillips said. “Sound emanates in a sphere. Think of two balloons being blown up next to each other. As they expand, they do so in the shape of perfect spheres."

     “OK.” Vanderwaal was in the dark, but could see a pinpoint of light in the distance that he hoped would get bigger.

     Phillips was lecturing now. “If when these two balloons get big enough to touch each other and are able to pass right through each other, what would be the shape of their intersection?”

     “If I’m following this,” Vanderwaal said, “where they touch will form a circle that will get bigger as they continue to grow.”

     “Yes!” Phillips said excitedly. “And if we fill that circle in, we’d be filling in a plane -- a sheet -- a flat piece of space that we could define with x-y coordinates.”

     “OK.” Vanderwaal didn’t sound as sure as he did a moment ago with his previous “OK,” but he was still in the game.

     “Along this plane would be eight-cycle sound. As the circle grew to where it hit the ground, it would define a line.” Phillips was thinking out loud, ahead of his illustration.

     “Hold it,” Vanderwaal said. “I think I’m tracking, but I want to be sure. So, if one speaker was blasting 104 cycles a second from the center of the goalpost at one end of the field, and another speaker was blasting 112 cycles per second from the center of the other goalpost, you’re saying anyone standing on the 50-yard line would drop like a rock?”

     “Precisely!” Phillips said. “But if that’s what happened, people in the stands would start dropping, too. They got trampled, but they didn’t drop.”

     He was running his fingers through his hair. It helped him think. “What if we changed the frequencies to 208 and 312?”

     “Don’t tell me,” Vanderwaal said. “OK -- no -- OK. I got it! One hundred four cycles per second would hit the 50-yard line!”

     “You rock, Grasshopper! -- Oh, sorry,” Phillips said “Now hold that thought. What if a speaker at the left side of the 50-yard line was blaring, oh, say 224 cycles a second, and another speaker on the other side of the 50-yard line was blaring at 336?”

     “Wait,” Vanderwaal said, scrunching his forehead into wavy wrinkles -- which may or may not have anything to do with helping him think -- he just did that whenever he was deep in thought.

     “OK. A line right down the center of the field from one goalpost to the other would vibrate at 112 cycles per second?”

     “And--,” Phillips encouraged.

     “I know -- well, at least I think I do: Where these two lines crossed each other, a person would drop!” Vanderwaal slapped his knee, he was so proud of himself!

     “With all the speakers in that stadium, points of eight-cycle sound could be defined and sustained by a single computer that could communicate with individual speakers,” Phillips crowed.

     “You’ll prob’ly find individual iPhones with substantial amps like the ones you found in Denver and San Francisco tied in somewhere between the main PA board and each individual speaker.

     “I’m guessing they could be synchronized using Wi-Fi.”