SO WHEN WE last left off,” Professor Phillips intoned, “I believe I was convincing you that lower notes can stimulate higher ones, and that yelling can stimulate the equivalent piano strings to make similar sounds.”

     “And that Cs can stimulate G strings,” Vanderwaal added.

     “Yes,” Phillips acknowledged. “But can higher notes stimulate lower notes? That’s really the question. Because that’s what your terrorist would have to do in order to get his conical speakers to generate eight-cycle sound.”

     “Give it your best shot.” Vanderwaal was sitting behind his desk with his hands clasped behind his neck.

     “Let’s say this wavy line here represents middle C.” Phillips drew a nearly perfect wave on Vanderwaal’s white board. “It’s technically called a sine wave. Think of it as a series of hills and valleys. If you jump from the top of one hill to the top of the next one, you’ve just defined a single cycle. Middle C has hundreds of cycles that make up one second’s worth of sound.

     “When a speaker cone moves in and out at that exact number of cycles per second, we hear middle C. In the case of the piano, the middle C strings -- all three of them -- go back and forth exactly that many times per second after being hit by that hammer for us hear middle C.

     “Now I’m going to draw another wavy line -- this one under the first one,” which he did as he was talking. “This represents high C. What can you tell me about these two?” he asked, as he turned back to face Vanderwaal.

     “Well, the second one has twice as many hills and valleys as the first one.” Vanderwaal didn’t know if that was the answer that Phillips was looking for.

     “That’s correct,” Phillips said, smiling. “You might also have noted that the tops of each middle C hill -- those on the top wavy line -- line up with the tops of every other hill on the bottom.”

     “That’s why you had me push my daughter on every other swing cycle?” Vanderwaal was trying to connect the dots.

     “You got it.” Phillips verbally patted him on the head. “Now, I’m going to draw a third line that we get if we were to combine these two. What do you see?”

     “Well, it looks less smooth,” Vanderwaal offered. “The lower C part of it is higher and lower than the original middle C, but also narrower.”

     “Very observant, Grasshopper,” Phillips beamed. “These waves are being added together, so their new peaks and valleys are, indeed, higher and lower than the originals.

     “Now for the money shot: Let’s apply what we’ve learned to create eight-cycle sound from higher sounds. Remember, we said speakers were not really effective below fifty to one hundred cycles.”

     “Cycles per second,” Vanderwaal elaborated.

     “Thank you, Grasshopper.” Phillips said this with an even bigger smile. “What we’ll need are frequencies that are evenly divisible by eight. Let’s arbitrarily say frequencies above one hundred. Does eight divide equally into one hundred?”

     “No,” Vanderwaal correctly assessed. “The first one would have to be 104. The next one would be 112, then 120, then 128, 136, 144, 152, 160. How many do you need”

     “That’s enough for now.” Phillips was writing the numbers on the board. “If we were to line up these frequencies under each other like we did with our Cs, what do you think we’d find?”

     “Well?” Vanderwaal had to think about that one for a moment. “If they’re all evenly divisible by eight, and they all started at the same time -- then I’d expect to find alignment with all of the hills every eighth of a second. I’m not sure how one would make that happen, but I think that is, in fact, what would happen.”

     "What happened when we added the two C frequencies together?” Phillips asked.

     “The strength of those aligned hills doubled?” Vanderwaal offered.

     “So,” Phillips encouraged, “if we align those eight hills?”

     “Logically,” Vanderwaal said -- starting out with that word might add credibility to his answer, or so he thought -- “we should have eight times the height or strength, or rather volume, of the sound represented by that specific frequency.”

     “In real life,” Phillips commented, “it doesn’t actually double when different sounds are added together, but it definitely is louder.”

     “So all our terrorist has to do is to generate and line up frequencies that have a factor of eight, and he can make eight-cycle sound as loud as his speakers can take?” Vanderwaal concluded.

     “You get a gold star, Grasshopper!” Phillips beamed.

     “How ‘bout we just knock off the ‘Grasshopper’ bit instead,” Vanderwaal suggested with a forced smile.