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Re: sympathetic vibrations--my husband says that when a
troop of armed forces is on the march, they are ordered to
march out of step going over a bridge, because the sympathetic vibrations can cause some bridges to collapse.
Also--in one of Clive Cussler's works of fiction (can't remember which, but can LIU if anyone's really interested),
the enemy had developed a weapon of incredibly destructive power that used sound waves. Bad literature, but theoretically "sound" technology, I think. [groaning at her
own unintended pun emoticon]
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Sympathetic vibration
I too have heard the marching-over-a-bridge theory, but I don't believe that would be sympathetic vibration as it is defined. However, a common, and benign (no H.G.Wells stuff) everyday use of sympathetic vibration is in tuning instruments.
To tune a piano, organ or harpsichord (and they need tuning more often than most people would think, especially a harpsichord which can go out of tune in a half hour) you don't need a tuning fork or electronic pitchpipe for each note in the scale. A friend of mine who bought a harpsichord asked our organist how to tune it easily. The answer: use a tuning fork to tune middle C; then tune C one octave above and one below, then tune E, G, and B-flat from there, all by holding down those notes and striking hard on C. You tune the octaves of C by listening to the beating of the two notes. Once you have tuned middle C with the tuning fork and you know that one is in tune, you tune the octaves from the beating sound you hear when you play both together. The slower the beating, the farther off; as you get the one more into tune with the first, the beating gets faster, and when you don't hear beating, they are in tune. The beating comes from the sound waves being out of synch and colliding with each other in the air before they reach our ears.
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Our Man in Ballmer suggests: use a tuning fork to tune middle C; then tune C one octave above and one below, then tune E, G, and B-flat from there etc. This will produce a tuning that sounds great in C, in fact even better than the normal tuning in which you tune each note individually to a standard reference. As you depart from the key of C around the circle of fifths (C-G-D-A-E-B-F#/Gb-Db-Ab-Eb-Bb-F-C) the tuning will get worse and worse until you hit F#/Gb at which point it will start to get better until you get back around to C. If you are interested in this tuning see http://www.dnai.com/~jinetwk/.
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Tuning
I daresay either I grossly oversimplified the tuning process or my organist friend did in his advice. What you describe is, of course, the problem of natural tuning, which was solved by tempered tuning, which became the standard method thanks largely to its championship by J.S. Bach nearly 300 years ago. Doing your own tuning takes a good ear at least and a good sense of relative pitch. Perfect pitch sense would be very useful, but few people have it.
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I'm sure you have seen the effect of "harmonic resonance" on "Those amazing disaster shows" where a cabled-suspension(?) bridge twists and torques in the wind. Just a bit too "sympathetic". 
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BobY adds: ...tempered tuning, which became the standard method thanks largely to its championship by J.S. Bach nearly 300 years ago.It gets even more complex than that. The change didn't happen all in one swell foop. There were bunches of partially tempered systems with all kinds weird names like 1/4 comma mean tone that fit in the general category of non-just non-equal temperaments ( http://www.ixpres.com/interval/dict/temp.htm). The system we use today is called equal temperament; the relation between the frequencies of any of the adjacent notes in the 12 tone scale is identical. Thus Freq(E):Freq(F)::Freq(A#):Freq(B). Mozart did not have the benefit of this temperament scheme and some of his music, in my opinion, shows it. I had never been a great fan of Mozart until I went to a Malcolm Bilson concert in which he was showing off his new pianoforte. He had it tuned in the temperament that Mozart would have used and was pretty much restricted to playing in three keys. He played one piece that he had introed by saying that most composers would repeat a motif maybe two or three times but that Mozart had done it 13 times in this piece. Mozart had taken the little three or four note theme and dragged it kicking and screaming around the circle of fifths watching it get more and more discordant until it made it halfway around at which point it slowly came back to harmonious accord. Ever since then I have listened for that sort of thing in Mozart's music (which I am getting more and more appreciative of Hi E) wondering what it would sound like if they were playing in his temperament.
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The system we use today is called equal temperament; the relation between the frequencies of any of the adjacent notes in the 12 tone scale is identical. Thus Freq(E):Freq(F)::Freq(A#):Freq(B). Mozart did not have the benefit of this temperament scheme and some of his music, in my opinion, shows it. That's the problem with being non-musical. I recently watched a series that I really enjoyed called "Howard Goodall's Big Bangs", on the history of Western music, and, not being remotely musical myself, I lapped it all up. Imagine my disappointment on learning that he had sold me a pup with his statement that J.S. Bach's "The Well-Tempered Clavier" was largely responsible for our "equal temperament" system, even specifically mentioning Mozart and saying that some of his music would not have been possible without it.   Once again, the TV has lied to me.
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The suspension bridge that is shown in most "disaster" TV shows in the Tacoma Narrows bridge– and it fail because of an engineering flaw.
Engineers where trying to be build lighter and "more elegant" bridges. The goal was strong, lightweight bridges. Reducing the weight meant smaller anchorages, and less maintenance
The bridge design was fine for supporting weight– but its light weight design had trouble coping with the high winds in the Tacoma Narrows gorge. It was high winds that caused the failure. The bridge start falling faster than gravity– since it wasn't so much rising and falling as it was oscillating.. The oscillation eventual (with in minutes once they really got going) destroyed the bridge.
The same engineering company used almost the same design for the Whitestone bridge across the East River connecting the Bronx to Long Island. After the failure of the Tacoma bridge, the Whitestone bridge was "improved" with new structural steel, in the form of triangular bracing. (Since the design was excellent for downward stress/weight, the bridge could carry the load.) This was installed in what was original the pedestrian pathways. The Whitestone bridge is still pretty bouncy it's 'interesting' to drive over when there are heavy winds. – but it's close to 70 years old, and still carries a heavy volume of traffic.
But some bridges have failed because of rhythmic stress of humans, not wind. Back in the 70, and again in the 80's, "Walkways" in malls and hotels have failed when a number of dancers, all moving to the same beat, caused sympatric vibrations. And the NY Marathon has rules about the number and speed of runners on the Verrazano Bridge- at the start of the race. So the is something to it.
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NY Marathon has rules about the number and speed of runners on the Verrazano Bridge- at the start of the race. So the is something to it.
I have shot off a note to Army pal for clarification on present practice ...back atcha' when I get an answer. He's used to my whacky questions.
wow
EDIT : about 4:45 p.m. EDT
My Army chum says : "I have never read anything in any manuals, I know that it isn't in FM22-5 (Drill & Ceremony).
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MaxQ complains: he (Howard Goodall) had sold me a pup with his statement that J.S. Bach's "The Well-Tempered Clavier" was largely responsible for our "equal temperament" system
It certainly led the way. Before any of the temperament schemes the just intonation system, which was based on the harmonic tuning, would not have allowed for the playing of any key on a keyboard instrument such as a a harpsichord or, later, a piano. In a well temperament the various keys *do have different qualities, the thirds and fifths are all slightly different. If Mozart had tried to write his music in just intonation the romp around the circle of fifths I described above, rather than leading to increased and then decreased discordance (and it wasn't really *bad, just a little off color) would have ended in a train wreck when he came back home.
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