Every rigid object in existence has a natural structural resonance frequency, a frequency at which it, metaphorically speaking, wants to shake more than any other. If you vibrate an object at its resonant frequency, it will gradually shake more and more wildly, like a child on each swing being pushed higher and higher on each swing, until eventually it shakes itself to pieces. This is how you can shatter a wineglass at a distance by singing the right note - though you have to hold the note for several seconds so that the vibrations build up, and you have to sing superhumanly loudly, or else have your voice amplified.

The resonant frequency of an object depends on its structure. The larger the object, generally, the lower that frequency is. For things as large as buildings, architects have to design them so that normal use of the building, and natural weather conditions (wind/rain), can never combine to shake the building at one of its resonant frequencies. Failure to do this has proven hazardous in the past. In 2000, the London Millennium Footbridge was found to sway unnervingly in time with people's footsteps when many people walked across it - it had to be closed and redesigned. Going back further, in 1850 the Angers Bridge in France was shaken to pieces when 400+ soldiers marched across it in lockstep. Lessons well learned. Some very tall structures in earthquake zones are now built with tuned mass dampers at the top, to cancel out such resonance.

(Conspicuously absent from these examples is the Tacoma Narrows Bridge collapse. The Tacoma Narrows Bridge wasn't, strictly speaking, demolished by forced structural resonance, because there was no periodic disturbance on any of its natural vibrational modes - just constant wind. Rather, aerodynamic effects caused the road surface to act not unlike an aeroplane's wing, twisting on its axis and repeatedly catching the wind, then stalling and falling.)

From here we move a little way into the realm of urban legend. Famed physicist Nikola Tesla is purported to have once created a small, pneumatic device which would automatically and mechanically seek out the structural resonance frequency of any solid object to which it was attached. Supposedly, he once took this device to a building site and attached it to the girderwork of a half-started building, causing it (after some minutes) to start vibrating so alarmingly that the builders abandoned it, fearing an earthquake. Tesla never used the machine again, out of fear. For the Earth is solid. The Earth has a structural resonance frequency. Could not his "earthquake machine" - which, let's be serious here, is probably fictitious - be used to vibrate the Earth to pieces?

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In the collective mind's eye, the process would go something like this. First, figure out the resonant frequency of Earth. This can be done by triggering a test earthquake and using seismometers to measure the passage of the quake through the Earth's interior. Time how long it takes for the quake's waves to reach the other side of the Earth, bounce back and return to the epicentre where they originally emanated. Let's say it takes about 60 minutes for one complete cycle. When the echo returns, trigger another quake at that instant and at the same location, thereby adding the two quakes' energy together. Wait for the echo, stronger this time. Do it again. And again. Trigger quake after quake until the Earth itself can be seen to bulge and distort from space. Keep going. Finally, the whole Earth spontaneously breaks apart, separating permanently into pieces. The Earth is thus destroyed. Right?

Well, no.

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The first problem is one of energy dissipation. During each oscillation, some energy is lost. Earthquakes are still detectable after more than one round trip through the Earth's interior, but they don't echo on forever. We know this, because otherwise the ground beneath our feet would be constantly shaking. After the quake has gone all the way around the Earth, the energy it retains has diminished hugely: by a factor of a hundred (or a million, but let's call it a hundred and be charitable). When you trigger the second quake, you don't have 2 quakes echoing around the world now. You have 1 quake, plus 0.01 of a quake. When you trigger the third, you're up to 1 quake (the third one) plus 0.01 of a quake (the echo of the second one) plus 0.0001 of a quake (the echo of the echo of the first one). As it goes on, the amount of energy stored in the Earth rapidly levels off to a strictly finite quantity - even though you're constantly adding more energy.

Now here's the rub: does the Earth fall to pieces before we reach this point? Or is it just going to shake angrily for ever and ever but stubbornly refuse to be destroyed?

You'll notice that the total amount of energy retained is only slightly more than the energy from the original quake. If we want this total to be big enough to destroy the Earth, that means that the energy from our original earthquake has to be almost big enough to destroy the Earth already. No small pneumatic device can generate an earthquake this big. No real earthquake can be this big. We are looking at a massive, hugely destructive, cataclysmic impact which would almost totally destroy the Earth in one shot, leaving essentially nothing for the second impact to even connect with! In effect, we're looking at physically blowing up or pulverising the Earth with a blunt instrument with our very first shot. Resonance and earthquake machines become an irrelevance.

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Secondly. I've made some simplifications and assumptions here, to make the point clearer. Restore these extra complications and the situation becomes even more untenable. Even an object as simple as a solid sphere actually has multiple vibrational modes - different ways in which it can shake. For more complicated structures, the number of vibrational modes becomes very large and not all of them can be known. In the case of the Millennium Bridge, mentioned above, the reason for the alarming vibration was the discovery of an entirely new vibrational mode which had never been seen in any previous structure and which the architects had not tested against.

It's not at all clear which of the Earth's modes we would be aiming to agitate. Real earthquakes generate at least three distinct types of wave, which pass through the planet's interior on different paths and at different speeds, and it's not clear which of these waves we would be attempting to build up, either. Also, Earth's interior is not uniform and solid, but layered and partially liquid, meaning that it doesn't vibrate anything like an ordinary sphere would. This complicates the situation still further.

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And lastly, it's far from clear what happens at this mysterious point where the Earth finally has enough energy to shake itself to bits. Ordinarily, a building or bridge shaken to pieces would naturally collapse to the ground, under the force of gravity. More abstractly, it would seek a state which is more stable, a state with lower energy-- in other words, a state where the structure has its constituent parts nearer the ground. What does this even mean for a gigantic ball of rock, which is bound together incredibly strongly by gravity, and therefore already in pretty much the most stable configuration possible? How could such a thing "collapse"? If we broke the Earth in two with our vibrations, what would happen to those pieces? Under gravity, naturally, they'd fall back together again! And we'd be right back where we started.

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If Tesla's machine did ever exist, then he would have known much, if not all of the above facts, and known that his device presented no danger to the world at large, and known that he did not need to destroy it. Likewise, we know that there is no point in creating one. We must find another way to destroy the Earth.