Q: Can resonance be used to destroy anything? Is the “brown note” possible?

Physicist: Nope!

“Resonance” is a “driven harmonic oscillation“, where the driving force pushes and pulls at, or near, the “resonant frequency” of whatever it is that doing the resonating.  There are two big issues involved with destroying stuff using sound, or gentle taps, or whatever you’re using to drive the motion.  The first is that nothing in the world “rings” perfectly, and the second is that every example of harmonic motion you’re likely to come across is actually an example of “damped harmonic motion”.

Here’s what “not ringing perfectly” means.  You can make anything bounce around and shake, but “harmonic motion” is something very specific.  Harmonic motion is a vibration that takes place at just one frequency.

Something that moves harmonically moves back and forth in a sine wave.  Maybe more or less spread out or taller than this, but always in exactly this shape.

When something is oscillating back and forth there has to be a “restoring force” to bring it back to center.  For a swinging pendulum that force is gravity, for a spring or a wine glass it’s the springiness of the material.  The one requirement for harmonic motion is that the restoring force is linear, and proportional to how much the object has been pushed from center.  For example, if you pull on a spring by a distance X and it pulls with a force F, then if you pull it twice as far, 2X, it will pull with twice the force, 2F.  This is the famous “Hooke’s law“: F = -kX.  The negative here means that the force points in the opposite direction of the displacement, so if a pendulum has swung to the right, then gravity is pulling it to the left.

When you change things by just a tiny bit the response is almost always linear.  Or at least very nearly linear.  If this weren’t the case, then physicists would barely be capable of doing any calculations at all.  This is called a “first order approximation” or “linearization”, and it’s really just the statement that things (in a mathematical sense) are smooth.

A typical force response for a spring.  If you don’t stretch it too far it reacts linearly (straight line), but if you stretch it too far then it doesn’t (and eventually gets turned into a straight wire or breaks).

Most physical systems have the “linear on a small scale” property.  It’s just a question of when it breaks down.  This is why clock pendulums don’t swing very far, for example.

So this is the first big problem; if you push something hard enough, or if the oscillation gets too large, then the restoring force won’t be linear.  As a result the system starts to lose all of the nice properties that make it a harmonic oscillator.  One way for this to happen is for the object to break (huzzah!), but most of the time the oscillation frequency starts getting wonky, the wave stops being pretty (not just one frequency), and trying to induce resonance just sorta stops working.

The second big problem is dampening.  Nothing’s perfect so over time every oscillator loses energy.  Pluck a guitar string, and it’ll eventually go mute.  Stop pushing a kid on a swing, and they’ll eventually just sit there.  And maybe get hungry.

Damped harmonic oscillation looks like this.

When you try to make something resonate you’re adding up a bunch of these waves (one for every time you “push the swing”), but because of dampening the waves from earlier don’t add as much as the waves from later.  Even if the system is still pretty linear, this puts a cap on how big an oscillation you can get for a given amount of pushing.

So, just to make sure the point on this is too fine, even if the frequency is dialed in perfectly, most things can not be destroyed by resonance.

If the dampening is bad enough, then instead of resonating at all the object will just “ooze” back to where it started.  When things are “over-damped” patience completely stops being a virtue and you really need to get all of the energy in place all at once.  For example, if you’re using sound you’d need to replace the speaker or your voice with a bomb or a hammer.

The “brown note” is a sound that supposedly resonates intestines and makes a mess.  However, there are issues.  Entrails don’t ring like bells, they flop like meat.  Even if you’re careful to stay in the “linear regime” (very, very small forces and oscillations), you’ll find that meat is usually critically damped, although not always.

There are better ways to induce trou dropping: laxatives, roller-coasters, away-toilet situations, etc.

While it is true that the (American) military has spent some money looking into the brown note (lots of people have), it’s also true that they’ve spent some money looking into almost everything.  Statistically speaking, you can’t spend $680,000,000,000 without buying something useless.