scott wrote:
>> How is all this related to standing waves?
> 
> A standing wave is just a wave that stays in position, ie it's frequency 
> and spatial phase remain constant.  If you see a resonant mode by 
> itself, it will look like a standing wave and you can easily identify 
> the stationary points that remain at rest.

I see... So the resonant modes of something are all possible standing 
waves that can be created in that space?

>> Presumably a 2D surface like a drum head would have quite a lot of 
>> possible resonant modes?
> 
> Yep, see these photos that show the stationary points, try to imagine 
> how the non-stationary parts would bend up and down.
> 
> http://www.meta-synthesis.com/webbook/34_qn/2d_waves.jpg

Ooo... so there are both concentric and radial modes, and combinations 
thereof?

Presumably for a square surface it would be different though.

>> Ah yes - just curving the paper makes it behave quite differently. 
>> (See corrigated sheet metal.) But why, I wonder?
> 
> To curve a flat sheet of paper all you need to do physically is to 
> stretch the top half of the thickness by a tiny amount and compress the 
> bottom half by an equally tiny amount.  Once you start curving it in 
> other directions as well the amount of stretching and squashing you need 
> to perform becomes orders of magnitudes higher.  You can see this just 
> by looking at the geometry of a curved piece of paper and thinking about 
> how it must deform in order to curve it in the other direction.

So it happens due to the infintesimal but finite thickness of the sheet? 
Interesting....

>> Similarly, a hollow tube responds differently to a solid rod.
> 
> For a hollow tube and a rod of the same weight, the tube will always be 
> stronger and stiffer because it will have a higher 2nd moment of area. 
> Actually the I-beam is pretty much the best useful shape you can have, 
> which is why they are so common (tubes are not so easy to join together 
> and funny angles).

"2nd moment of area"? ._.

Well anyway, it strikes me that a hollow tube of the same weight would 
be very much larger, and hence obviously stronger.

The reason I mention this is that apparently trees that have rotted 
hollow tend to survive storms better. Thus, the fungi that rot trees 
actually *prolong* their life, not shorten it. (It's amazing what you 
can find out from wildlife books!)

>> And again, a tear tends to propogate along a sheet, but if you put a 
>> hole in the sheet, it actually stops the tear. WTF?
> 
> It's to do with the radius at the tip of the tear (very small) compared 
> to the radius of the hole (much bigger).  It's why the windows in 
> aeroplanes are round and not square (they had some square ones to start 
> with but they got cracks at the corners once the pressure difference 
> went up).

How interesting...

>> Presumably there's some molecular-level *reason* for all of this?
> 
> Yeh, something to do with dislocations in the crystal lattice from what 
> I remember.

Ouch. "Doctor, I've got a dislocated crystal in my lattice. It hurts 
when I bend over..."

>> ...because the Laplace transform turns differential equations into 
>> algebraic equations?
> 
> Yup.

I see...

>> Did you see Scrappy Races, where the same guy tried to "tune" a 6 L 
>> diesel engine, and the govener fell off? (I don't know what a govener 
>> is, but it sounds important...)
> 
> No I didn't see that one, the governer stops the engine going too fast 
> if there is no load.  Diesel engines are quite capable of 
> self-destruction if you try to run them flat out with no load.

Guy was tinkering with this fuel-injected diesel engine. (The engine is 
about the size of a small bath tub.) Obviously the camera crew weren't 
paying too much attention. (This was during build time.) Suddenly we got 
a shot of the machine in the distance with ****-off black Clouds Of Doom 
spewing out of the exhaust pipes, and the team of hill-billies running 
for their lives in every direction like startled rats.

Apparently the governer just fell off, and the engine is now running at 
full tilt in neutral with the fuel injection system at non-default settings.

A few moments later the team run back to the machine. One of them is 
trying to stuff his shirt into the air intake. Another is trying to 
physically rip the leads off the battery. (Uh, how does that help?) 
Another is trying to squeeze the fuel lines with his hands. I don't know 
quite what happened, but after a few moments the machine fell silent.

Cut to a scene 20 minutes later, and they've taken the head block off 
the engine, and the team captin is holding up bits of pistol rods, and 
other items which look like they really ought to be *attached* to 
something, but clearly aren't...

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