Diffraction in lasers

I met an amazingly smart 2nd year grad student on my geology trip. A bit of background - two dozen geologists (actually, two astro kids and a Mech E were included in the jovial bunch) traipsed their way down to Baja California -- not in such a lackadaisical manner, with more intent -- but we made our way down to explore a bunch of rock formations, paleomag, a bit of cretaceous-dinosaur-fauna things, etc.

A few of us had wandered down to the beach at the Salton Sea the first night. When lasers came out, there was a period of comparing the power output of the lights and the distance this corresponded too.

But when you put a slight obstruction in the beam of the light -- we were using green lasers, best for seeing in the dark and for pointing out stars -- you get a diffraction pattern in the output light. There is a very obvious pattern of dark and light. Now, apparently, these are the Fourier transform of a single slit diffraction pattern! The light is blocked for exactly a top-hat function, and the resultant pattern has an intensity (not energy) that we see. The transform of the energy is the sinc function, but the intensity of the light is the sinc function squared.

Now, imagine two obstructions -- two hairs. On the ground, then, you see the pattern repeated, the standard diffraction of light and dark that progressively gets dimmer farther from the center. This is the first and the second fourier transform.

Yay :) You can do this for any number of obstructions!

Reflecting the laser off your teeth, your are confronted with a large number of organic shapes that have an apparent depth to the layers. They form moving shapes that crawl in green shadows across the ground, changing and disappearing. These are the fourier transform of...saliva. Ha ;)

Hello.

Why, hello!

This isn't really relevant to astronomy, and it's not exactly an introduction of myself. Instead, this is more a sorta collapsed version of both in the form of a little question! I was reading up on some astrophysics things over the summer, to better understand my SURF with our Professor ;) and I came across this wonderful class, the results of which are here. One project, by an Aaswath Raman, was really helpful, and there are many others that I plan on reading and blogging about :)

But that's not the point. Earlier - maybe a day or so ago, one of my fellow Lloydies and I were fiddling around the piano when he asked a question that was amazingly simple but that I had no idea of how to answer. How, indeed, does a piano create sound? On the first order it's hitting a key with your fingers, and it's obvious from glancing in any piano that this movement depresses a lever that in turn hits a string; the string vibrates, and sound is produced.

We're taking Ph12 right now, and just took our first class today. It's a class on waves. So when that hammer hits the string, how does the string vibrate to create sound? We determined it's probably not a torque driven oscillation, that the string must vibrate either up-and-down or side-to-side, or some combination of both.

Then again, the sound waves coming towards us are longitudinal waves. which compress the air and (with help of a search engine) the opposite, which is a process you might recall as rarefaction.

The motion of the strings is (mostly) transverse. This is a wave where oscillations move perpendicular to the energy being transferred (like our light waves). The string vibrates so that the energy associated with that note being struck is transferred through a wooden bridge to a soundboard, and it is the velocity of the soundboard, in turn, that actually produces the sound waves.

Interestingly, the string vibrates both in a transverse and longitudinal mode, which is, in retrospect, perfectly reasonable for something being struck. There's a nonlinear coupling of the transverse string modes to longitudinal modes (the strings are damped, as well as stiff), but in the strings of the piano the longitudinal speed of sound in piano strings are ~20 times that of the transverse oscillations (i.e., their frequencies are therefore too high to hear most the time).

See also

A piano waveform (a chord)

• Modeling a piano here, effectively on the basic wave equation for strings in one dimension. This was my foremost reference
• This is an artistic thingy centered on burning pianos :( sad, but interestingly epic for a short while
• Strings have very pretty sounds: listen to this gorgeous piece! Heartbeat, by Jake Shimabukuro
• This page is nice: it has a simple explanation of the math of music (with really cool stuff on the waveforms of instruments)
• Numerical simulations of piano strings, an article written back in 1993 that describes the vibrations of a string by a set of differential and PDEs. Another useful reference.