Differential equation for carbon dating
FT’s even show up in quantum computers, as described in this shoddily written article.Mathematicians tend to be more excited by the abstract mathematical properties of Fourier transforms than by the more intuitive properties.
It’s not until a speaker has to physically play the sound that the FT is turned back into a regular sound signal. For example, when you adjust the equalizer on your sound system, like when changing the bass or treble, what you’re really doing is telling the device to multiply the different frequencies by different amounts before sending the signal to the speakers.The Fourier transform of a sound wave is such a natural way to think about it, that it’s kinda difficult to think about it in any other way.When you imagine a sound or play an instrument it’s much easier to consider the tone of the sound than the actual movement of the air.So when the base is turned up the lower frequencies get multiplied by a bigger value than the higher frequencies.However, acoustics are just the simplest application of FT’s.= cos(θ) i sin(θ), you can compress this into one equation: , .
There are some important details behind this next bit, but if you expand the size of the interval from [0, 2π] to (-∞, ∞) you get: and Here, instead of C, you have and instead of a summation you have an integral, but the essential idea is the same.
Physicists jump between talking about functions and their Fourier transforms so often that they barely see the difference.
For example, for not-terribly-obvious reasons, in quantum mechanics the Fourier transform of the position a particle (or anything really) is the momentum of that particle.
be recorded (this is what a “.wav” file is), but more often these days the Fourier transform is recorded instead.
At every moment a list of the strengths of the various frequencies is “written down” (like in the picture above).
If you could see sound, it would look like air molecules bouncing back and forth very quickly.