The Fourier series is a powerful technique that can be used to break down any repeating waveform into sinusoidal components, based on integer number harmonics of a fundamental frequency:
Video:
This is already amazing, that by summing up harmonically related sinusoids (with suitable amplitudes and lagging) we can analyse any repeating waveform as a sum of components. This then extends to any non-repeating pulse, once we go to an integral, which brings in the idea of a continuous spectrum where some wave “energy” is found at every particular frequency in a band.
However, something subtler lurks: As the illustration based on clips from the video shows, a sinusoid can be seen as the projection of a rotating vector (= a phasor). Thus, what the Fourier series is doing is that it is adding up a set of harmonically related rotating vectors instant by instant, yielding the overall result of the periodic waveform (here, a square wave). And indeed, we can get the rotation by adding up two perpendicular oscillating vectors, one on oX, the other on oY. This then gives us two summed harmonic motions, which allows us to bring in the forces and inertia at work. (Where, yes, we can then go on to elliptical, parabolic, hyperbolic and even linear motion.)
Rotating vectors should ring a few bells for those who have been following our discussion recently.
Yes, we can start with the von Neumann construction of the natural counting numbers, N:
{} –> 0
{0} –> 1
{0,1} –> 2, etc
And from that define integers (Z), then rational numbers (Q) then reals, R, eventually the surreals:
That is interesting, but let us take the reals, R, and define a vector rotation operator i* so that it rotates R through a right angle anticlockwise pivoting on 0. So, any positive x in R is transformed to i*x, along i*R, going upwards. (Yes, the y-axis, oY.) Now, apply i* again, so we see i*i*x = – x, on the negative reals. That is, we have a natural interpretation of i, i.e. i^2 = -1, i is sqrt(-1). The rotating vectors approach makes for a more coherent understanding of so-called imaginary numbers. Thus, complex numbers, C are actually 2-d vectors that help us to do all sorts of interesting things, especially once they get to rotate.
We can go on from here, using power series to define a complex exponential form of the rotating vectors approach, which then yields the famous expression 0 = 1 + e^(i*pi).
All of this then comes back to the Fourier analysis, as the video demonstrates by showing rotating vectors added up tip to tail to form complex wave forms. Which, then points to how we can move freely between the time domain and the frequency domain to better understand phenomena in the real world.
Where, the underlying message clearly is that the logic of structure and quantity — mathematics — is deeply embedded in reality. And no, this is not flogging a dead horse, it is awakening human (as opposed to equine) understanding. END
The Fourier series & rotating vectors in action (with i lurking) — more on the mathematical fabric of reality
Nice math, and nice application to the physical world. I used to use a simple bicycle wheel on a stand to introduce the creation of the sine and cosine wave from rotational motion.
H, loss of net services. Praythee, tell me, how does hearing work, specifically the cochlea? KF
F/N: On how hearing creates a frequency domain transform of sound inputs, driving the onward processing, Wiki is a handy reference:
In short, sinusoidal frequency domain decomposition of sound waves is a key mechanical phenomenon exploited by our hearing system, leading to in effect a frequency domain transformation of the temporal pattern of compressions and rarefactions that we term sound. This is of course closely related to the patterns we explored and discovered using Fourier power series and integral analysis of oscillations and transient pulses.
Where, on the mechanical side, harmonic motion is tied to elastic and inertial behaviour. Which in turn is directly connected to a rotating vector analysis — leading straight to the complex exponential analysis that draws out the full power of complex numbers, form Z = R*e^i*wt, w being circular frequency 2* pi*f (in radians per second), f the cycle per second frequency. All of this ties back to the fundamental frequency cycles and integer-multiple frequency harmonic epicycles in the OP above.
Again, Mathematical study turns out to reflect quantities, structures and linked phenomena which are embedded in the fabric of our world.
KF
PS: Notice, not a few design subtleties?
PPS: The vocal tract, in effect a wind instrument, also exploits fundamentals and harmonics to create auditory, frequency-based patterns as well as transients.
Allow me to point out that these sums of sine and cosine functions constitute the building blocks of the standard Hilbert space L2. Each element f of L2 is a (usually infinite) sum of different sines and cosines. But these functions f are the wave forms of quantum mechanics that model reality at the micro-level.
As BA77 often reminds us, the wave forms appear to “collapse” into measurements of reality only after choices are made by observers. QM is reminding us of Pythagorus’ dictum that “all is number” (or rather sums of sines and cosines).
MG, interesting observation. I for preference go to complex exponentials as characteristic functions for most things of interest but sinusoids are just as valid. The complex exponential vector z = R*e^iwt on polar axis 0X, captures the sine and cos picture in a power series based framework. Where, of course, cos wt + i* sin wt is giving us x and y axis components at right angles that then give the position vector from the origin as r — the hypotenuse — of length defined through pythagoras. KF
PS: Some notes: https://www.math.upenn.edu/~deturck/m426/notes02.pdf and a vid: https://www.youtube.com/watch?v=jgi8hbOmUmk Sadun goes on to introduce Fourier series in his next vid: https://www.youtube.com/watch?v=PTm1H40g-UI (I am thinking his channel is a trove of very well introduced Math: https://www.youtube.com/channel/UC3IZYipZ6A_Rs5v0GdwUM7Q/videos?sort=dd&view=0&shelf_id=1 I am looking at his over- the- shoulder- vid of partly covered sheets of paper approach as an interesting video-ised rendering of the chalk talk or transparency roll on an OHP approach also, cf. Khan Academy.)
KF, I also prefer the standard basis of complex exponentials but “dumbed down” my observation for the benefit of readers who have limited math backgrounds.
MG,
Okay.
My thought is, it’s not hard to think in terms of vectors and rotations, which gets us to a lot of useful results, including a reasonable view of i.
The cos-sin view and magnitude angle to polar axis view is a next step, as is the concept of angular velocity, which can be demonstrated with a rotating wheel. Indeed, projecting to the axes (similar to the vid) brings out the sinusoids.
The power series representations can probably be motivated on Taylor or the like, or taken on faith by authority. In that context, the relationship between sin, cos and the complex exponential can be brought in.
Going beyond, I find it useful to see how standing waves on a string give a natural way to get to quantum states. The particle in a box context pops up and we are in the quantum world. Along the way, Bohr’s orbits may be useful.
All along, we should be observing how deeply embedded structure and quantity are in the physical world.
I think immersion in facts may well be the best antidote to the sort of nominalism that keeps cropping up.
Merely pointing out incoherence seems to have limited impact on today’s ultra-modernism indoctrinated mind. As in, they doubt the principle of explosion and think that their doubt trumps logic and evidence to the contrary.
As we have seen.
KF