It’s been quite a while since I took trigonometry in high school. But I do still find the subject fascinating. And one question still perplexes me:
The Sine wave is found all throughout nature. The average temperature follows a sine-wave pattern, if you graph it out over the year. The daily sunrise and sunset also does this. And there are more complex versions of the sine wave that account for things the graph of a beating heart, etc. But where exactly in nature (or any place else for that matter) do you find the tangent wave? Because it is very peculiar you have to admit, at least as compared to the sine wave. And to-date, I have never heard of something that resembles a tangent wave, when graphed out. Are there any examples of this?
If you had a beam projected from a rotating source onto an infinitely large groundplane, measuring the distance between the projected spot and the point on the plane nearest to the rotating object, then plotting this over time, would result in a graph that resembles what you call a ‘tangent wave’.
Rotating sources exist in nature (Pulsars? or is it Quasars?), but infinite groundplanes do not.
You missed the obvious appearances of the sine wave – rotating and oscillating objects display beautiful sine waves, much closer to the ideal form than the examples you cite (most of which are periodic phenomena that can be decomposed into a superposition of sine waves. Fourier analysis is a wonderful thing).
I wouldn’t call a tangent function a “wave”. It’s be hard to have anything follow a function that zips off to plus or minus infinity. That said, tangents do asppear in physical situations, but not in the obvious way you seem to suggest. Things don’t move along as if they were beads on a tangent function-shaped wire, but you can describe motion in terms of it – For instance, a particle mobving along a sttraight line is a distance a * tan (theta) from a point a distance a from that line.
Or, there are physical situations where the solution to the equations describing it are x = a tan x, and all lie at the intersections of that tangent curve with a straight line running through the origin.
Just to nitpick a bit…keep in mind that there is an important difference between a periodic wave (general case) and a sine wave (very specific case defined by the ratios involved in circular rotation). Many periodic waves that may look like sine waves are not, although they can be approximated very closely with a sine wave or the superposition of multiple sine waves.
The answer is never. You never see sine waves, exponentials, circles or straight lines, either. These only occur as varyingly approximate solutions to varyingly approximate mathematical models of physical phenomena that may or may not be completely understood.
Sure, if they are perfectly linear, which not springs are, have no damping, which no springs do, and there is not drag/friction in the moving object. The sine waves are the idealized case.
Yes, if it were exactly monochromatic, and travelling through a perfect vacuum.
Sines and cosines show up in all sorts of idealized situations, but of course nothing in the Universe is truly ideal. Some real physical situations come closer to ideal than others.
