By regular, I mean having a constant period and amplitude: the waves arrive like the ticking of a clock, and each is exactly as tall as the next.
When waves combine, their amplitudes can be either added or subtracted depending on how they align: two peaks add to make a taller peak, but a peak and a trough cancel out. So two regular wave patterns with different periods will make a new pattern of waves with a new period.
You can see this with a graphing calculator. Here are the sine curves for 2sin(x) (red), 2sin(\frac{x}{2}) (green), and 2sin(\frac{x}{3}) (black):
(Multiplying each by 2 increases the amplitude, i.e. makes the ups and downs bigger, which will make the pattern more visible in subsequent steps.) Sine of x has a period of 2\pi, sine of \frac{x}{2} has a period of 4\pi, and \frac{x}{3} has a period of 6\pi – i.e. each has a period of 2\pi times the denominator.
Now here’s the curve for those three curves combined, i.e. 2sin(x)+2sin(\frac{x}{2})+2sin(\frac{x}{3}):
It has a period of
12\pi, which is
2\pi \times 2 \times 3. The pattern repeats in those cycles, but in each cycle the pattern is irregular (though if you look closely you can see that the second half is the first half repeated backwards and upside down).
If we keep layering on curves, we’ll get a longer and longer period with more and more irregular waves in between. Here’s (part of) 2sin(x)+2sin(\frac{x}{2})+2sin(\frac{x}{3})+2sin(\frac{x}{5})+2sin(\frac{x}{7}):
This doesn’t show a full cycle, because the period is 2\pi \times 210 (i.e. 2\pi \times 2 \times 3 \times 5 \times 7). I chose those denominators because they are prime; the period will always be the product of the prime factors of the denominators, so for example 2sin(x)+2sin(\frac{x}{2})+2sin(\frac{x}{4})+2sin(\frac{x}{8}) only has a period of 2\pi \times 8.
The pattern looks pretty irregular, downright random, but it’s just a stack of perfectly regular sine waves. And if you keep going, it becomes indistinguishable from randomness. Here’s a chunk from a plot using the first 100 prime numbers as denominators (multiplying each by 100 so we can zoom out and still see the ups and downs).
Again, perfectly regular patterns, summing to produce something nearly indistinguishable from noise – I say nearly indistinguishable because there’s a clearly something non-random around zero:
Notes
I used LaTeX to make the functions, here is an example:
$2sin(x)+2sin(\frac{x}{2})+2sin(\frac{x}{3})+2sin(\frac{x}{5})+2sin(\frac{x}{7})$
Unfortunately LaTeX doesn’t work in hidden sections, but when you put that into a post it looks like this:
I used this site for graphing, which is super fun to play with:
Desmos | Graphing Calculator
To make the sum using the first hundred primes, I got a list from wikipedia and used the list function on that site to make a sum:
List:
L=[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541]
Function (that sites support latex, so you can copy-paste this):
y=\sum_{n=1}^{100}100\sin\left(\frac{x}{L\left[n\right]}\right)
L[n] is the nth element from the list L.
