What is the function for a 'fractal sine wave'?

Question

Maybe I abused the word fractal here. I was wondering what's the function ( if not functions ) for this wave:

My attempt was this function, It looks the same, but It's not. The second sine wave is following the envelope of the first, somewhat following the 90-degree angle of the first sine wave.

$y=\frac{sin(200*x)}{10}+sin(x)$

Answer 1

Probably not what you had in mind, but this is what you get with the parametric equation below:

$$x=t-\frac{\cos(t)}{\sqrt{1+\cos^2(t)}}0.15\sin(12t)\\ y=\sin(t)+\frac1{\sqrt{1+\cos^2(t)}}0.15\sin(12t)$$

Answer 2

Perhaps you are looking for functions of the form

$y=a\sin(bx)+c\sin(dx)$

with $a,b,c,d$ positive constants.

For example: Try $a=1,b=1,c=.3,d=12$ to start with -- and play around with it till it matches your purpose.

Answer 3

If you are actually looking for a fractal sine wave, you probably mean this function from Weierstrauss:

$$f(x)=\sum_{n=0}^{\infty}\left(\frac 23\right)^n\cos(9^n\pi x)$$

The main fame of this function is that it is continuous everywhere but differentiable nowhere.

Here are the first four partial sums of that series, for $n=0,1,2,3$. The graph you show is similar to the second one, which has the equation

$$f1(x)=\cos(\pi x)+\frac 23\cos(9\pi x)$$