What is the average of many sinusoids with similar frequencies?

Question

Suppose I have a large number of harmonic oscillators all producing slightly different pitches in a certain range. What is the resulting waveform?

I tried modeling this as $$\lim_{q\to\infty}\frac1q \sum_{i=0}^q \sin\left(\left( 1+\frac iq\right)x\right) $$

and somewhat to my surprise the limit seemed to exist, at least according to Desmos:

Then I thought “oh, this will just be the Fourier transform of the rectangular pulse from 1 to 2” but no, it appears not, that's the sinc function, which is even, and this is odd. (I don't understand Fourier transforms very well.) It looks very similar if I sum from $-q$ to $q$.

1. Is this real or is it an artifact of floating-point calculations?
2. If it's real, does this function have a name? Is anything known about it?
3. If I replace the $1+\frac iq$ with $0+\frac iq$ and sum from $-q$ to $q$ it seems to sum to the everywhere-zero function, which seems less surprising to me. Why does it do this for $0$ but not for $1$?

Here is a link to Desmos: https://www.desmos.com/calculator/jqtilqhv4p

Note that when I originally asked the question I pasted a screenshot of the wrong graph.

Answer 1

$$\sin\left(\left(1+\frac iq\right)x\right)=\sin (x) \cos \left(\frac{ix}{q}\right)+\cos (x) \sin \left(\frac{i x}{q}\right)$$ Making the summations $$S_q=\sum_{i=0}^{q}\sin\left(\left(1+\dfrac{i}{q}\right)x\right)=\sin \left(\frac{3 x}{2}\right) \sin \left(\frac{(q+1) x}{2 q}\right) \csc \left(\frac{x}{2 q}\right)$$ Expanding for large $q$ $$\frac {S_q}q=\frac{\cos (x)-\cos (2 x)}{x}+\frac{\sin (x)+\sin (2 x)}{2 q}+\frac{x (\cos (x)-\cos (2 x))}{12 q^2}-\frac{x^3 (\cos (x)-\cos (2 x))}{720 q^3}+O\left(\frac{1}{q^4}\right)$$

Answer 2

We have $$s(x) = \dfrac{1}{q}\sum_{i=0}^{q}\sin\left(\left(1+\dfrac{i}{q}\right)x\right)$$

Let $t = \frac{i}{q}$, $\triangle t = \frac{1}{q}$

So, this is like a Riemann sum $$s(x) \approx \int_{0}^{1}\sin((1+t)x).dt\\ = \dfrac{1}{x}(\cos x - \cos 2x) = \dfrac{2}{x}\sin\left(\dfrac{3x}{2}\right)\sin\left(\dfrac{x}{2}\right)$$

This is an odd function just like you observed

If it's real, does this function have a name? Is anything known about it?

As far as I am aware, this function does not have a name

If I replace the $1+\frac iq$ with $0+\frac iq$ and sum from $-q$ to $q$ it seems to sum to the everywhere-zero function, which seems less surprising to me. Why does it do this for $0$ but not for $1$?

The Riemann sum then becomes $$\int_{-1}^{1}\sin(tx).dt = 0$$

This is because the integrand is just an odd function

But, if we consider the original sum $$\int_{-1}^{1}\sin((1+t)x).dt \not= 0$$

As you may have observed, the integrand is not odd and therefore, the value is unfortunately not $0$