period of a sine function in the form of $F(x) = \sin(ax) + \sin(bx)$. where the amplitudes of the two individual sine waves are the same?

Question

As the title says, I'm working on a research paper for my IB diploma, that is trying to find a link between patterns in music and their waves. but I've encountered a problem when trying to mathematically calculate the period of a function that is a sum of two different sin functions with the same amplitude but different frequencies. What I have been doing is plotting the function on Desmos and finding it manually, which doesn't help me with what i am currently trying to do. I looked far and wide on the internet but I found nothing.

I'm doing this to see how the change in the difference between the frequencies (b/a) effect the period of a function. and would appreciate any help!

Does this answer your question? Is the sum of two sine waves periodic? - found using an Approach0 search. There are also other basically duplicate questions, e.g., Period of $\sin(ax)+\cos(bx)$, When is $f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)$ periodic?, etc. — John Omielan, Jan 30 '23 at 08:06
@Bouncyy Are there any restrictions on the coefficients $a,b$? — PierreCarre, Jan 30 '23 at 08:20
@JohnOmielan From the looks of it, it's helpful i'll have to take a moment to try it out — Bouncyy, Jan 30 '23 at 08:21

MathDona · Accepted Answer · 2023-01-30T08:26:24.467

3

If $a/b$ is real rational, then the period of $f(x)$ is LCM$\left(\frac{2 \pi}{a}, \frac{2 \pi}{b}\right)$.

If $a/b$ is irrational then $f(x)$ is not periodic.

edited Jan 30 '23 at 08:26

answered Jan 30 '23 at 08:20

MathDona

632

score 1 · Answer 2 · answered Jan 30 '23 at 08:14

If the ratio $b/a$ is not rational then the resulting function will not be periodic. For example, the tritone (e.g. C to F#) in well tempered tuning has a ratio of $\sqrt 2$.

If the ratio is rational, $\frac{b}{a} = \frac{p}{q}$ where $p$ and $q$ are integers with no common factor then $p$ cycles of the $a$ wave will match $q$ cycles of the $b$ wave. The sum will be periodic. Generally, this will the period but they may be exceptional cases, e.g. if $a = -b$ then the sum will be $0$.

E.g. a perfect fifth in just tempered tuning has the ratio $\frac{3}{2}$ so $2$ cycles of the lower frequency will match 3 cycles of the higher.

As well as assuming the same amplitude, you are assuming that the waves start together. This won't be true in general.

Yes, i have encountered a problem with well-tempered tuning. and thus I have done all calculations based on just tempered tuning. you'll have to give me a moment for me to check it out. — Bouncyy, Jan 30 '23 at 08:28

period of a sine function in the form of $F(x) = \sin(ax) + \sin(bx)$. where the amplitudes of the two individual sine waves are the same?

2 Answers2