TITLE: Is the sum of two sine waves periodic?
QUESTION [11 upvotes]: Is the sum of two sine waves periodic? Specifically, what will be the period of $\sin(mx)+\sin(nx)$ where $m$ and $n$ are real constants?

When $m$ and $n$ are integers, it is not difficult to show by definition that $\sin(mx)+\sin(nx)$ is periodic. But I don't know how to handle the general cases. Can anyone help?

REPLY [2 votes]: The period of $\sin(n_1x)$ is $T_1 =\tfrac{2\pi}{n_1}$ and that of $T_2 =\sin(n_2x)$ is $\tfrac{2\pi}{n_2}$. These are the lengths on the real line after which the functions repeat. For combined period, we need the smallest value so that
$$n_1T_1 = n_2 T_2$$
where $n_1, n_2 \in \Bbb Z$. Thus as Oleg said, we need $\tfrac{T_1}{T_2} \in \Bbb Q$ for the sum to be periodic.