TITLE: Condition for true periodicity
QUESTION [0 upvotes]: In Vibrations and Waves, French writes that

The condition for any sort of true periodicity in the combined motion is that the period of the component motions be commensurable- i.e., there exist two integers $n_1$ and $n_2$ such that $$T=n_1T_1=n_2T_2$$

But what is the justification for this?

REPLY [2 votes]: You really should put sufficient details in the question that someone can understand the context without actually going to the book in question.
Here we have a combined signal such that $$x=x_1+x_2$$ where $$x_1=A_1 \cos(\omega_1 t)$$ $$x_2=A_2 \cos(\omega_2 t)$$ so $x_1$ and $x_2$ are periodic with periods $T_1$ and $T_2$ respectively. This implies that $$x_1(t+n T_1)=x_1(t)$$ $$x_2(t+nT_2)=x_2(t)$$ for all integers $n$. Also, if $n_1$ and $n_2$ are integers then $n_1 n$ and $n_2 n$ are also integers. So if $x$ is periodic then $$x(t+nT)=x(t)$$ $$x_1(t+nT)+x_2(t+nT)=x_1(t)+x_2(t)$$ $$x_1(t+nT)+x_2(t+nT)=x_1(t+n_1 n T_1)+x_2(t+n_2 n T_2)$$ so we have $$nT=n_1nT_1$$ and $$nT=n_2 n T_2$$ and thus $$T=n_1 T_1 = n_2 T_2$$
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