Problem:

Let $\displaystyle f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)$ where $\displaystyle \omega_1 t = 2 \pi f_1$ and $\displaystyle \omega_2 t = 2 \pi f_2$.

When is $\displaystyle f(t)$ periodic?

My Attempt:

Let $\displaystyle f(t)$ be periodic.

Let the time period of $\displaystyle f(t)$, $\displaystyle \sin(\omega_1 t)$ and $\displaystyle \sin(\omega_2 t)$ be $\displaystyle T$, $\displaystyle T_1$ and $\displaystyle T_2$ respectively.

Since $\displaystyle \sin$ has period $\displaystyle 2\pi$, $\displaystyle \sin(\omega t) = \sin(\omega_1 t + 2\pi) = \sin(\omega_1(t + \frac{2\pi}{\omega_1}))$.

So $\displaystyle T_1 = \frac{2\pi}{\omega_1}$ and by the same logic $\displaystyle T_2 = \frac{2\pi}{\omega_2}$.

So

$\displaystyle \\f(t) &=& f(t+T) \\ &=& \sin(\omega_1 (t + T) + \sin(\omega_2 (t + T) \\ &=& \sin(\omega_1 t + \omega_1 T) + \sin(\omega_2 t + \omega_2 T) \\ &=& \sin(\omega_1 t + 2\pi) + \sin(\omega_2 t + 2\pi)\\$.

Not really sure that this gets me anywhere...

As far as I am aware in order for $\displaystyle f(t)$ to be periodic then the time periods of $\displaystyle \sin(\omega_1 t)$ and $\displaystyle \sin(\omega_2 t)$ must have a rational LCM, so $\displaystyle f(x)$ is periodic if there exist $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle a T_1 = b T_2 = r$. Is this correct? If so how can I show this?

Thanks in advance for your help!