# When is $f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)$ periodic?

Problem:

Let $$f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)$$.

What condition must $$\omega_1$$ and $$\omega_2$$ satisfy for $$x(t)$$ to be periodic? When $$x(t)$$ is periodic, what is its period?

My Attempt:

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

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

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

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

So

\begin{aligned}\\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).\\\end{aligned}

Not really sure that this gets me anywhere...

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

• Hint: If $f$ is periodic with period $T$ and has a second derivative, then $f''$ is also periodic with period $T$. Use this to prove that if $\omega_1^2 \ne \omega_2^2$, then the period of $f$ is also a period for both sines. Aug 6, 2017 at 20:23
If $f$ is periodic with period $T$ and has a second derivative, then $f''$ is also periodic with period $T$. Use this to prove that if $\omega_1^2 \ne \omega_2^2$, then the period of $f$ is also a period for both sines
Note that $\omega_1 T$ and $\omega_2 T$ are not necessarily equal to $2\pi$. In fact, $\omega_1 T = 2\pi k_1$, and $\omega_2 T = 2\pi k_2$ for some positive integers $k_1$ and $k_2$. Since $\omega_1 = \frac{2\pi}{T_1}$, and $\omega_2 = \frac{2\pi}{T_2}$, $\omega_1 T = 2\pi k_1$, and $\omega_2 T = 2\pi k_2$ imply $T = k_1 T_1$, and $T = k_2 T_2$. Lastly, we have to find the smallest $T$ that is a multiple of both $T_1$ and $T_2$. Does LCM ring any bell?