Thread: [TEX]f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)[/TEX]

1. [TEX]f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)[/TEX]

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?

Is the question asking you to determine the characteristics of the functions $f_1$ and $f_2$ to determine when the function $f$ will be periodic? Or is it looking for the period of the function? Because those are very different questions.
Is the question asking you to determine the characteristics of the functions $f_1$ and $f_2$ to determine when the function $f$ will be periodic? Or is it looking for the period of the function? Because those are very different questions.
I have just realised that it is unclear: $f_1$ and $f_2$ are not functions they are constants (representing frequency). I believe that part is irrelevent to the math.
The question is asking which conditions $\omega_1$ and $\omega_2$ need to satisfy in order for $f(x)$ to be periodic.