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

  1. #1
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    [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?

    Thanks in advance for your help!
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  2. #2
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    Re: [TEX]f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)[/TEX]

    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.
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  3. #3
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    Re: [TEX]f(t) = \sin(\omega_1 t)+\sin(\omega_2 t)[/TEX]

    Quote Originally Posted by SlipEternal View Post
    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.
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