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I have the function

$$f(t)=\sin(t)+\sin(\sqrt2t)$$

I would like to calculate the fundamental period of $f(t)$. I know that the period of $\sin(t)$ is $2\pi$ and the period of $\sin(\sqrt2t)$ is $\sqrt2\pi$. I sense that I must work out the lcm of $2$ and and $\sqrt2$ but I'm unsure on how to do this.

kingW3
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  • Proper notation is $\sin t$ or $\sin(t)$, not $sin(t)$. It's coded as \sin t or \sin(t). The backslash not only prevents italicization but also provides proper spacing in things like $a\sin b$ and $a\sin(b)$. I mention both of these last examples so that you can see the context-dependent nature of the spacing: less space appears to the right of $\sin$ in $a\sin(b)$ than in $a\sin b.\qquad$ – Michael Hardy Feb 12 '17 at 17:09

2 Answers2

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Your equation boils down to solving for some $a,b\in\Bbb{Z}$ $$2a\pi=\sqrt{2}b\pi$$ But this is clearly not possible because then $b=\sqrt{2}a$,but $\sqrt{2}$ is irrational and $b$ isn't.So $f(t)$ is not periodic.

kingW3
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When two numbers have an lcm ‒ say $18$ and $24$ ‒ there are integers by which you can multiply both of them to get the lcm, thus: \begin{align} 18 \times 4 & = 72 \\ 24 \times 3 & = 72 \\[10pt] \text{so } \operatorname{lcm}(18,24) & = 72 \end{align} and those two multipliers are $4$ and $3$. What would be the multipliers in the case of $2$ and $\sqrt 2\text{ ?}$ \begin{align} 2\times n & = \operatorname{lcm} \\ \sqrt 2 \times m & = \operatorname{lcm} \end{align} so $\dfrac m n = \dfrac 2 {\sqrt 2} = \sqrt 2$ and $m$ and $n$ are integers and the fraction $m/n$ is in lowest terms.

A bit of algebra shows that $\dfrac{2n-m}{m-n} = \sqrt 2$ and then this expresses that fraction ins lower terms. That is impossible if it was already in lowest terms.

Conclusion: there can be no such integers; therefore $\sqrt 2$ is an irrational number.

Further conclusion: There is no such lcm.

And the punch line: Therefore, this function is not periodic.

Michael Hardy
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