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In my book, I have read that the frequency of sound is inversely proportional to the time period i.e., $1/T = \nu$. So does that mean $$\text{frequency} \times \text{time period} =1$$ i.e., is $\nu \,T = 1$ unit??

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Yes. The frequency of $\nu$ of any monochromatic wave is defined as $\nu=1/T$, so that $\nu \, T\equiv 1$ is an identity by definition.


As for the initial version of your question: the product of the speed of sound $c$ and the period $T$ of a (monochromatic) wave is $$ c \, T = \lambda, $$ the wavelength of the wave.

The fact that a quantity $a$ is "inversely proportional to" some other quantity $b$, denoted $a\propto 1/b$, means that there exists some constant $k$ such that $a=k/b$, not that $a=1/b$ with that constant equal to unity. Moreover, if the product $ab$ is not dimensionless (like the product $cT$, a speed times a time, which gives a length) it is completely impossible for that constant to be unity or any other dimensionless number.

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  • $\begingroup$ Actually I meant frequency* time period. $\endgroup$ – user179174 Feb 23 '18 at 13:13
  • $\begingroup$ @AnkitBehera Then have a downvote for posting before checking your post. $\endgroup$ – Emilio Pisanty Feb 23 '18 at 13:14
  • $\begingroup$ I admit that I hadn't checked before posting but please answer my question $\endgroup$ – user179174 Feb 23 '18 at 13:15

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