Why monochromatic waves exist even theoretically

I was reading this question. Where the author made a statement that

If I take a wave with period T, it is also true that it has a period 2T, 3T and so on. That is, it has frequencies of 2π/T, but also 2π/2T and so on.

I know if I take a (quasi) monochromatic wave of frequency $\omega$ it do not contain the sub-harmonics ($\omega/2,\omega/3...$).

I also believe that the statement (as I have quoted above) given by the author should not be correct but can not figure out why it should be wrong.

My question is: what physical property of the wave prohibit the multiples of time period to be taken as the time period of wave, and if this statement is not true what prevent the presence of sub-harmonics in that wave.

• It's just semantics - when physicists refer to the period of a wave, it's understood that they mean the smallest amount of time it takes for a waveform to repeat itself. In the physics sense, a wave of period 5 seconds is a different wave, and does not contain as a frequency component, a wave of period 10 or 15 seconds. So, while yes, a wave of period 5 seconds does also repeat every 10 seconds, it's not correct in a physics context to say the period of the wave is also 10 or 15 or 20 seconds. – Brionius Jun 18 '16 at 16:13

Let's back up.

How do you know that a monochromatic wave of frequency $\omega$ doesn't contain any component at $\sqrt{2} \omega$? It's because these two frequencies are not commensurate: if you plot $\sin(\omega t)$ and $\sin(\sqrt{2} \omega t)$ they'll have no clear relation. The peaks of one look like random points in the other. Then it's clear their overall overlap must be zero.

Now let's compare $\omega$ and $\omega/4$, which I've plotted below.

They are commensurate, but every point where they line up is cancelled by a point where they are opposite. The overall amount of $\sin(\omega t/4)$ in $\sin(\omega t)$ is zero.

Physically, if you had an ideal oscillator with resonant frequency $\omega / 4$, and imposed an external force like $\sin(\omega t)$, you'd be pushing it the 'right' way and the 'wrong' way half the time, so the oscillator doesn't do anything.

Mathematically, this occurs because sinusoids are orthogonal. We can choose a different basis for periodic functions which isn't orthogonal, but it wouldn't be useful (because everything would have sub/superharmonics, as you said) or physically realistic.

• In principle I agree with the gist of your answer however the sum of two waves is not zero but will look like as the high frequency wave is riding over the low frequency wave. My doubt is if the wave of frequency $\omega$ do not contain its sub harmonics, but one can take 2T, 3T as possible periods then is the connection between frequency of the wave with its time period is really fundamental in nature. – hsinghal Jun 18 '16 at 18:10
• @hsinghal It's not about the sum. I'm talking about the inner product. – knzhou Jun 18 '16 at 18:45
• The product will give $\omega_1\pm omega_2\$. Anyways it is another topic and we are in different plans. – hsinghal Jun 19 '16 at 5:24

To say that a wave, say with amplitude given by f(t), has a period $T$ means that not only $f(t+T) = f(t)$, but also that $T$ is the smallest value that has this property. Given that $f(t+T) = f(t)$ then it follows that $f(t + nT) = f(T)$ where $n$ is an integer (for example $f(t+2T) = f(t+T+T) = f(t+T) = f(t)$).

• Do you mean that although the time periods having multiple of the shortest periods can also be treated as the time periods but the wave can not contain the corresponding sub harmonic frequencies. – hsinghal Jun 18 '16 at 18:19