# Can frequency be equal to 0?

Is correct to speak about frequency equal to 0 ?

$$f= \frac{1}{t}$$

If $t\rightarrow\infty$ can I consider that the frequency is equal to 0 ?

• well I would call static electric (magnetic ) fields as the limit for wavelength going to infinity of the electromagnetic field Commented Feb 11, 2014 at 12:18
• In fact, there is no periodic process, because it means that it wold continue for ever. Commented Feb 11, 2014 at 12:33
• A non-periodic signal may be assumed as limiting case of a periodic signal where the period of the signal approaches infinity. It is wrong to say $f=0$. $T \to \infty \implies f\to 0$ The concept $f→0$ is used in Fourier transform analysis. It would be good to see here an answer from the experts. Commented Feb 11, 2014 at 12:50
• Are you actually asking if the universe will someday end? Because strictly speaking it looks as if the smallest frequency you can ever measure at any give time is 1/Tbigbang. In case of an electric circuit, you can't have 'direct current' (a term I an using here to mean f=0), the best you can have is a step-like signal starting with the powering up (or building) of your circuit. Commented Feb 11, 2014 at 17:04

Yes. For example, the frequency of times you go to space is zero.

• Yes for what? For"Is correct to speak about frequency equal to 0 ?" OR "f t→∞ can I consider that the frequency is equal to 0 ?" I guess yes is the answer to the former not later. Commented Feb 11, 2014 at 13:02
• @anupam Manly for the first question, but you could apply it to the last part. Commented Feb 11, 2014 at 13:24
• No the last line"If t→∞ can I consider that the frequency is equal to 0 ?" We cant consider $f=0$ We can say $(\lim_{T \to \infty}f)\ =0$. Commented Feb 11, 2014 at 13:36

A quantity oscillating with frequency equal to zero would simply be static or constant.

EDIT

When $T$ goes to infinity, it is not possible for an observer to see that the phenomenon is periodic. Think about $T=\text{a few times the age of the universe}$, for instance. If there is no observable periodicity the concept of frequency is not physically relevant.

Now, it is possible to give a meaning to $f=0$ that is consistent with the usual concept of frequency. Since frequency is the mean number of occurences of an event per time unit, if this event never happens, the frequency can be considered as equal to zero.

More technically, a quantity $A$ that is periodic, for instance the motion of a pendulum, can be expressed as a sum of harmonics like $$A(t)=\sum_{n=0}^\infty a_n\cos(\varphi_n+2\pi n\;ft).\tag{1}$$ This a called a Fourier series. It is a sum of sinusoidal curves with frequencies $nf$. The coefficients $a_n$ and $\varphi_n$ depend on the nature of the movement. In the mathematical theory of Fourier series, the expression (1) starts with $n=0$, which can be understood as a contribution of frequency $0\times f=0$. The zero frequency term in a Fourier series is constant (equal to $a_0\cos\varphi_0$ in (1)) and is the time average of $A$.

All of this is a matter of interpretation. My conclusion is that it is correct to speak about frequency equal to zero if this is consistent with other definitions of the frequency and if it makes what you are doing clearer.

Frequency of a wave is equal to the inverse of the period. If the period goes to infinity, that means that the wave has only a single crest and trough. In some sense in this limit it stops being a wave, but just a single "disturbance". This is certainly possible, at least in principle, in some media.

You can say

$$\lim_{t \to \infty} \frac{1}{t} = 0 ,$$

so when the period tends to infinity, in the limit the frequency is 0.

• It would be better to say "frequency tends to $0$". Commented Feb 11, 2014 at 11:41