Why can a wave only have a single frequency if it goes on to infinity? I have read that the only way a wave can have a single frequency is when it goes on for all infinity.
Could someone explain me in plain english the reason?
 A: The fundamental reason is to be found in the context of Fourier analysis, but I'll try not to use that.
Imagine you have a plane wave of a single frequency $\omega$. In 1D for example, it can be written as $A \cos (\omega t - k x)$. This mathematical function takes values for $x \in \mathbb{R}$ and $t \in \mathbb{R}$.
If now you take a signal of frequency $\omega$, but that doesn't "go on forever", you have a function $A \cos (\omega t - k x)$ that takes values for $x \in \mathbb{R}$ and $t$ in a certain finite interval, say $t \in [0, \tau]$. I think you'll agree with me that this is not the exact same function : it takes value $A \cos (\omega t - k x)$ for $t \in [0, \tau]$ and zero otherwise. This is the reason why we cannot say, strictly speaking, that this signal has frequency $\omega$ : in fact, it is composed of a spectrum of many different frequencies that are all very close to $\omega$.
However, if the time $\tau$ is long enough (by this i mean much greater than the signal's period $T = \frac{2 \pi}{\omega}$), the frequencies different than $\omega$ will have a very small contribution to the signal. In that case, you can say that your signal is a good approximation of a plane wave of frequency $\omega$.
A: I think a single frequency is an oxymoron. If by single frequency you mean one cycle then try to imagine a pendulum on a clock swinging one time only. A frequency needs a duration. It's easy to imagine one swing per second but try to imagine one swing forever.
