What is meant by a 'pure' wave?


A sine or cosine wave has just one frequency and it called a pure wave for that reason. If you have a periodic function with a different shape you can Fourier analyze it and get some number of different frequencies. These are not pure waves because of the multiple frequencies.

A "pure" sine (or cosine) wave is a wave of the form $y(t)=c_1\sin(\omega t+\phi)$ or $y(t)=c_2\cos(\omega t+\phi)$. Essentially, it's a wave that may have been translated, scaled, or have its period modified, but at it's core it's still a sine or cosine wave.

This is contrary to sums of various waves. A core result of a field known as Fourier Analysis is that any periodic function can be approximated as a sum of the "pure" sine and cosine waves. Here, the sines and cosines are of the form $y_n(t)=b_n\sin(nt)$ or $y_n(t)=a_n\cos(nt)$ for certain series $a_n$, $b_n$ known as the fourier series of your function.

So, given a periodic function $f(t)$, you can find that $f(t)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n\cos(nt)+\sum_{n=1}^\infty b_n\sin(nt)$. So, you can use an infinite number of "pure" sine/cosine waves to represent any periodic function.

Note that a pure wave isn't just having certain properties sine/cosine have - even waves that look fairly similar to sine to the untrained eye (like the triangle wave - while it has sharp boundaries, it (can) have similar period and amplitude) can be very complex when written in terms of sines/cosines. While you (usually) need the full infinite number to perfectly describe a wave, you can get surprisingly good approximations relatively quickly (here $n$ is the number of sine terms, and there are no cosine terms).

In electrical technology any periodic motion can be expressed as a sum of the fundamental wave and several harmonics whose frequencies are multiple of the fundamental, expressible in Fourier series. If the wave is pure it means that amplitudes other than the fundamental are zero. The other harmonics are seen as "contaminating" the pure sine wave.

It is pure or aka simple harmonic, as is represented by the harmonic dynamic time differential function which strictly has a time period $T$ where $ \omega T = 2 \pi $ in:

$$ \ddot x + \omega^2 x =0. $$

A square wave for example has several non-fundamental additives to the pure signal. It consistes of third, fifth,.. order harmonics:

Other Harmonics in Square Wave