Why does a square wave have more harmonics?

The harmonics of a square wave exist because the rate of change (first derivative) of a square wave consists of very high, sudden peaks; infinitely high spikes, in the limit-case of a so-called perfect square wave. Real physical systems can’t follow such high rates, so the signals get distorted.

A square wave behaves the exact same way as a sine wave, in that as its fundamental frequency increases, you will see more cycles in a given amount of time . Square waves theoretically have infinite bandwidth.

3: A square wave consists of a fundamental frequency sinusoid and the superposition of the odd harmonics of the fundamental. The amplitude of each harmonic is equal to the amplitude of the fundamental scaled by the inverse of the harmonic index.