# If human hearing upper limit is 20 kHz then how does a 15 kHz sine-wave sound so remarkably different from 15 kHz square-wave ?

The first (even, sine-wave) harmonic of 15 kHz is, by definition, 30 kHz, more than the supposed ‘upper limit’ of human hearing.

The first (odd, square-wave) harmonic of 15 kHz is, by definition, 45 kHz, also more than the supposed ‘upper limit’ of human hearing.

The 15 kHz fundamental of both sine and square is the same, but the overall sound is wildly different between the two modes.

How can this be?

The audible difference between a 15 kHz sine wave and a 15 kHz square wave can be significant due to the harmonic content of each waveform. Understanding this difference requires a basic grasp of what sine waves and square waves are, along with the concept of harmonics.

### Sine Wave

A sine wave is a pure tone with no additional harmonics. It represents a single frequency with a smooth, periodic oscillation. In this case, a 15 kHz sine wave would produce a very high-pitched tone that some people might find difficult to hear, especially as human hearing declines with age and typically struggles with frequencies above 15 kHz. The sine wave’s tone is pure and may not be audible to older adults or those with hearing loss at high frequencies.

### Square Wave

A square wave, on the other hand, is rich in harmonics. In addition to the fundamental frequency (in this case, 15 kHz), a square wave includes odd harmonics (3rd, 5th, 7th, etc., of the fundamental frequency). These harmonics add complexity to the sound, making the square wave sound harsher and more “buzzy” compared to the pure tone of a sine wave.

### Harmonics of a 15 kHz Square Wave

For a 15 kHz square wave, the harmonics would be at multiples of the fundamental frequency that are odd integers: 45 kHz (3rd harmonic), 75 kHz (5th harmonic), 105 kHz (7th harmonic), and so on. Most of these harmonics are beyond the range of human hearing, which generally tops out at around 20 kHz under the best of circumstances. This means that, while theoretically richer in content due to these harmonics, the practical audible difference for a listener from the fundamental frequency might be minimal or non-existent due to the high starting frequency.

### Audible Differences

Given the fundamental frequency of 15 kHz, the primary audible difference between the sine and square wave might not be as pronounced to a human listener due to the limitations of human hearing. Young listeners with exceptionally good hearing might perceive the square wave as slightly harsher or more textured compared to the sine wave, assuming they can hear the 15 kHz frequency to begin with. However, the harmonics that typically characterize the timbral differences of these waveforms would largely fall outside the audible range for humans.

In summary, while a square wave contains harmonics that add complexity and texture to its sound, the high fundamental frequency of 15 kHz means that many of these harmonics will be inaudible to humans. The result is that the difference between a 15 kHz sine wave and a 15 kHz square wave might not be significant or even perceptible to most listeners.

So, one can believe their own hearing, or believe the vague dismissal of technology and reason.