What are harmonics?

A harmonic is a sound wave that has a frequency that is an integer multiple of a fundamental tone. The lowest frequency sound that can be produced on the tube is the fundamental tone frequency. The frequency twice that of the fundamental tone is the second harmonic, and the frequency three times the frequency of the fundamental tone is its third harmonic.
For example, if the fundamental tone is C, then its second harmonic is C in the next octave up, and the third will be G an octave and a perfect fifth higher, and the fourth will be C two octaves up. From low to high, the harmonics will be C, C, G, C, E, C, A♭.
Wind instruments can produce harmonics with all pitches, but the volume at which various harmonics are produced varies based on the instrument, the player, and the pitch. This combination of harmonics is the very thing that creates the timbre of the instrument.
One representative example of something that creates a pure sound, or only a fundamental tone with no harmonics, is a tuning fork. For more about harmonics, we recommend Yoshinori Ando's Instrument Acoustics and Atsushi Hashimoto's The Science of Instruments (Blue Backs series).