In addition to being a faithful soldier and producing beautiful sound, the flute can help us understand the way that harmonics work after we know how it produces sound. The flute can be thought of as an open pipe. The pressure at the ends will be atmospheric, but the pressure inside the tube will not be.
The maximum variations in air pressure are the anti-nodes, and we can see in this image, which is the longest standing wave that can support the zero atmospheric pressure condition at the end of the flute.
The wave to the right represents the displacement of air molecules, and it makes sense that the molecules are most free to move at the ends where there is no pressure variance.
The longest standing wavelength corresponds to the lowest note the instrument can produce, which is typically a C4 in scientific pitch notation. My flute looks a bit different because it has B footing, which means it can play the B below C4, or a B3. In truth, I’ve only ever played this once or twice in a concert setting and we don’t have to worry about this bonus note too much. We’ll use C4 as our base note moving forward.
Music technique analogies can get a bit hairy, especially when being described in text instead of in person where it is a bit easier to demonstrate an idea. Bear with me as I make a few comparisons to help explain how a note is produced. When we want to play a low C, we use all of our fingers—except for one pinky—to press down the keys, and we blow with a relaxed embouchure that creates slower, warm air. This makes an air jet stream that will then cause the flute to vibrate at the proper frequency to produce the C4, which flautists will call a low C. To change the fundamental frequency, we can change the configuration of keys we press down, and this will shorten the length of the air column inside the flute, creating a higher pitch.
However, each sound we hear from the flute is not comprised of just one standing wave at one frequency. There are other frequencies that are integer multiples of the fundamental frequency that are present as well, and these frequencies are called harmonics. In fact, every note you hear from every instrument is comprised of multiple different harmonics in addition to the fundamental frequency.
The comparison graphs to the right came from a study you can find here considering possibilities for searching through databases of music. All instruments are playing the same note, C4, but you can see that the flute has a different sound pressure response than both the piano and trumpet have. The number and intensity of harmonics present in each sound wave is what allows our ears to distinguish one instrument from another, even if they are all producing the same pitch. As a function of acoustic pressure, each harmonic higher will have one more pressure anti-node, and they behave in a sinusoidal pattern, which this diagram demonstrates.
Another way we can visualize the harmonics present in the composition of the sound of a low C on a flute is through their placement on the musical staff (at right). Also, I can play these harmonics on my flute. I was able to play the first 6 harmonic intervals with decent tone. There are other fingerings that allow for the higher pitches to be played more clearly, and the tone gets to be a little icky in the upper register when you are only adjusting your embouchure and air speed on the low note fingering configuration, and nobody wants to hear squawky flute, not even in the name of science.
You can see I didn’t change the configuration of my fingers, but I was able to produce those different tones by just blowing with a more focused and faster column of air and creating a tighter embouchure. This allowed the flute to hone in on the other frequencies present in that larger composition of waves that make up the sound of C4. It’s kind of crazy to imagine that every single note we hear is composed of a tiny symphony of frequencies, but each time we play a note on any instrument, we’re conducting the ensemble of frequencies to work with us.
Some graphs above courtesy of Research Gate.