First let's make some approximations: we'll pretend a flute and clarinet are the same length. For the moment we'll also neglect end corrections, to which we shall return later. The next diagram (from Pipes and harmonics) shows some possible standing waves for an open pipe (left) and a closed pipe (right) of the same length. The red line is the amplitude of the variation in pressure, which is zero at the open end, where the pressure is (nearly) atmospheric, and a maximum at a closed end. The blue line is the amplitude of the variation in the flow of air. This is a maximum at an open end, because air can flow freely in and out, and zero at a closed end. These are what we call the boundary conditions.

Open pipe (flute). Note that, in the top left diagram, the red curve has only half a cycle of a sine wave. So the longest sine wave that fits into the open pipe is twice as long as the pipe. A flute is about 0.6 m long, so it can produce a wavelength that is about twice as long, which is about 2L = 1.2 m. The longest wave is its lowest note, so let's calculate. Sound travels at about c = 340 m/s. This gives a frequency (speed divided by wavelength) of c/2L = 280 Hz. Given the crude approximations we are making, this is close to the frequency of middle C, the lowest note on a flute. (See this site to convert between pitches and frequencies, and flute acoustics for more about flute acoustics.)

Note that we can also fit in waves that equal the length of the flute (half the fundamental wavelength so twice the frequency of the fundamental), 2/3 the length of the flute (one third the fundamental wavelength so three times the frequency of the fundamental), 1/2 the length of the flute (one quarter the wavelength so four times the frequency of the fundamental). This set of frequencies is the (complete) harmonic series, discussed in more detail below.

Closed pipe (clarinet). The blue curve in the top right diagram has only quarter of a cycle of a sine wave, so the longest sine wave that fits into the closed pipe is four times as long as the pipe. Therefore a clarinet can produce a wavelength that is about four times as long as a clarinet, which is about 4L = 2.4 m. This gives a frequency of c/4L = 140 Hz – one octave lower than the flute. Now the lowest note on a clarinet is either the D or the C# below middle C, so again, given the roughness of the measurements and approximations, this works out. We can also fit in a wave if the length of the pipe is three quarters of the wavelength, i.e. if wavelength is one third that of the fundamental and the frequency is three times that of the fundamental. But we cannot fit in a wave with half or a quarter the fundamental wavelength (twice or four times the frequency). So the second register of the clarinet is a musical twelfth above the first. (See clarinet acoustics for more detail.)

* It's worth adding that the flute is not entirely open at the embouchure: the hole across which the player blows is smaller than the cross section of the pipe. This narrowing does have an acoustic effect. Nevertheless, it is sufficiently open that large oscillating flows of air can enter and leave the pipe with very little pressure difference from atmospheric. Low pressure, high flow: this boundary condition is a low value of acoustic impedance. The clarinet is not completely closed by the reed: a small, varying aperture is left, even when the player pushes the reed towards the mouthpiece. However, this average area is much less than the cross section of the clarinet so the reflection of the acoustic wave is almost complete, and the acoustic flow is very small, in spite of the large acoustic pressure produced by the vibrating reed. High pressure, low flow: it is a high value of acoustic impedance. See Flute acoustics and Clarinet acoustics for details.

Air motion animations

A few notes about these animations. First, the density variations are not to scale: sound waves involve variations of density that are a tiny fractional