Waves in Pipes

This is far from obvious but - the mathematics of sound waves (harmonics) in organ pipes and waves (harmonics) on a string are mathematically identical.  But here are some distinctions:

Waves on a string are TRANSVERSE - this means that the wave vibrates in a direction perpendicular to the direction of wave travel.  These are traditional looking waves.

Sound waves are LONGITUDINAL (also known as COMPRESSIONAL) - this means that the wave vibrates in a direction parallel to the direction of wave travel.  See the second set of illustrations above.

When you speak, you are oscillating the air around your mouth.  It vibrates BACK AND FORTH (not up and down).  Each air molecule vibrates the air molecules next to it and the impulse/wave travels at the speed of sound - which in room temperature dry air is around 345 m/s.

Now tubes that are open on both ends are forced to produce waves that have anti-nodes on both ends - meaning that there is nothing for the sound to bounce off of.  This is similar to strings which have nodes on both ends - something to bounce off of both ends.  In both cases, the wavelength is the same for the resonant frequency:

wavelength (for n=1) = 2L

And the sequence of harmonics is exactly the same as for strings.

Waves in tubes LOOK different than waves on strings, but they act very similarly and the mathematics are the same.

(If the tube is closed on one end, you are forced to have an anti-node on one end only.  This is trickier.  See the top 2 images of figure 1 above.)

Another image that depicts the sound in organ pipes.  Below are 6 pairs of images.  The first 3 pairs depict the waves formed in organ pipes open at both ends.  Pairs 4-6 depict the waves formed in organ pipes capped on one end.  There is a major difference with tubes capped at one end - since you are forced to have a node at one end and an antinode at the other, you only get ODD harmonics.  The wavelength is also doubled (compared to the same harmonics in tubes open at both ends).  Since the wavelengths are twice as long, the frequencies are half as much.  This means that a resonant frequency (n=1) for a tube open only on one end is half as much (one octave lower) than the same length tube open on both ends.

In other words, if you cap a tube on one end, the tone produced is one octave lower.