The behavior of sound waves is controlled by boundary conditions. Sound waves combine two varying parameters: air displacement and pressure. The air displacement we can see by letting the air push something; the pressure displacement is invisible and can only be measured with barometric probes.
When sound encounters a rigid barrier, the air molecules next to the barrier cannot cooperate with the wave motion because the wall is in the way. Instead, the air molecules pile up against the wall making higher pressure, and then thin out near the wall, making lower pressure. This illustrates the fact that a displacement node (where the air cannot move) becomes a pressure antinode (where the pressure goes up and down a lot).
When a sound wave traveling down a pipe encounters an open end, you might think it simply escapes the pipe. But it still partially reflects — and with a pressure reversal. A a high-pressure wavefront, reaching the end of the pipe, spews out in all unconfined directions, leaving behind a LOW pressure region which reflects back down the pipe as a low-pressure wavefront.
When a low-pressure wavefront reaches an open pipe end it pulls in air molecules from outside the pipe, producing a hig- pressure region that reflects back down the pipe as a high-pressure wavefront.
If BOTH ends of the pipe are open, the wave reflects back and forth, creating a fundamental motion pattern in which the ends of the pipe are motion antinodes but stay at ambient pressure, whereas in the middle of the pipe the air molecules crowd together and form a pressure antinode.
NOW, imagine three perpendicular pipes with open ends that meet in a common center. Along each axis there will be displacement antinodes at the open ends, and all three pressure antinodes will coincide at the common center, making one stronger pressure antinode.
A wave travelling down a pipe whose cross-section narrows will eventually reflect when it gets too crowded for its wavelength. Thus, if we had three perpendicular pipes with narrowed ends, the same sort of standing waves will occur as for open ends.
The hollow rhombic dodecahedron, to sound, looks like an empty cube shared by three perpendicular square narrowing ducts that are the pairs of pyramids on the x, y, and z faces of the empty cube. Thus, the fundamental standing wave pattern looks like displacement antinodal planes perpendicular to the positive and negative x, y, and z axes, and a pressure antinode at the very center where all three axes intersect.
Another way to think of it is as a pulsating cube of air whose faces are vibrating with a point in the middle where the pressure goes up and down with the sound frequency.
