The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions are derived from the physical understanding of how the standing waves are sustained, that is, how the vibrating particles of the medium behave at the boundaries imposed on them.
For a tube open at one end and closed at the other filled with air, the modes are such that there is always an antinode at the open end and a node at the closed end. Interestingly, the resonant frequencies depend on the speed of sound, of which depends on temperature. This can pose noticeable problems, for example, for organs in old unheated cathedrals. It is also why musicians commonly bring their wind instruments to room temperature before playing them.
The harmonics for a tube can be similarly derived. First, the boundary conditions must be noted; the air molecules are free to vibrate at both ends. As a result, both ends are antinodes. This can be used to determine the mode's wavelength which is equal to an integer multiple of half the tube's length. The harmonics follow the same mathematical pattern as that of standing waves on a string, which has nodes at both ends.
