Wide pipes are weak in upper harmonics

Evidence

The length of the resonator controls the f0 of the air reed (see Figure 4.15), and the natural modes of the pipe are the frequencies that the pipe can support in its output. The amplitude relationship between the pipe modes is governed by the material from which the pipe is constructed and the diameter of the pipe with respect to its length. In particular, wide pipes tend to be weak in upper harmonics. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1182