What I think has been missing so far is a discussion of harmonics, i.e. multiples of the fundamental frequency of the whistle. If the fundamental frequency is 1000 Hertz, there are also some other frequencies present, such as 2000, 4000, etc. These are all pressure waves that need some space to do their thing - i.e. resonate. A wider whistle will provide the space for more pressure waves of varying harmonic frequencies and hence a "richer" tone. A narrow whistle will only provide space for the fundamental frequency without any overtones - hence a shrill sound. The foregoing is a bit of a hand-wave, but I believe it is correct.
Indeed, as Hansel says short and stubby = high pitched, and long and skinny = deep pitched, but I'll add that wide = rich and skinny = shrill.
-John