Does the shape or speed affect the frequency of a sonic boom and if so in what way?
Is the frequency of a sonic boom always the same? If so, what frequency is it and why is it that frequency?
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"Pitch" is an attribute defined by the human sensation of sound. Therefore it is highly non trivial to connect it with a physical phenomenon very firmly (read: don't do that). The process of hearing is not fully understood, but if there is a signal processing operation roughly approximating it, it would be the cross-(or auto-)correlation in time, i.e. real-time comparing of short samples. In such procedure the fourier-like approach is inherently present without switching to the frequency domain.
From this very rough approach follows that periodicity is the key to a "constant sensation" of sound on which the pitch relays. But the sonic boom (as a shock-wave) is purely non-periodic. That is the issue.
But there are some more "buts". I am not covering all of them certainly.
Context (perception). What was just before the sonic boom? Silence? Or build-up of loudness? Does something sound simultaneously? Is it tone-like or noise-like? Recall the correlation. It really matters.
Context (physics). The pure shock wave is only an idealization. There are multiple approaches and scenarios counting viscosity and severe entropy changes in (see e.g. Hamilton and Blackstock: Nonlinear Acoustics). And don't forget the effects of sound propagation. Air isn't very dispersive but we are talking about very loud sounds. Also the resonant effects of buildings and terrain in the nearby is to be mentioned. It can affect a "pitch" of pulse like sound decisively.