How to calculate acoustic impedance of a megaphone?

Question

I'm working on a physics problem that asks how a passive megaphone (simple cone shape) can optimise the transfer of the human voice, compared to not using one at all. I understand that some of this is due to the megaphone simply directing the sound waves in one direction, but I've also come across explanations involving acoustic impedance matching. I understand the basic premise as: if a sound wave meets an interface between two media with vastly different acoustic impedances, then most of the sound will be reflected, instead of transmitted.

I've come across the formula Z = p/U for acoustic impedance, where p = sound pressure in pascals and U = volume flow rate. Here's where some of my confusion lies. If I use this to calculate the acoustic impedance of the megaphone, then won't louder sounds simply increase the acoustic impedance? And how does impedance matching work between the end of the megaphone and the open air? And can anyone give me an accurate calculation of the acoustic impedance of the human vocal tract?

niels nielsen · Answer

see my comment above. I'll try to add some useful points here.

@farcher is right, there is a well-developed model of impedance transformation called horn theory which contains everything you need to get a handle on this.

A megaphone can be modeled as an impedance transformer that has one characteristic impedance at its entry port and another at its exit. Ideally, the entry point is a match to the impedance of the vocal tract and the exit point is a match to the impedance of free air.

To get you started on the impedance analysis, consider that from a volume displacement standpoint, the megaphone exhibits continuity- that is, any volume displacement at one end has to be matched by a corresponding displacement at the other end, and that those displacements are linked by the geometry of the megaphone.

How to calculate acoustic impedance of a megaphone?

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