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Consider a common dynamic microphone, where the membrane moves an attached coil in the field of a permanent magnet.

I believe that the voltage produced by the coil as the membrane is moving is proportional to its velocity and not the position relative to the magnet.

Thus, is the output voltage proportional to the rate of change of the input sound waveform and not the shape of the waveform itself?

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It is my understanding that said microphone does record the first derivative of position. This is why you get little low frequencies in the recording, the microphone acting like a high frequency pass filter.

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Microphones can be divided into two general classes: velocity and pressure types. Generally speaking, most velocity mics have a coil of wire attached to a flexible diaphragm which moves slightly when a sound wave strikes it. A magnet near the coil then induces a current in the moving coil that is proportional to the velocity at which the coil is moving through the magnet's field. Because the diaphragm is mechanically soft, it moves a significant (but still microscopic) distance when the wave strikes it. Depending on the details of its construction, the output of a velocity mic can be either a current with only an infinitesimal amount of voltage "pushing" it (in which case it is a low-impedance source) or a high-impedance source (see below).

A pressure mic is one in which the active element is a slab of piezoelectric material so stiff that it moves almost not at all in response to a sound wave, yet still produces an electrical signal which in this case is proportional to the pressure exerted upon it by the impinging wave. That signal is a voltage with only infinitesimal current "behind" it and hence is a high-impedance source.

There are hybrid mic designs out there that blend the characteristics of moving-coil elements and piezos which I can describe if you are interested.

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