Cylindrical Magnet Travelling Through a Coil Inductor

[c_mitra]

c_mitra's claim that a magnet passing completely through a coil will display only the characteristics of a single pole is demonstrably incorrect. As you point out, it is bipolar and the result will be sinusoidal in nature. Unless he's using monopoles, in which case I congratulate him for his discovery!

You are not understanding what I was trying to tell. The lines of force for a bar magnet can be approximated as the overlap of two equal and opposite poles. At a large distance, the magnetic field changes as r^(-3) (and the theta) but a coil sees only the magnetic field and not the magnetic poles. At short distance, the magnetic field is a complex function of distance and position. You will get a dispersion like voltage (if the magnet moves with constant velocity through the coil.

Unless the magnet motion is sinusoidal (simple harmonic or circular), the voltage produced will not be a sine graph.

By mathematical definition, the bipolar plus is unquestionably sinusoidal. And it does have a specific frequency. A period of one cycle does not negate the basic equations or strip it of a frequency The wavelength is of a specific duration and the formula applies.
Cylindrical magnets passed through in series, separated by small distances, at a constant velocity create a very pure sinusoidal waveform of multiple cycles.
So, there we have it!

What kind of mathematical definition you are talking about?