If the person is shaking her hand up-and-down 12 times per second, what is the wave speed?

We, first of all, it would take incredible muscular development and control to be able to do that, and I don't believe it's actually humanly possible.

But for Math and Physics problems, that's OK.  We don't mind suspending our disbelief, accepting a temporary alternate reality, and working with the hand that is dealt.

The speed of a wave doesn't depend on how the wave is created.  A puppy wagging its tail, a fly batting its wings, or a person shaking her hand up and down, are moving the air.  The wave that travels away from the vibration is a sound wave in air.  Its speed depends only on the characteristics of the air it travels through.

For some typical combination of temperature, pressure, and humidity, this speed (of sound) is taken to be 343 meters per second.

Notice that the 'sound' of shaking her hand up and down will not be 'heard' by anyone, no matter how close she stands to them.  12 Hz (12 per second) is not a fast-enough wiggle to be sensed as sound by human ears.  If the person senses the wave at all, it will only be as some kind of pulsating breeze.

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explanation: