In some practical problems, the quantity of interest is not the measured signal, but the power of the measured signal. In electrical engineering, the measured quantity may be voltage, and for resistive loads power is proportional to voltage squared. For mechanical vibrations, we recall energy = force*displacement. So power=force*velocity. For a viscous damper, force = c * velocity. Thus power dissipated = c * velocity^2. By similar means we can get a signal proportional to power by squaring the signal of interest.
