Galvanometer critically damped

Exercise. The response of a critically damped ballistic galvanometer to a current pulse at t = 0 is u(t) = ct e-yt. Find the spectral density of the response to a stationary stream of independent random pulses. 

The final case is that of critical damping, in which the quantity inside the square root in Eq. (8.41) exactly vanishes. This case is not likely to happen by chance, but it is possible to construct an oscillating object such as a galvanometer mirror or a two-pan balance beam that is critically damped by a magnetic field. The condition for critical damping is... 

For theoretical purposes the mean value of the current oscillations, recorded with a critically damped galvanometer with a period of oscillation of about 10 sec must be measured, because most equations have been derived to incorporate this quantity. For analytical purposes, on the other hand, the upper or lower peaks of the current oscillations of a critically damped galvanometer can be measured as well. It is important that all waves to be compared are measured in the same way. The measurement of the peak current obtained with an undamped galvanometer cannot be recommended. 

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