text: The bias error in particle tracking discussed in the previous section is evident in the frequency domain, but it can be difficult to appreciate in cases where the particle motion is being tracked to perform flow field measurements (like in particle image velocimetry). A simple but insightful solution to the above-mentioned differential equation is possible when the forcing function  is a Heaviside step function; representing particles going through a shockwave. In this case,  is the flow velocity upstream of the shock; whereas  is the velocity drop across the shock.
  The step response for a particle is a simple exponential:
  To convert the velocity as a function of time to a particle velocity distribution as a function of distance, lets assume a 1-dimensional velocity jump in the  direction. Lets assume  is positioned where the shock wave is, and then integrate the previous equation to get:
  Considering a relaxation time of  (time to 95% velocity change), we have:
  This means the particle velocity would be settled to within 5% of the downstream velocity at  from the shock. In practice, this means a shock wave would look, to a PIV system, blurred by approximately this  distance. 
  For example, consider a normal shock wave of Mach number  at a stagnation temperature of 298 K. A propylene glycol particle of  would blur the flow by ; whereas a  would blur the flow by  (which would, in most cases, yield unacceptable PIV results).
  Although a shock wave is the worst-case scenario of abrupt deceleration of a flow, it illustrates the effect of particle tracking error in PIV, which results in a blurring of the velocity fields acquired at the length scales of order .