The Stokes number provides a means of estimating the quality of PIV data sets, as previously discussed. However, a definition of a characteristic velocity or length scale may not be evident in all applications. Thus, a deeper insight of how a tracking delay arises could be drawn by simply defining the differential equations of a particle in the Stokes regime. A particle moving with the fluid at some velocity  will encounter a variable fluid velocity field as it advects. Let's assume the velocity of the fluid, in the Lagrangian frame of reference of the particle, is . It is the difference between these velocities that will generate the drag force necessary to correct the particle path:
The stokes drag force is then:
The particle mass is:
Thus, the particle acceleration can be found through Newton's second law:
Note the relaxation time  can be replaced to yield:
The first-order differential equation above can be solved through the Laplace transform method:
The solution above, in the frequency domain, characterizes a first-order system with a characteristic time of . Thus, the −3 dB gain (cut-off) frequency will be:
The cut-off frequency and the particle transfer function, plotted on the side panel, allows for the assessment of PIV error in unsteady flow applications and its effect on turbulence spectral quantities and kinetic energy.