Some physical understanding of the meaning of \(\Delta\) can be gained from Figure 5 where we have plotted the estimated probability density functions calculated in generate the previous figures. Each figure shows \(\phi(dP)\) and \(\phi(dP_+ + dP_-)\) as blue lines (solid and dashed) and \(\phi(\rho cdU)\) and \(\phi(dP_+ - dP_-)\) as red lines (solid and dashed). The centre figure shows the distributions calculated for \(c = 4.5\) m/s which corresponds to the minimum value of \(\Delta\), the left figure shows the distributions for \(c = 3.5\) m/s and the right figure shows the distributions for \(c = 5.5\) m/s. \(\Delta\) is the sum of the Jensen-Shannon distance between the blue distributions and the red distributions.

Since \(\phi(dP)\) does not depend upon \(c\), the solid blue line is the same in each of the figures. The other 3 distributions depend on the value of \(c\) and so they are different in the different figures. For smaller \(c\) the distribution of \(\ho cdU\) gets narrower and the peak is higher and for larger \(c\) the distribution of \(\rho cdU\) is broader and less high. Subjectively it is possible to see that the similarity between the solid and dashed lines of both colours is greatest in the central figure and reduced in the figures to the left and right. This is quantified by the Jensen-Shannon distance.