Analytical expressions for effective weighting functions used during simulations of water hammer

For some time, work has been underway aimed at significant simplification of the modelling
of hydraulic resistance occurring in the water hammer while maintaining an acceptable error.
This type of resistance is modelled using a convolution integral, among others, from local
acceleration of a liquid and a certain weighting function. The recently completed work shows
that during efficient calculations of the convolution integral, the effective weighting function
used does not have to be characterised by large convergence with a classical function (according
to Zielke during laminar flow and to Vardy-Brown during turbulent flow). However, it
must be a sum of at least two or three exponential expressions so that the final results of the
simulation could be considered as satisfactory. In this work, it has been decided to present
certain analytical formulas using which it will be possible to determine the coefficients of
simplified effective weighting functions in a simple direct way.