where A (SI unit: m2) is the cross-section area of the pipe, ρ (SI unit: kg/m3) is the fluid density, and u (SI unit: m/s) is the tangential fluid velocity.

where Kρ is the bulk modulus of the fluid (the inverse of its compressibility), and KA is the effective bulk modulus of the cross section area. A0 and ρ0 are the reference area and reference density at a given pressure p0.

The Water Hammer wave speed c (SI unit: m/s) is given by a combination of fluid and structural material properties

The effective bulk modulus for the cross-sectional area KA (SI unit: Pa) is given by the pipe’s material properties

where E is the Young’s modulus, dh is the hydraulic diameter, and wth is the pipe’s wall thickness. This is the so-called Korteweg formula (Ref. 1)

The Korteweg formula can also be extended to pipes suffering from axial stresses. In this case, a more general formula would include Poisson’s ratio, ν, of the pipe’s material

where ζ = 1 for pipe with zero axial stress (this is Korteweg’s original formula for a pipe furbished with expansion joints), ζ = 1 − ν/2 for a pipe anchored at one end, and ζ = 1 − ν2 for a pipe anchored at both ends.

here, fD is the Darcy friction factor, normally a function of the Reynolds number, the surface roughness and the hydraulic diameter, as described in Expressions for the Darcy Friction Factor.

This set of equations are normally acknowledged as the Water Hammer equations, and are they mainly written in the literature as (consider gravity forces, so F = ρg)

The available boundary conditions are Closed, Pressure, Velocity, and Additional Flow Resistances.