As described in Wavespeed and Communication Time, the speed at which a wave can travel through a fluid is limited by its bulk modulus of elasticity and density. When constrained to a pipe, the wavespeed is further reduced as some of the wave energy is lost to deforming the pipe. An in depth derivation of wavespeed is presented here, along with discussion of the meaning of the Pipe Support constant.

Derivation Including Pipe Effects

Consider again an instantaneous valve closure. When a valve at the end of a pipe closes, a number of things happen. As we have seen, a wave is propagated upstream as discussed in the Conceptual Example. However, what happens to the pipe itself?

Fluid continues to flow into the pipe, but not out of it. This causes both radial and axial strain - the pipe may be deformed in both directions, depending on how it is supported. This increase in fluid mass continues until the wave is reflected, or in other words, for a time equal to the time required for the wave to travel the pipe:

During the entirety of this period, the pipe is being deformed. If we consider the pipe to be supported only at the upstream end, it is free to expand lengthwise. The total axial increase in length occurs at the end of the period.

Further, this means that the closed valve has a certain velocity - it moves the distance Δx over the time period L/a.

The fluid velocity at the face of the valve decreases from the initial steady value V₀ to this V_valve, making the change in fluid velocity:

The total fluid mass in the pipe increases during this period as well, by the steady rate.

Combining the previous two equations:

There are three mechanisms that allow the mass to increase in the pipe - the pipe strains axially, strains radially, and the fluid compresses.

Simplifying and reorganizing:

With the instantaneous waterhammer relationship,

the velocity change term can be eliminated, and the result can be solved for a²:

The bulk modulus of a fluid relates the change in volume (or density) to an applied change in pressure:

Multiplying the a² equation through by K/ΔP:

The area/pressure term is a measure of the elastic response of the pipe - its value will depend on how the pipe is constrained. For convenience, common terms that show up in this analysis are separated, and a new constant c₁ is defined.

Pipe Support Constants

The value of c₁ varies based on how the pipe is constrained. As an example, a thin-walled pipe fully anchored at the upstream end is discussed here, with other values of c₁ listed below.

The change in internal area of the pipe A, is equal to the change in radius multiplied by the circumference. The change in diameter is equal to the radial strain multiplied by the diameter.

Dividing by area:

The radial strain is affected both by axial and circumferential strain, which are related to one another via Poisson's Ratio ν:

Further, strain is related to stress with Young's Modulus E:

Combining the above and introducing the ΔP term:

The two stresses are related to their respective forces and areas:

Adding these to the previous equation and simplifying:

Which is able to give an expression for c₁:

Note: The above value of c₁ is only valid for a pipe anchored only at the upstream end. See below for other values of c₁ for differing constraints.

• Thin-Walled Anchored Upstream (detailed above)

• Thin-Walled Anchored Throughout

• Thin-Walled With Expansion Joints

• Thick-Walled Anchored Upstream

• Thick-Walled Anchored Throughout

• Thick-Walled With Expansion Joints

• Circular Tunnel (unlined)

• Circular Tunnel (lined) / Buried Pipe