Reflection of pressure waves

The way in which a pipeline is terminated is decisive for the reflection of pressure waves in an hydraulic network.

If the characteristic Impedance of a pipeline changes in the direction in which a wave propagates, reflections occur. Generally speaking, a reflection leads to the emergence of two new waves - a transmitted and a reflected wave. Changes in the characteristic impedance can occur due to:

  • Closed ends
  • Open ends
  • Changes in the pipe diameter
  • Changes in the pipe material (e.g. from steel to a hose)
  • Discrete flow resistances such as orifices, valves etc.
  • T-junctions and branch conduits
  • Changes in density and speed of sound of the fluid (e.g. due to thermal effects)

Reflection coefficient

The ratio of the reflected pressure wave \(\delta p_r\) to the incident pressure wave \(\delta p_i\) is referred to as the reflection coefficient \(r\). The reflection coefficient can be calculated from the characteristic impedances before (\(Z_1\)) and behind (\(Z_2\)) an impedance jump: $$r = \frac{\delta p_r}{\delta p_i} =\frac{Z_2 - Z_1}{Z_2 + Z_1}.$$ The reflection factor \(r\) is plotted against the impedance ratio \(Z_1/Z_2\) and the diameter ratio \(D_2/D_1\) in the following diagram:

MathJax TeX Test Page

Open end

For the special case of an open end, the impedance ratio \(Z_1/Z_2\) tends to infinity. Hence, the reflection factor approaches \(r \approx -1\). The reflected waves absolute value is equal to the incident wave. The opposite sign indicates that the reflected wave travels exactly opposite to the incident wave. Hence, the sum of incidend and reflected wave equals zero.

Closed end

Since the impedance ratio \(Z_1/Z_2\) tends to zero for the special case of a closed end, the reflection factor approaches \(r \approx 1\). Hence, the reflected wave is equal to the incident wave in amplitude and sign. In this case, the sum of reflected and incident waves equals \(2\delta p_e\).