Reflection of Pressure Waves in Hydraulic Pipelines
If the characteristic impedance of a pipe changes in the direction in which a wave propagates, reflections occur. In general, a reflection results in the generation of two new waves - a transmitted wave and a reflected wave. Possible causes of impedance changes can be:
- Closed pipe ends
- Open pipe ends, e.g. pipe leading into a large reservoir
- Changes of pipe diameter
- Change of pipe/hose material (e.g. from steel to rubber hose)
- Discrete flow resistances like orifices, valves etc.
- T and Y junctions
- Changes of density and/or speed of sound (e.g. due to thermal effects)
Reflection Factor
The ratio of the reflected pressure wave \(\delta p_r\) to the incoming pressure wave \(\delta p_e\) is referred to as the reflection coefficient \(r\). It can be calculated from the characteristic impedances before (\(Z_1\)) and after (\(Z_2\)) the impedance change: $$r = \frac{\delta p_r}{\delta p_e} =\frac{Z_2 - Z_1}{Z_2 + Z_1}. $$ The variation of the reflection coefficient \(r\) versus the impedance ratio \(Z_1/Z_2\) or versus the diameter ratio \(D_2/D_1\) is shown in the following figure:
Open End
Since the impedance ratio \(Z_1/Z_2\) tends to infinity for the special case of an open end, the reflection coefficient is \(r \approx -1\). Thus, the reflected wave is exactly equal in magnitude to the incoming wave. The minus sign indicates that the reflected wave travels exactly opposite to the original wave. The returning wave (sum of reflected and incoming wave) results to zero for this case.
Closed End
Since the impedance ratio \(Z_1/Z_2\) tends to zero for the special case of a closed end, the reflection coefficient is \(r \approx 1\). Thus, the reflected wave is equal to the incoming wave with respect to both its magnitude and its sign. The returning wave (sum of reflected and incoming wave) results for this case to \(2\delta p_e\), i.e. twice the magnitude of the incoming wave.
