Centrifugal Pumps Design and Application 2nd ed - Val S. Lobanoff, Robert R. Ross (Butterworth-Heinemann, 1992)

438 Centrifugal Pumps: Design and Application
Half-Wave Resonance (Open-Open and Closed-Closed)
The first three modes for an open-open pipe are shown in Figure 18-7.
Resonances may also occur at integer multiples of the half-wave frequency.
For a closed-closed pipe, the formula also applies since both elements
have a standing wave that is one-half of a sine wave even though
the peaks occur at different locations. The pressure mode shapes of the
first three modes are also shown for the open-closed configuration. The
length should be corrected for entrance and exit effects (add approximately
80% of the pipe inside diameter) to calculate the half-wave resonance
of open-open configurations. The end correction factor becomes
very important in short pipes.
Quarter-Wave Resonance (Open-Closed)
The first three modes for an open-closed pipe, commonly referred to as
a "quarter-wave stub" are depicted in Figure 18-7. The stub has its resonant
frequencies at odd integer multiples of the fundamental quarterwave
frequency. Examples of a quarter-wave stub include a bypass line
with a closed valve or a test connection with a pressure gauge.
A quarter-wave resonance can cause erroneous measurements [13]
when obtaining dynamic pressure data. A typical test connection, depicted
in Figure 18-8, with a short nipple and valve connected to a main
line is an acoustical quarter-wave stub. This length can tune up to pulsations
in the main line and cause the needle on a pressure gauge to wobble
or indicate severe pressure variations that do not actually exist in the
main line. Similarly, the data from a dynamic pressure transducer can be
misinterpreted.
Figure 18-7. Acoustic resonances of piping elements.