Mechanics of Fluids

560 Unsteady flow
of conditions in the pipe – represented by the line XX in Fig. 12.2c. On the
valve side of XX the fluid has stopped and has been compressed; also, unless
it is perfectly rigid, the pipe is slightly expanded.
On the other side of XX, however, the fluid is still moving with its original
velocity, and the pressure and the pipe diameter still have their original
values. As each successive element of the fluid is halted, the discontinuity
XX moves farther away from the valve. In this way the change of velocity
is transmitted along the pipe: the closing of the valve directly stops only the
fluid that comes in contact with it; the remainder has to be brought to rest
by a ‘message’ passed along the pipe from one fluid particle to another, each
‘telling’ the next that it must stop. The travelling discontinuity, known as a
pressure wave or pressure transient, is in fact the message.
(Pressure waves are also transmitted through the material of the pipe
walls; their effect on pressure changes in the fluid, however, is almost always
negligible, so we may disregard them here.)
We now see that neglecting the compressibility of the fluid – in other words
assuming that all the fluid particles change velocity together – is legitimate
only if the time of travel of the pressure wave is negligibly small compared
with the time during which the change of velocity takes place.
When a fluid is suddenly stopped its behaviour is closely similar to that of
a train of loosely coupled railway wagons. If the locomotive suddenly stops,
the wagon immediately behind it compresses the buffer springs between itself
and the locomotive. The force in the buffer springs increases to a value
sufficient to stop the wagon. The second wagon then behaves likewise; it
is stopped by the compressive force in the springs between itself and the
first wagon. This process takes place successively along the whole length of
the train. The compression of the buffer springs in the train is analogous
to the compression of the fluid in the pipe-line. When a wagon has stopped,
the force in the buffer springs at the front must equal the force in the springs
at the rear (otherwise there would be a net force on the wagon and it would
move). Similarly, a pressure wave in a fluid suddenly alters the pressure as
it passes a particular point; but after that the pressure there stays at its new
value.
There are, incidentally, many other examples of the transmission of a
wave through an elastic material. Shock waves (considered in Chapter 11)
are transmitted through gases. A sharp blow applied to one end of a long,
weak coil spring causes a deflection of the coils that may be seen to travel to
the opposite end of the spring. A stretched rubber tube may behave similarly.
And if a series of similar, more gradual, disturbances occur regularly several
times a second, musical sounds may be produced as, for example, when a
succession of waves travels to and fro along the length of an organ pipe or
other wind instrument.
12.3.1 The velocity and magnitude of pressure waves
We now need to consider the rate at which a change of pressure is transmitted
through the fluid. Sections 11.4 and 11.5 dealt with the transmission of