Experiments with Supersonic Beams as a Source of Cold Atoms

which for steady state flow in 1D can be reduced to
wdw = −dP
ρ
. (2.14)
Lastly, energy is conserved in the flow. This means that
D
h +
Dt
w2
= 0 (2.15)
2
which can be simplified to
2.1.2 Mach Number and Supersonic Beams
wdw +dh =0. (2.16)
A final concept that must be introduced before addressing the flow problem
is information’s propagation in the gas. Information is transmitted in the flow as a
pressure disturbance or pressure wave, and the speed of propagation is simply the
speed of sound. In an ideal gas, the speed of sound is
γP
vs = . (2.17)
ρ
This equation assumes that there is no collective motion in the gas. This can be
addressed by simply changing coordinate systems into one that is moving along with
the flow. Thus information will travel in the static lab frame at a velocity vs ± w,
where the sign depends on whether the pressure wave is traveling with the flow or
against the flow. If the flow is faster than the speed of sound, then information cannot
travel backwards in the flow, and the flow is supersonic. The ratio of the flow velocity
to the speed of sound is called the Mach number M,
M = w
. (2.18)
Since the speed of sound depends on the mean molecular speed, the Mach number
can be thought of as a measure of the collective motion of the beam relative to the
thermal motion of the individual atoms in the ensemble.
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vs