Sound waves are traveling pressure waves in a fluid (gas or a liquid).
A consequence of this is that there can be no sound propagation
in vacuum. We verify this by showing that an otherwise noisy
personal alarm device becomes completely silenced when placed in
a vacuum.
Corresponding to the pressure wave is a displacement wave
s(x-vt) of atoms from where they would be if there had
been no sound propagating. The relationship between
the displacement wave and the deviation from average pressure
is a characteristic property of the fluid:
Note that ds/dx rather than s appears on the right hand side of
the equation because ds/dx measures the deformation of the
fluid from equilibrium: A constant s simply corresponds to an
overall displacement of the fluid. The negative sign is there
because a contraction ds/dx<0 gives an increase in pressure and thus a positive p.
B is called the bulk modulus
of the fluid and has dimensions of a pressure. It can be shown that
where
is the ratio of the constant pressure to the constant volume specific
heat. We shall return to Eq. 12 later in the course.
To derive the wave velocity in a fluid
we write Newton's second law for a cylindrical slice of thickness
and area A. The force on this slice is
We also calculate
Newton tells us to equate F and ma and this gives us the wave-equation
From which we conclude that the speed of sound is
Neglecting the 
we could immediately have
written down this formula based on dimensional analysis or an analogy
with the previously derived expression for the wave-velocity on a
taut string. The formula was first derived by Newton himself.
Putting in numbers we get for air at ambient pressure
(
, 
, 
)
which is indistinguishable from the measured value 
. Note that we have derived the interesting result that at constant pressure the velocity of sound is greater in a light than
a heavy gas.