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There are many different types of wave motions in physical situations.
Sound waves occur in fluids and non-magnetic plasmas and arise when
there are variations in the pressure. Consider a fluid or plasma
satisfying, the equation of motion
 |
(6.5) |
the equation of mass continuity,
 |
(6.6) |
and an equation of state which we take as the gas law
 |
(6.7) |
where the gas constant,
, and the temperature,
, are taken as
constants and the fluid is isothermal.
Now the basic situation is regarded as a uniform medium at rest
(
) with a uniform pressure (
) and uniform density
(
) satisfying
Note that the other equations, (6.5) and (6.6) are
satisfied automatically. We assume that there is a small change in
pressure,
, which introduces a small velocity,
, and causes small change in the density,
. These
perturbations
are small in magnitude and much smaller than the magnitude of the
initial uniform medium values of
and
. Hence,
Thus, we can write the actual density as
and similarly for the pressure. Hence, (6.5), (6.6) and
(6.7) are expressed as
and
Now we use the fact that
and
are uniform, neglect
all products of small quantities to give
 |
(6.8) |
where
is the square of the isothermal sound speed
(typical values for the sound speed are 300m/s for air, 270m/s for
hydrogen, 1410m/s for water and 5600m/s for glass).
Notice that these equations are linear equations in the perturbed
quantities and that they have constant coefficients.
We can eliminate the velocity
and
from the
equations to derive a single equation for the density perturbations,
. Hence,
 |
(6.9) |
This is a general wave equation. If we consider a 1-D problem, so
that
, then
and we have the usual wave equation. Solutions to this wave equation
represent sound waves propagating with speed
along
the
-direction. If we write,
and
is the constant amplitude of the wave.
More generally, a single wave mode in three dimensions has
and we consider
 |
(6.10) |
which represents a wave travelling in the direction of
with wavenumber
Substituting (6.10) into (6.9) gives the dispersion relation
 |
(6.11) |
Note
- Sound waves propagate equally in all directions, there is no
preferred direction.
- From (6.8)
and so
is in the same direction as
. This
means that sound waves propagate longitudinally.
Next: Electromagnetic Waves
Up: Waves
Previous: 2D Vibrations of a
Prof. Alan Hood
2000-11-06