Point Sources
If the source of sound is
much smaller than the wavelength of sound it emits, it can be represented
by a 'point source' or 'monopole'. It will tend to radiate sound equally
in all directions, that is to say, with 'spherical symmetry'. Imagine a
tiny spherical source which generates sound by rhythmically expanding and
contracting. When it expands a pressure wave is transmitted outwards in all
directions. The pressure pulse is followed by a rarefaction pulse. The
resulting sound field (due to successive compressions and rarefactions of
the surrounding fluid) looks the same in all directions.
The pressure field produced
by a point source is shown in the animation below. The air particles move
back and forth as the spherical wave expands outwards. In practice, the
point source model is a good approximation for the sound field
radiated by a loudspeaker in a sealed box at low frequencies.
The following colour plot shows the
pressure field produced by a point source. As usual with colour maps, red
coloured regions represent high pressure areas
and blue coloured regions correspond to low pressure areas. If you move
your mouse
over the animation, you can also visualize the particle displacements
associated with the wave motion.
Inverse Square Law
As one moves further from a
source of spherical waves, the amplitude of the sound at your location
gets less. This is because of the ways in which, in travelling over the
distance between the source and you, the intensity
I of the wave
decreases. The intensity I
is the power W
in the wave divided by the area A
over which it is spread:
I = W/A
Assume that none of the
sound wave power is absorbed (i.e. converted to heat) as it propagates
from the source to you. If you are a distance
r
from the source, the area of the imaginary sphere over which the spherical
wave is spread is
A = 4π r^{2}.
Substituting this into the previous equation gives
I = W/4 π r^{2}
where
W
is the power of the acoustic source. As you move further from the source,
r
increases and the intensity of the wave decreases. From the above
equation, the intensity decreases as an "inversesquare law" with distance
r, that is like
1/r^{2}.
The soundpressure amplitude of a travelling simple spherical wave is
proportional to the squareroot of its intensity. Therefore, in a
spherical travelling wave, acoustic amplitude is proportional to
1/r.
This physical law is explained by the figure below.
In the absence of absorption, the
pressure amplitude of spherical sound
waves decays as 1/r.
