Figure 9
represents two spheres or simple sources separated
by a distance B. The assumption here is that B is always much,
much less than the radiated wavelengths. If this condition
occurs, than the two point sources will generate double the
pressure and the directivity is still that of a single point (omni).
This is a simple and intuitive case where two radiating sources
simply generate twice the pressure of the single source.
Figure 10
shows these two point sources separated by a
distance of 12 inches. The polar response shown is that of
those two point sources radiating 100 hz signal. Again, the
space in B is much, much less than the wavelength, and as a
result, the radiation continues to be that of an omni-directional
condition. (Again, this is only a theoretical case, as point
sources do not exist in practice.) This representation is
extremely useful when we look at
Figure 11
, which is the same
two point sources as that of
Figure 10
. The distance continues
to be 12 inches, but now the frequency has been raised to 630
hz. (B approximately equivalent to 1/2 of the wavelength.)
Examination of
Figure 11
shows that at 0 degrees on axis
and at 180 degrees the radiation is summing coherently and
the radiation at –90 degrees and +90 degrees (-y/,+y on the
Array Show polar plot) is experiencing cancellation. The
radiation of +x and –x, or that of the radiation on axis, has
seen a 3 dB gain in pressure associated with the pressure
addition of the two sources.
Figure 11
begins to illustrate the
principles underlying successful application of a continuous line
of vertical sources (that of a line array).
Figure 12
is extremely interesting as well as it explains
the “historical” applications where line arrays were limited
bandwidth devices, such as those referenced in
Figure 1,
Figure 2
and
Figure 3
earlier in this discussion. The two point
sources continue to be spaced by 12 inches, but now the
frequency has been raised to 2500 hz. In this case, the space
B is equal to twice the wavelength. Examination of the polar
response shows substantial polar lobing errors. It describes
exactly the response of any group of sources, whether they are
vertically oriented or horizontally oriented when the wave-
lengths become shorter than the device spacing.
Figure 12
is a clear representation of difficulties that
system designers face when trying to provide full bandwidth
radiation (i.e. greater than 16 kz) with real world radiating
sources. The peaks and nulls in the diagram of
Figure 12
are
easily heard in real world applications and have always been
taken as a “necessary evil” when orienting sources. The
previous polar diagrams also require some explanation.
In definition of terms,
Figure 13,
the beamwidth is
defined as the included angular separation between the –6 dB
points, reference to the 0 db (+x) axis. The term Q is the ratio
of the acoustic intensity on that reference axis at some
reference distance to a true point source radiating the identical
acoustic power. Again, the true point source is useful from a
mathematical standpoint to enable us to define the acoustic
intensity ratio of real world devices to theoretical omni
Figure 9
Figure 10
Figure 11
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