Graphs of the radiation
impedance at a circular opening in an infinite baffle were produced
by performing the numerical integration in equation (3.25) for a
number of dimensionless frequencies ().
In order to keep the general applicability of the results, as is standard
practice, we will normalise the radiation impedance by dividing through by
rather than choosing a particular value of
.
Remembering that the radiation
impedance is a matrix whose element
gives the pressure amplitude
of the
th mode due to a given volume velocity amplitude of the
th mode,
it is useful to distinguish between the
and
elements.
The
elements are referred to as direct impedances since they give the
contribution to a pressure mode by the velocity mode with the same amplitude
distribution.
Figure 3.2 shows the real and imaginary parts of the first three direct radiation impedances for a cylindrical opening in an infinite baffle. The real part is known as the radiation resistance and a large positive value for this indicates that acoustic energy is radiated efficiently from the opening. The imaginary part is called the radiation reactance and a positive value for this indicates a mass loading of the air column [40] pp.191-192, or equivalently a length correction [45] pp.180-181. At low enough frequency the radiation impedance is effectively zero. In this case, no matter how large the velocity amplitude is, no pressure is produced, indicating the presence of a pressure node at the open end. The ideal open end condition then holds.
At low frequencies, and the
impedance is small and imaginary. The very low radiation resistance means
that almost no sound is radiated from the instrument. Nearly all the sound
is reflected back down the tube. The small imaginary value of the
impedance means that the velocity produces a small pressure, 90 degrees out
of phase in the time domain, as is the case close to a pressure node in a
tube supporting standing waves. A pressure node is therefore present, but
has been shifted slightly from the end of the tube, which is why a correction
must be made to the tube length when calculating the length of the standing
waves.
At intermediate frequencies the resistance becomes larger than the reactance.
The oscillatory look of all the graphs which follow in this chapter
result from local maxima which occur as
the wavelength becomes comparable with the tube width. In the high frequency
limit the radiation impedance converges to the
real value 1 (or before normalisation) which is the characteristic
impedance of plane waves in free space. This indicates that the waves are not
reflected at the opening, but propagate out of the tube
undisturbed and with 100% efficiency. This agrees with the
intuitive behaviour of wave diffraction from an opening; high
frequency waves are transmitted in a beam of the same cross-section as the
opening. Standing waves cannot be set up in
this regime as no energy is reflected back to contribute to resonance.
Note how the direct impedances of the modes converge more slowly as
increases. The normalised characteristic
impedance of the mode
from equation (2.36)
is
which tends to 1 from above when
. It is therefore
observed that the radiation impedance tends to the characteristic impedance
termination value, which in turn tends to 1, more slowly as
(and therefore
) increases.
Next we consider the elements of the impedance matrix for which .
These are referred to as coupled impedances since they give the
contribution to a pressure mode by a velocity mode with a different amplitude
distribution.
Figure 3.3(a) shows the radiation impedance resulting from the coupling of
the plane wave pressure mode (
) and the
th velocity mode for
and
. Figure 3.3(b)
shows the radiation impedance resulting from the coupling of the pressure
mode with one nodal circle (
) and the
th velocity mode for
and
.
At the zero frequency limit, the coupled radiation impedances go to zero,
indicating that there is no component of the th pressure mode due to the
th velocity mode and therefore no coupling for
.
At intermediate frequencies we can see non-zero impedance terms (less in
magnitude than for direct impedances) which indicate a certain amount of
inter-modal coupling is taking place. In the high frequency limit we observe
the radiation impedance tending to zero.
The infinite pipe termination or characteristic impedance condition has
no inter-modal coupling, or equivalently the
elements of the
characteristic impedance matrix have a value of zero. The radiation impedance
matrix therefore
tends to the characteristic impedance matrix at high frequencies for all
elements, for those with
in addition to those with
discussed earlier.