206 Chapter 8
strength electrical signals. In this circumstance, if the
observation point, o, is located in the median plane
where Tis zero, the acoustic pressure at any radial
distance, r, is just double that which would be produced
by either source acting alone. This is true because the
path lengths are equal so that the two pressure signals
undergo the same inverse distance loss and phase lag
and thus arrive with equal strength and in phase at the
observation point. Now if one considers those observa-
tion points where r is always much larger than d and if d
is small compared with the wavelength, then the ampli-
tude difference between the two signals as well as the
phase difference between the two signals will be insig-
nificant at all such points and again the total pressure
will be nearly twice that of a single source acting alone.
Such observation points are located in the far field of
the combined sources as would be the case in a stadium
for all medium or long throw devices. This instance of
pressure doubling at all far field observation points only
occurs at low frequencies where the wavelength at the
operating frequency is significantly larger than the
device separation. Consider the case where the oper-
ating frequency is such that. In the far field
the amplitude of the signal from each source is again
essentially the same but now there will be a phase
difference for all values of the angle Tgreater than zero.
This is most obvious for distant points on the vertical
axis where T= ±90°. At such points the phase differ-
ence between the two sources is 180° and the acoustic
pressure is zero. The two sources are now exhibiting a
frequency-dependent directivity function as a result of
their physical placement one above the other. If the two
sources are horns rather than point sources, then there is
an additional directivity function associated with the
horn behavior that is a function of both the azimuthal
angle M as well as the vertical angle T. The acoustic
pressure amplitude for all points in the far field for both
sources being driven equally and in phase can be calcu-
lated from
(8-2)In the two equations for the pressure amplitude, pm,
A is the source amplitude factor, Dh(TM is the horn
directivity function, and Da(T) is the directivity function
brought about by arraying one source above the other.
The vertical braces | | denote absolute magnitude of the
enclosed quantity. This is necessary as the two direc-
tivity functions can independently each be positive or
negative dependent upon the frequency and the pressure
amplitude is always positive. The quantity, k, is the
propagation constant and is related to the wavelength
through. It is important to note
that the directivity behavior brought on by arraying one
device above the other depends only on the angle T and
that the horizontal directivity of the devices in the far
field is not influenced by the physical placement of one
above the other. This behavior in the vertical plane is
depicted in Fig.8-6A through E where the individual
devices are 40° vertical by 60° horizontal horns having
small mouths.
Figs. 8-6A through E illustrate both the desirable and
undesirable attributes of arraying devices in a vertical
line. A illustrates the directivity in the vertical plane for
each device. The device directivity has a magnitude of
0.5 at ±20° indicating that the vertical coverage angle is
40°. The minimum vertical spacing between the devices
is limited by the mouth size and in this instance is
0.344 meter. This corresponds to O/2 at a frequency of
500 Hz. As shown in Fig. 8-6B, the pressure on-axis is
indeed doubled and the overall shape closely follows
that of the horn directivity function with just a small
narrowing of the vertical coverage angle. In Fig. 8-6C,
where the operating frequency is now 1000 Hz, the
on-axis pressure is again doubled but now the central
lobe is noticeably narrower and small side lobes are in
evidence. This trend continues in Fig. 8-6D where the
operating frequency is now 2000 Hz. Side lobes are
now much in evidence and the central lobe is narrowed
even further. Finally, at 4000 Hz, as depicted in Fig.
8-6E, another pair of side lobes appear, the original side
lobes, while having narrowed, are considerably
stronger, and the central lobe is narrower still while
maintaining double on-axis pressure. In all instances the
overall envelope containing the vertical directivity
behavior of the stacked pair has the same shape as that
of the individual device directivity function.
One is not limited to stacking just two devices in a
vertical line. Any number, N, of identical devices can be
so arranged and when several discrete devices are so
arranged the combination is called a line array. The
qualitative behavior of such an array as observed in the
far field is quite similar to that of the stacked pair
discussed above. Directional control does not appear
until the length of the array becomes comparable to the
wavelength, the on-axis pressure in the far field is N
times as great as that of a single device, and as the oper-
ating frequency increases, side lobes appear and the
central lobe becomes narrower and narrower as the
operating frequency increases. This assumes that all ofd=Oe 2pm2 A
r=------ -Dh TM Da T2 A
r------ -Dh TM kd
2= cos ---sin Tk== O e 2 S 2 Sfce