|
Plane Circular Piston
in Infinite Baffle (Beranek
Equation)
| where |
ZM |
= mechanical impedance
in newtons-seconds per meter |
| |
a |
= radius of piston in
meters |
| |
r 0 |
= density of gas in kilograms
per cubic meter |
| |
c |
= speed of sound in meters
per seconds |
| |
 |
= mechanical resistance
in newtons-seconds per meter |
| |
|
The German  indicates that the resistive component
is a function of frequency |
| |
XM |
= mechanical reactance
in newtons-seconds per meter |
| |
k |
= w / c = 2 p / l = wave
number |
| |
J1 K1 |
= two types of Bessel
functions |
designates that it is a function
of w and is normalized with respect to the radius of the driver as ka. It
contains the Bessel function of the 1st.order and to understand the response,
it is necessary to examine the Bessel J1 function.
|
|
Fig 1
click image for larger
chart
|
The plot in Fig. 1
is that of the Bessel function. The response is overlayed with the Bessel/
Z, normalized by
Note that when the function is scaled in dB's the result is the same as the Beranek
function for .
|
|
The normalizing variable
ka is converted to frequency so that the Rd frequency is displayed
to understand how it is related to Sd.
|
|
|
Sd
|
a
|
Rd
f peak
|
Rd
-6dB
|
Fb
|
| Driver Type |
(cm2)
|
(meter)
|
(Hz)
|
(Hz)
|
(Hz)
|
|
|
| KEF B110 |
92
|
0.0541
|
406
|
170
|
63.6
|
| P17WJ-00-08 |
136
|
0.0658
|
334
|
|
|
| 21W8554 |
200
|
0.0798
|
270
|
130
|
54.8
|
| M26WR-09-08 |
337
|
0.1036
|
207
|
95
|
60
|
| NHT1259 |
520
|
0.1287
|
167
|
95
|
20
|
|
The tabulated data
shows that the *Rd f peak, corresponding to the Bessel ripple peak in
the computed Rd, shifts inversely with Sd and corresponds to
about 0 dB on the Qtc=0.5 response curve. Since Rd has only
Sd of the drivers parameters it tracks but poorly the Qtc=0.5
response as shown by the Fb data point (-6 dB point in the Qtc=0.5
response).
However the slope of the Rd function matches that of the Qtc
= 0.5 model.
|
|
Fig 2
click image for larger
chart
|
The Fig. 2 plot illustrates
that the Qtc=0.5 response tracks reasonably well the driver's near field
response in the TL but that to use the computed Rd() response one would
have to shift the curve in frequency and ignore the ripple of the bessel function.
Since the phase response would be calculated as a 1st derivative of the
frequency response, the response of the Bessel function would produce a similar ripple
in the phase as in the magnitude at the reference ka point. The plot data
shows that the measured driver's near field does not show any similar ripple for
the Qtc=0.5 alignment. This result seems to indicate that the use of the
concept of Pistonic Minimum Phase must be restricted to the small signal regime and
has very limited utility in the large signal TL response modeling.
* Rd refers to the values of the Y axis used in the figure. It is the
normalized real part of Beranek's impeadance analogy:
Zm
= /Xm
. Thus Rd = Zm/(pa2rc)
|
|
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