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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