PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Effective Radiation Area (Sd) for Axisymmetric Loudspeakers
Radiating in an Infinite Baffle Using a Near Field Analysis
Angelo VELARDE1; Jorge MORENO2
1

Pontificia Universidad Católica del Perú-Engineering Department, Perú2

Pontificia Universidad Católica del Perú-Physics Department, PerúABSTRACT
In this paper, the effective radiation area (Sd) is obtained (theoretically and experimentally) for different
loudspeakers. For this purpose, is considered a near field approach. This method is theoretically based inprevious works that use the geometrical properties of different axisymmetric pistons and calculate theeffective radiation area by positioning the pistons in an infinite baffle. In the case of loudspeakers, thecalculations are redefined with the new geometrical conditions, in order to increase the precision of the results.
Finally, these results are compared with the listed effective radiation area of the loudspeakers data sheets,
and other techniques, in order to set the accuracy of the measurement.
Keywords: Axisymmetric Piston, Effective radiation area.

1. INTRODUCTION
Exact calculation and experimental determination of the Effective Radiation Area (Sd) has always
been a very important topic in order to establish loudspeaker parameters. For that purpose, different
approaches have been found. In previous work, theoretical analysis has been developed using
axisymmetric pistons with different shapes, and then solving the Rayleigh -Sommerfeld equation inorder to obtain a simplified expression for this value.
Based on a previous work from Moreno (1) it has been verified that a relation between pressure
and acceleration can lead to the value of the diameter used to obtain the Effective Radiation Area (Sd).
With the purpose of verifying this relation, two different mathematical methods have been tested. The
first method consider a mathematical simplification of the equations based on the cylindricalsymmetry of the pistons (2). The second one correspond to an adaptation of a previous publicationfrom Alba (3) that used multiple concentric rings as an approximation for circular pistons. Bothmethods were adapted for a near field analysis for the calculation of the Sd.
However, in both methods were detected that for conical and convex pistons the value of the error
can exceed the 10% of the projected area of them. And for that reason, even when this method probe
to work for concave pistons with a small error, it must be tested in real speakers.

2. EFFECTIVE RADIATION AREA(Sd)In accordance with Beranek (4) (5), the Sd parameter helps to determine the sound pressure on theaxis and total acoustic power output. A precise value of Sd is required to calculate the Thiele-Smallparameters with a certain degree of reliability.
As Klipple (6) mentioned, in order to determine the Sd, it is desired to find the radius of the flat
piston that generates the same sound pressure at a distant point (far field) than a generic axisym metricpiston. This piston radiates independently from the occupied volume in a uniform way, always
considering the components in the direction of the symmetry axis.
127302angelo.velarde@pucp.pe
jmoreno@pucp.edu.pe

Moreno (1) also says, that some “ruler method” exist, that consist, in taking the diameter in themiddle of the surround or edge area.
2.1 Near-Field Theoretical Analysis
For the theoretical analysis, here is presented an extract from (2) and (7). The general model forthe sound pressure produced by the vibration of a piston mounted in an infinite baffle is called theRayleigh-Sommerfeld Integral. If the piston shape is arbitrary and the normal velocity component is‫ݑ‬௣ ; we will have a time-dependent arbitrary function.

‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬Ǣ ‫ݐ‬ሻ ൌ ߩ଴ නௌ‫ݑ‬ሶ ௣ ቀ‫ ݔ‬ᇱ ǡ ‫ ݕ‬ᇱ Ǣ ‫ ݐ‬െ ʹߨܴܴܿ ቁ ݀ܵ(1)Where R is the distance to the point where the sound pressure L will be determined.
This calculation is an integral, usually very complex and is often solved with numerical calculation
methods.
It is important to analyze the axisymmetric piston generically, and for this purpose, two graphics
have been prepared to see the piston from different perspectives. Then, from this analysis, the integralof Rayleigh-Sommerfeld can be evaluated.

Figure 1: Circular axisymmetric piston. Ring with a width dσ at a height d on the XY plane
From the figures, we can define the characteristics of the Rayleigh -Sommerfeld integral for theplane axisymmetric piston. That expression can be seen below in equation (2)‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬Ǣ ‫ݐ‬ሻ ൌ ߩ଴ නௌ‫ݑ‬ሶ ௣ ቀ‫ ݔ‬ᇱ ǡ ‫ ݕ‬ᇱ Ǣ ‫ ݐ‬െ ʹߨܴܴܿ ቁ ݀ܵ(2)Based on these graphics it can be proposed the following analysis.
2.2

Calculation methods

2.2.1 Integral simplification
Using the expressions from Hasegawa (8) (9) an applying them for the near field the equation (2)can be transform into:7303

మஶమି୨୩ඥ୰ᇱ ା୰ ିଶ୰୰ᇱୡ୭ୱஓඥԢଶ ൅  ଶ െ ʹԢ ɀൌ  ෍ ሺʹ ൅ ͳሻ୫ୀ଴ ୫ାଵൗ ሺԢሻ ୫ାଵൗ ሺሻଶଶξԢξ୫ ሺ ɀሻ(3)In order to use different geometries we need to include terms  and ɐ with the expressions thatrelate these with ‫ݎ‬Ԣ and ܿ‫ ߠ ݏ݋‬ᇱ. After these modifications we have an expression that can be used toobtain the relation between pressure and acceleration. This expression, which have been called Beta(β) is shown here:Ⱦ୤୪ୟ୲ ୮୧ୱ୲୭୬ ൌɏ଴ ή (4)Now, for the near field it can be obtain the Beta expression for a uniform speed:ஶୟሺଵሻȾ୮Ǥୟ୶୧ୱ୷୫୫ୣ୲୰୧ୡ ୒୊ ൌ െ ෍ ሺʹ ൅ ͳሻ୫ ሺሻ න ɐ୫ ቀඥଶ ൅ ɐଶ ቁ ୫ ൬൰ ɐଶξ ൅ ɐଶ
଴୫ୀ଴(5)

2.2.2 Numerical approximation
Using the work from Alba (3), and the same geometry, it will be assumed that the piston radiation
can be decomposed as the addition of the pressure generated by two differential rings applying thesuperposition principle, but considering both velocity in opposite direction. Even when this principle
is applied originally for the far field analysis, it can be used also in the transition from near to far fieldwith the Blackstock (10) equations.
It is considered that the pressure expression for a piston in the z axis:మమൌ ܲ଴ ሾ݁ ௝ሺఠ௧ି௞௥ሻ െ ݁ ௝ሺఠ௧ି௞ඥ௥ ା௔ ሻ ሿ(6)Then, it can be define:௔xThe rings have a width: οߪ ൌx

The formation rule for the pistons consider the radius ߪ௜ ൌ οߪ ൈ ݅ where i goes from 1 to #ringsx

The z position of the ring is:͓௥௜௡௚௦‫ݖ‬௣௥௢௠ ൌ‫ݖ‬ሺߪ௜  ሻ ൅ ‫ݖ‬ሺߪ௜ିଵ ሻʹFinally, for the near field it can be define the Beta expression for a uniform speed:

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(7)

Ⱦ୮Ǥୟ୶୧Ǥ୒୊͓௔௡௜௟௟௢௦ൌଶଶଶ෍ ቊܿ‫ ݏ݋‬ቆ݇ට൫‫ ݎ‬െ ‫ݖ‬௣௥௢௠ ൯ ൅ ߪ௜ ଶ ቇ െ ܿ‫ ݏ݋‬ቆ݇ට൫‫ ݎ‬െ ‫ݖ‬௣௥௢௠ ൯ ൅ ߪ௜ିଵ ଶ ቇ െ ݆‫ ݊݁ݏ‬ቆ݇ ට൫‫ ݎ‬െ ‫ݖ‬௣௥௢௠ ൯ ൅ ߪ௜ ଶ ቇ݇
௜ୀଵ(8)ଶ൅ ݆‫ ݊݁ݏ‬ቆ݇ ට൫‫ ݎ‬െ ‫ݖ‬௣௥௢௠ ൯ ൅ ߪ௜ିଵ ଶ ቇቋ

These expressions can be used to verify that the radius that lead to the effective radiation area canbe obtain from the relation between the pressure and the acceleration. It should be also consider thatthe expression for Beta is related to a surface and not for a single point. For the experimental part itwill be necessary to measure these values and obtain the relation accurately enough to compare theresults with the speakers data sheets or another technique.

3. EXPERIMENTAL DETERMINATION
For the purpose of the paper it have been developed two set ups in order to compare the effective
radiation area or the radio that lead to its calculation.
For the validation process, the following speakers were measure. The Tables 1, 2 and 3 shows theThiele-Small (T-S) parameters of the speakers:
Table 1: T-S Speakers parameters 12" Speaker

Free Air
Sd(m2)

Empty Box 1

Empty Box 2

Vol = 86.45l

Vol=156l

0.04750.04750.0475Re(Ohm)5.95.95.9fs(Hz)

50.26

62.04

56.87

Bl (Txm)

15.89

16.23

16.23

Qms

2.90

3.47

3.30

Qes

0.46

0.57

0.53

Qts

0.39

0.49

0.46

Mms (g)

61.55

64.70

65.00

Rms (Kg/s)

6.73

7.26

7.04

Cms (m/N)

0.000163

0.0001020.000121

Vas (l)

51.90

32.00

37.90

No (%)

1.41

1.29

1.287305

Table 2: T-S Speakers parameters 6.5" Speaker

Free Air
Sd(m2)

Empty Box 1

Empty Box 2

Vol=10.5l

Vol=44l

0.01350.01350.0135Re(Ohm)

7.5

7.5

7.5

fs(Hz)

94.26136.67

101.19Bl (Txm)

5.28

5.15

5.49

Qms

3.80

6.13

3.88

Qes

1.00

1.55

1.05

Qts

0.79

1.24

0.82

Mms (g)

6.33

6.43

6.80

Rms (Kg/s)

1.00

0.91

1.11

Cms (m/N)

0.000455

0.000211

0.000388

Vas (l)

11.37

5.27

9.20

No (%)

0.92

0.84

0.88

Table 3: T-S Speakers parameters 8" Speaker

Free Air2

Sd(m )

Empty Box 1

Empty Box 2

Vol=10.5l

Vol=44l

0.02190.02190.0219Re(Ohm)

6.2

6.2

6.2

fs(Hz)

59.30

123.92

78.01

Bl (Txm)

8.88

8.62

8.84

Qms

1.87

3.90

2.35

Qes

0.43

0.91

0.55

Qts

0.35

0.74

0.44

Mms (g)

14.63

14.13

14.50

Rms (Kg/s)

2.96

2.83

3.02

Cms (m/N)

0.000494

0.000117

0.000299

Vas (l)

33.77

7.97

19.60

No (%)

1.60

1.62

1.65

3.1 Near-Field method
The first set up is based in the theoretical approach that is explained in the previous part of thepaper. So, is necessary to measure the pressure and the acceleration of the speaker, and for that reason
the next configuration has been deployed:7306Figure 2: Block Diagram and image for measuring pressure and acceleration
The value of Beta is measured for 3 different speakers using this setup, for different distances fromthe surface in the z axis and for different frequencies. It has been taken into account the resonancefrequency as a reference. Finally, the value of Beta has been measured along the radius of the speakerin order to calculate its mean in the projected surface of the speaker at the edge.
3.2 Differential Volume method
As a way to verify the value of the Effective Radiation Area it has been performed the measure of
the speaker parameters in different volumes, according to Klippel (6). In this case, however, it has
been measured the compliance of the speakers in free air, and also in the two volumes in order toeliminate the error of measuring the additional speaker volume when it is installed in a box.
By using the measuring method proposed by Moreno (11) and the considerations of Jonsson (12)the Effective Radiation Area (Sd) is obtained.

4. RESULTSFirst, it will be shown the graphics from measuring Beta along the z-axis and the measuring ofBeta along the radius, for the 12” and 6.5” speakers as an example.

Figure 4: Beta vs. Distance in z-axis from the Speaker (a) Case of 6.5” Speaker, (b) Case of 12” Speaker.

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Figure 4: Beta vs. Distance along the radius, from the center of the Speaker (a) Case of 6.5” Speaker, (b) Case of12” Speaker
From this values, we can obtain the radius that lead to the Effective Radiation Area (Sd) andcompare the results with the calculated by using the volume method.
Two approaches have been used. The first one is related with the previous work from Moreno (1),measuring the value of Beta in the surface of the speaker, but with the addition on this paper, of themeasuring of Beta along the axis, as it is showed in Figure 4 (a) and, (b).
The second approach, consider the measuring of Beta along the radius and obtaining from thesevalues (See Figure 4) the surface average, in order to obtain a value that represent the effect all overthe surface from the center until the edge.
The results can be seen in the next table:

Table 4 - Beta comparison with different approaches.(All values in mm.)Average
fromerror
(%)

error
100Hz

(%)

150Hz

error

Compliance

(%)

Boxes

Aproximation

50Hz

Exponential (z-axis aprox)

69.73-10.6169.77 -10.68

71.50-13.42

6.5" Speaker Polinomial (z-axis aprox)

69.78-10.7069.84 -10.80

71.51-13.45aprox)

60.94

3.33

63.92

-1.40

59.57

5.49

Exponential (z-axis aprox)

83.96

-1.41

82.72

0.08

87.50

-5.69

Polinomial (z-axis aprox)

85.49

-3.27

84.70

-2.30

89.27

-7.82aprox)

81.84

1.14

78.94

4.65

84.29

-1.81

Exponential (z-axis aprox)

110.2711.51

113.508.92

112.799.49

12" Speaker Polinomial (z-axis aprox)

108.5812.87

112.969.35

112.22

9.94

120.22

3.52

116.686.37

114.408.19(fs=94.26Hz) Surface average (along radius

8" Speaker(fs=59.3Hz) Surface average (along radius(fz=50.26Hz) Surface average (along radiusaprox)

7308

63.03

82.79

124.61

5. CONCLUSIONSAs can be seen, the inclusion of the Beta average over the surface increase the precision of thetechnique. Also, it is clear that the value is more precise when the frequency of the measuring is close
to the resonance of the speaker. Even when the z-axis approximation on the 6.5” and the 12” speakers
have an error around 10%, with the surface average in the worst case is near 3.5% around theresonance frequency. It is important to mention that in the case of the determination of the Effective
Radiation Area (Sd) with the volume technique, in the need of obtaining accuracy, the measuring mustbe performed many times because the value of the compliance is very sensitive to the amplifiercharacteristics. Finally, this paper succeed in proposing an improvement to previous techniques forthe determination of the Effective Radiation Area in a Near Field approach.

REFERENCES
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preprint; 1994; Amsterdan.
2. Velarde A, Moreno J. Effective radiation area (Sd) for an axisymmetric piston radiating in an infinite
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7. Velarde A, Moreno J. Análisis de Campo Cercano para el Cálculo del Area Efectiva de Radiación (Sd)en un Pistón Axisimétrico Radiando en un Plano Infinito. In ; 2018; Cádiz, España.
8. Hasegawa T, Inoue N, Matsuzawa K. A new theory for the radiation from a concave piston source.
Journal Acoustical Society of America. 1987; 82(2): p. 706-708.
9. Hasegawa T, Matsuzawa K, Inoue N. A new expansion for the velocity potential osf a circular concavepiston. Journal Acoustical Society of America. 1986; 79(4): p. 927-931.
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11. Moreno J, Jonsson S, Bog H. Measuring of Closed Box Loudspeaker Suyetem Parameters using a Laser
Velocity Transducer and an Audio Analyzer. In 92nd Convention AES; 1992; Vienna.
12. Jonsson S. [Application Note: Accurate Determination of Loudspeaker Parameters using AudioAnalyzer Type 2012 and Laser Velocity Transducer Type 3544].; 1996 [cited 2019 April. Available from:https://www.bksv.com/en/.

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