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Infinite Baffle.

A planar boundary or infinite baffle may be considered as a trivial case of a conical waveguide. In this section, the radiating characteristics of various diaphragm shapes on an infinite baffle are considered, using transient FEM. In each case, the radiator shape either is a spherical cap, defined by the interior angle, or is planar, in the case of the piston.

Planar piston radiator.

From a geometrical point of view, this is the simplest source since it shares the same geometry as the baffle. The resulting on axis frequency response, power response and spatial dispersion and impulse response contour are illustrated in figures 3,4,5 &6 respectively.

The characteristics of a rigid piston are well known [4]. Figure 2 shows a flat axial response with the response at 40 degrees off-axis rolling off to be 10dB down by 20kHz with lobes above this.

Figure 3 shows the power response, which rolls off smoothly above a few thousand Hertz reaching a slope of approximately 6dB/octave. The contour plot in figure 4 displays the directivity. This shows a main lobe that narrows from several thousand hertz; the side lobes are not displayed on the contour due to the range of 10dB. The impulse response contour plot in figure 5 shows a sharp impulse on axis that broadens off-axis. There is no apparent time smearing or oscillation.

The major practical difficulty with a planar diaphragm is that it has no geometric stiffness. Piston designs tend to be inefficient due to the high mass required to make such a structure sufficiently rigid.

Spherical cap radiators.

Hemi-spherical cap.

The geometry of the axially moving hemi-spherical cap is the same shape as the spherical waves we would like to create, however, the velocity profile is not. The resulting on axis frequency response, power response and spatial dispersion and impulse response contour are illustrated in figures 6,7,8 & 9 respectively.

Considering figure 6, we can see that in this case the onaxis response is far from smooth, with dips at 7kHz and 35kHz. The 40-degree off-axis curve is smoother than the piston, but rolls off faster than the on axis curve being over 10dB down at 20kHz. The power response in Figure 7 is smooth and rolls off more rapidly than the planar piston. The response contour plot, figure 8 confirms that at 7kHz significantly more level is produced off-axis, above this, the radiation narrows. The impulse response, figure 9, is interesting since it shows some oscillation.

While the hemispherical dome has excellent, geometric stiffness it’s acoustic performance is far from ideal. Spherical cap with 120 degree included angle. With a flatter dome, we might expect that the performance will move towards that of the planar piston.

The resulting on axis frequency response, power response and spatial dispersion and impulse response contour are illustrated in figures 10,11,12 &13 respectively.

The dips in the on axis response curve, shown in figure 10, at 10kHz and 40kHz are higher in frequency as well as deeper and wider in amplitude than the hemispherical dome. The axial response is not as smooth as the hemispherical dome but is only 3.6dB down at 20kHz compared to the piston. The power response, figure 11, is smooth and only a 2dB down at 10kHz compared to the piston. The contour plot, figure 12, confirms that at 10kHz more level is produced off-axis, above this the response narrows with slight lobing. The impulse response is much improved over the hemispherical cap albeit still with some oscillation present.

The 120-degree spherical cap has good geometric stiffness but like the hemispherical cap, its frequency response varies with direction.

Conical Waveguide

Spherical cap with 120 degree included angle We will now consider an axially moving 120-degree spherical cap, as used in the previous example, bounded by an 80-degree conical waveguide. In this case, the conical waveguide is much narrower than the infinite baffle giving a closer match between dome shape and the desired wavefront shape.

The resulting on axis frequency response, power response and spatial dispersion are illustrated in figures 14,15,16 &17 respectively.

We can see from figure 14 that the response below 12kHz is reasonably behaved, however at 19kHz & 23kHz large dips combine to give a very poor on axis response.

The directivity contour in figure 16 shows significant lobing. The impulse response in this case had to be normalized to the 40-degree off-axis response to avoid a non-causal filter. The result shown in figure 17 has significant oscillation.

The 120-degree spherical cap in the 80-degree waveguide provides a good response at low frequencies; however, the high frequency response is very poor limiting the bandwidth. Whereas the radiator is of a shape suitable for manufacture, the infinite waveguide is not realizable in practice.

Spherical cap with 80 degree included angle

In this instance, the centers of the spherical cap and conical waveguide are coincident; the radiator shape matches the desired wavefront shape and the velocity profile is a good approximation.

The on axis and 40 degree response curves in figure 18 are both smooth with only a 2dB difference between them up to 50kHz. The power response in figure 19 is very smooth. The contour plot in figure 20 has only one contour due to the lack of variation of level with angle.

The impulse response shows little oscillation or variation of impulse with angle. In practice an 80° angle spherical cap is possible to manufacture using suitable modern materials or structure. The acoustic performance is good but the infinite waveguide is not practical.

Finite waveguide with optimum dome shape

Taking a spherical cap radiator and conical waveguide with coincident centers as the base geometry we may truncate the waveguide and terminate it with a suitable smooth finite flare. The flare shape must be designed in the usual manner: to give smooth response and freedom from lobing. Accommodation for the surround must also be made to give a practical waveguide shape. The waveguide design illustrated here has a mouth diameter of 0.14m with a depth of 0.05m and is illustrated in figure 22.

Since the mouth diameter is 0.14m the 0.12m distance to sampling points is in the near field. The impulse response calculated at 0.12m shows the distance to the source varies along the arc of points. To avoid this type of error the steady state response has been calculated at a larger distance of 1m. This large model size necessitated the use of BEM & FEM requiring a direct solution in the frequency domain. Referring to figure 23 both on-axis and 40 degree responses are both smooth.

Due to the large size of the waveguide the directivity is controlled to a lower frequency and varies less strongly with frequency over the range of interest. The almost horizontal contours in figure 24 indicate an almost frequency independent directivity.

While the axial response is not flat like a rigid piston, the smoothness of the response and the similarity between off-axis and on-axis response make this an ideal candidate for equalization whether passive or active.

The increase in efficiency in the mid-band allows higher sound pressure levels to be reproduced.

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