In addition to having low coloration and low distortion, electrostatic loudspeakers enable us to control the directivity pattern produced by a single diaphragm in a way that is not possible with dynamic loudspeakers, by partitioning the stators into concentric annular rings, which are connected to tappings along the delay line shown in Fig. 15.1.

Because the membrane is flexible, each part can move more-or-less independently from the rest according to the signal on the nearest ring. Because of the delay line, the sound first emanates from the center, followed by the first ring, and then each successive ring in turn until it is radiated from the outermost ring by which time the sound from the center is already some distance away from the membrane. We just need to determine how to configure the delay to produce the optimum wavefront shape.

The traditional approach has been to arrange the delay to reproduce the wavefront of a virtual point source located behind the membrane [6]. Because of the finite size of the membrane, Walker [7] correctly pointed out that the delay line needs to be attenuated to prevent irregularities in the frequency response of the radiated sound. Because the far-field pressure response is the Fourier transform of the sound source, the attenuation may be regarded as a windowing function. It is essentially a shaded array.

Now imagine a massless sphere oscillating back and forth with constant velocity at all frequencies, thus radiating sound into free space. Such a sound source would have a constant figure-of-eight directivity pattern and at higher frequencies, where the wavelength is smaller than the sphere, constant power would be radiated because of the mainly resistive radiation impedance. Unfortunately, such a sound source is impractical to construct. Even if it were possible to make a large perfectly rigid hemispherical dynamic driver, it would need a lot of signal boosting at high frequencies to make it move with constant velocity, rather than constant acceleration, and thus radiate constant power.

Instead, we shall describe how to imitate an oscillating sphere using a planar circular electrostatic loudspeaker with stators partitioned into concentric annular rings that are connected to a delay line [8]. These rings reproduce the sound that would emanate from an oscillating sphere placed immediately behind the membrane (and in contact at the center) as it arrives at the membrane. It turns out that using a geometric approximation that assumes a plane wave traveling axially from the face of the imaginary sphere gives far superior results to reproducing the true magnitude and phase of the waves produced by the sphere using a finite membrane. We shall compare the effect of partitioning them into a finite number of rings with equal area, equal delay sections, and equal widths, using a continuously varying radial delay as an ideal reference. A significant benefit of using an analog delay line is that it converts the capacitive load of the electrostatic loudspeaker into an almost purely resistive one that is much easier for amplifiers to drive.

To begin with, we shall ignore secondary effects, such as the membrane mass and stiffness, the stator perforations, and stray capacitances, which are all considered later in the chapter. This will enable us to concentrate fully on how to control the directivity. In other words, we shall treat the membrane as a pure pressure source with zero mass and stiffness, aka a resilient disk as described in Section 13.8.