Discussion

The planar piston has very simple behavior in both time and frequency domain. This is a consequence of the source being coincident with the baffle. The planar piston may be thought of, and indeed analyzed, as an evenly spaced collection of point sources with equal intensity. Each source produces a perfect spherical wave; these waves combine to give the wave-shape of the piston. We can define the directivity of this source simply in terms of superposition. The impulse response exhibits no ringing and simply broadens off-axis due to the different path lengths to the sampling point.

The hemispherical cap radiator is less simply behaved; Were we to consider the radiator to be composed of evenly spaced point sources we must define them with lower amplitude near the periphery of the radiator as the normal velocity decreases. Furthermore, the boundary conditions are not met if we assume these sources radiate without diffraction. In this case, it is not helpful to consider the radiator as an ensemble of point sources. A more useful analysis of the far field radiation is to decompose the velocity of the radiator in terms of the spherical harmonic solutions to the wave equation. Thus, the directivity behavior and response will be determined by the combination of modes excited. [5]

In this context the presence of oscillation in the impulse response seems less anomalous.

The 120-degree spherical cap radiator in the 80-degree waveguide suffers from severe response dips at 18kHz and 24kHz. These dips are in a similar range, albeit more severe than those seen in [6] and would limit the frequency range of a transducer. The presence of oscillations in the impulse response does not seem surprising in the context of a non-planar waveguide.

The 80-degree spherical cap radiator in the 80-degree waveguide provides almost perfect coverage within the angle of the waveguide. Although the radiator shares the same geometry as the propagating waves, the normal velocity decreases towards the outside of the dome. Consequently, we should not expect a model based on point sources to be applicable in this case.

The finite waveguide allows the wavefront to ‘spread’ without introducing significant problems in either the frequency or time domain.

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