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It was pointed out in the section Nautilus tapered tubes that the technique is limited in bandwidth by the onset of cross-mode resonances as the wavelength gets smaller. It has long been known from horn theory that, if there is a cavity between the driver and the throat of the horn, high-frequency performance is limited. While undesirable in standard horn design, this effect is just what is needed to attenuate higher frequencies before they enter the Nautilus tube and cause any cross-mode resonances. Cavities, of course, have their own internal resonance problems. Indeed, it is to get away from these that we use tubes in the first place. However, for once physics is on our side. Extensive computer modelling and practical experimentation showed that, if the cavity is a sphere whose diameter had a particular ratio to the diameter of the driver diaphragm, and a hole directly opposite the driver which is the same diameter as its diaphragm and leads to a tapered tube, the internal resonances are to a large extent eliminated. Any residual effects are readily mopped up by wadding inside the sphere and tube. The sphere also has the ideal shape for avoiding diffraction effects on the outside of the enclosure, with consequent benefits to the imaging.