By John Borwick
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Additional resources for Loudspeaker and Headphone Handbook, Third Edition
12. Three-dimensional representation of the far-ﬁeld directivity of two identical monopoles spaced one-eighth of a wavelength apart. 111 0 0111 0111 0 0 111 20 Principles of sound radiation 1 ϩ 1 ϭ 4! Bringing a second source close to a ﬁrst effectively doubles the power output of both sources. At high frequencies (Fig. 11), the two sound ﬁelds sum in phase for approximately half of the angles where the radius is 2; for the other half they tend to cancel each other and the radius is near zero. The net result is approximately double the power output compared to a single source and we are back to 1 ϩ 1 ϭ 2.
The low frequency polar plot (Fig. 12) is almost spherical with a radius of 2. Integrating the square of this radius over all angles leads to a power output four times greater than that of a single source (a sphere with a radius of 1). 9. Far-ﬁeld directivity of two identical monopoles spaced three wavelengths apart. 10. Far-ﬁeld directivity of two identical monopoles spaced one-eighth of a wavelength apart. 11. Three-dimensional representation of the far-ﬁeld directivity of two identical monopoles spaced three wavelengths apart.
30) which is identical to that for a monopole on a surface. In practice, the high-frequency approximation is seldom used, but the lowfrequency approximation serves us well. This is partly because most loudspeaker diaphragms are pretty good approximations to perfect pistons at low frequencies, but not at higher frequencies, partly because it is the imaginary part of the radiation impedance that most affects diaphragm motion (and this vanishes at high frequencies) and partly because the total power output is not usually of much interest at frequencies where the diaphragm has a complex directivity pattern.