682 Chapter 23
The term Ka has special signifi cance: it relates the diaphragm radius to the wavelength of
sound at any particular frequency. It is numerically equal to
K
a
a2 π
λ, (23.17)
where a is radius of the diaphragm and λ is wavelength.
When the wavelength λ is greater than the circumference of the diaphragm, the
loudspeaker behaves substantially as a point source and the sound fi eld pattern is
essentially omnidirectional. At the same time the radiation resistance increases with
frequency. Thus at frequencies below Ka 1, the increase in radiation resistance
with frequency is exactly balanced by the reduction in velocity of the diaphragm with
frequency due to its mass reactance (assuming there are no resonances in the diaphragm)
and the sound pressure will be constant. At values above Ka 1, the radiation resistance
(neglecting the minor “ wiggles ” ) becomes constant, but because of focusing due to the
diaphragm dimensions being greater thanλ , the sound pressure on the axis remains more
or less constant. The velocity of sound in air is approximately 340 m/s, therefore a
150-mm (6 inch)-diameter diaphragm will behave as a point source to a limiting
frequency of about 720 Hz; thereafter it begins to focus. Various artifi ces (such as
corrugations) are used with paper cones to extend this range, with more or less success.
This was the classic premise that Rice and Kelogg postulated in 1925 when they
reinvented the moving coil loudspeaker, and it is still fundamental today.
To summarize, the loudspeaker should operate under mass-controlled conditions and
(neglecting directional effects due to focusing of the diaphragm) sound pressure will be
constant and independent of frequency; for a given magnet and coil system it will be
inversely proportional to the total mass of the diaphragm and moving coil system.
23.7 Electrical Analogue ...............................................................................................
The analysis of mechanical and acoustical circuits is made very much easier by the
application of analogues in which mass is equivalent to inductance, compliance
to capacitance, and friction to resistance. Using SI units, direct conversion among
acoustical, mechanical, and electrical elements can be performed.
The three basic elements (RCL) of acoustical, electrical, and mechanical, systems are shown
schematically in Figure 23.5. The inertance M of an acoustic system is represented by the