Page 15 - Spring 2007

Fig. 5. Mobility magnitudes (top), radiation resistances (middle), and radiated sound power transfer functions (bottom) of a 16.6 mm radius circular baffled pis- ton in air (black) and in water (blue).
ing, using Eq. 10), and in water (including fluid loading using Eq. 11). Next, we multiply Rο by the square of mobility to com- pute radiated sound power. Plots of the mobility magnitude, radiation resistance (in air and in water), and the radiated sound power for a unit force input are shown in Fig. 5.
The effect of mass loading on the piston in water is pro- nounced, shifting the piston resonance frequency downward, and the overall mobility amplitude downward. The radiation resistance (and reactance) of water is much higher than that of air (be sure to note the multiple scales used on the resist- ance comparison plot). Therefore, radiated sound power is quite different in air and in water, and the piston resonance peaks occur at different frequencies.
The radiated sound power curve in water illustrates the basic principle of loudspeaker design: adjust the piston prop- erties to set the fundamental resonance as low in frequency as possible, so that the piston response is controlled by the mass term in Eq. 11. Recall that the radiated sound power is the product of the radiation resistance Rο and the square of the pis- ton normal velocity. Since Rο at low ka is proportional to (ka)2, and the piston mobility above resonance is proportional to 1/ω2, the frequency dependencies cancel, leaving a nearly fre- quency independent radiated sound power/F2 response.
In our example in Fig. 5, the in-air radiated sound power transfer function is not flat above the piston resonance fre- quency because the radiation resistance is above the frequen-
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cy range where it is proportional to (ka) . To resolve this, a
speaker designer would simply reduce the radius of the pis- ton, shifting the radiation resistance curve further out in fre- quency. This solution comes with a cost though—the radia- tion resistance amplitude reduces with surface area, which will in turn reduce the radiated sound power.
Structural waves vibrating against an acoustic fluid
How well do structural waves (rather than rigid oscilla- tors) radiate sound? Since only structural motion normal to an object’s surface induces an equal motion in a neighboring fluid, we consider transversely vibrating, or flexural waves (we acknowledge, however, that longitudinal waves deform a structure transversely due to an elastic material’s Poisson effect, but do not focus on the sound radiated by longitudinal waves here).
How well flexural waves in a structure radiate sound depends on whether the waves, which essentially act as a source against the fluid, are subsonic (slower than the wavespeed in the fluid) or supersonic (faster than the wavespeed in the fluid). Supersonic waves radiate sound, and subsonic waves do not.
Many structural acousticians like to consider the sound radiated by structural waves in wavenumber space, and examine wavetypes on frequency-wavenumber plots. Consider the traveling flexural waves in an infinite plate shown in Fig. 6. The flexural and acoustic wavenumbers (computed by dividing radial frequency by the flexural and acoustic wavespeeds) are plotted against frequency in the top of the figure. Since the acoustic waves are non-dispersive, the wavenumber curve has constant slope. The flexural waves, however, are dispersive, causing a varying slope in the wavenumber curve.
At low frequencies, the structural wavenumbers are higher than those in the acoustic fluid, corresponding to sub- sonic structural waves. These flexural waves radiate no sound at all (this is only true for infinite plates—we will discuss the sound radiated by finite plates soon). This is because the par- ticle velocity in the fluid normal to the structure’s surface must match that of the structure. At low frequencies, acoustic waves are faster than structural ones, so their wavelengths are longer. This means the structure simply cannot induce a propagating wave in the fluid.
The frequency at which the flexural and acoustic waves have the same wavenumber (and wavespeed, and wave- length) is called the coincidence frequency, and the flexural
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